Wythoffian Skeletal Polyhedra

Home , Antiprism, Apeirogon, Archimedean solid, Bipyramid, Coxeter notation, Cube

Wythoﬃan Skeletal Polyhedra

by Abigail Williams

B.S. in Mathematics, Bates College M.S. in Mathematics, Northeastern University

A dissertation submitted to

The Faculty of the College of Science of Northeastern University in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy

April 14, 2015

Dissertation directed by

Egon Schulte Professor of Mathematics Dedication

I would like to dedicate this dissertation to my Meme. She has always been my loudest cheerleader and has supported me in all that I have done. Thank you, Meme.

ii Abstract of Dissertation

Wythoﬀ’s construction can be used to generate new polyhedra from the symmetry groups of the regular polyhedra. In this dissertation we examine all polyhedra that can be generated through this construction from the 48 regular polyhedra. We also examine when the construction produces uniform polyhedra and then discuss other methods for ﬁnding uniform polyhedra.

iii Acknowledgements

I would like to start by thanking Professor Schulte for all of the guidance he has provided me over the last few years. He has given me interesting articles to read, provided invaluable commentary on this thesis, had many helpful and insightful discussions with me about my work, and invited me to wonderful conferences. I truly cannot thank him enough for all of his help. I am also very thankful to my committee members for their time and attention. Additionally, I want to thank my family and friends who, for years, have supported me and pretended to care everytime I start talking about math. Finally, I want to thank my husband, Keith. Whenever I have doubted myself he has always been there with a hug and an encouraging word. Everyday I am thankful for his love and support.

iv Table of Contents

Dedication ii

Abstract of Dissertation ii

Acknowledgements iii

Table of Contents iv

1 Background 1 1.1 Introduction ...... 1 1.2 Abstract polytopes ...... 2 1.3 Regularity ...... 5 1.4 Operations on regular polyhedra ...... 8 1.4.1 Duality - δ ...... 8 1.4.2 Petrie duality - π ...... 9

1.4.3 Facetting - ϕk ...... 10 1.4.4 Halving - η ...... 11 1.4.5 Skewing - σ ...... 11 1.4.6 Mixing ...... 11 1.5 Realizations ...... 12 1.5.1 Blends of realizations ...... 14 1.6 Uniform polyhedra ...... 14 1.7 Fundamental regions ...... 19 1.8 Polygonal complexes ...... 20

2 Wythoff’s construction 21 2.1 Expanding Wythoff’s construction ...... 21 2.2 Properties of Wythoffians ...... 27

3 Finite regular polyhedra and their Wythoﬃans 33 3.1 Regular polyhedra with tetrahedral symmetry ...... 34 3.1.1 {3, 3} ...... 34

v 3.1.2 {4, 3}3 ...... 36 3.2 Regular polyhedra with octahedral symmetry ...... 38 3.2.1 {3, 4} ...... 38 3.2.2 {4, 3} ...... 39

3.2.3 {6, 4}3 ...... 41

3.2.4 {6, 3}4 ...... 42 3.3 Regular polyhedra with icosahedral symmetry ...... 44 3.3.1 {3, 5} ...... 44 3.3.2 {5, 3} ...... 46

3.3.3 {10, 5}3 ...... 47

3.3.4 {10, 3}5 ...... 49 5 3.3.5 {5, 2 } ...... 50 5 3.3.6 { 2 , 5} ...... 52 5 3.3.7 {6, 2 } ...... 54 3.3.8 {6, 5} ...... 56 5 3.3.9 {3, 2 } ...... 57 5 3.3.10 { 2 , 3} ...... 59 10 5 3.3.11 { 3 , 2 } ...... 61 10 3.3.12 { 3 , 3} ...... 63

4 Planar regular apeirohedra and their Wythoﬃans 65 4.1 Square tiling ...... 65 4.1.1 {4, 4} ...... 65

4.1.2 {∞, 4}4 ...... 67 4.2 Triangular tiling ...... 68 4.2.1 {3, 6} ...... 68

4.2.2 {∞, 6}3 ...... 70 4.3 Hexagonal tiling ...... 71 4.3.1 {6, 3} ...... 71

4.3.2 {∞, 3}6 ...... 72

5 Blended regular apeirohedra and their Wythoﬃans 75 5.1 Blended apeirohedra related to the square tiling of the plane ...... 76 5.1.1 {4, 4}#{} ...... 76

5.1.2 {∞, 4}4#{} ...... 78 5.1.3 {4, 4}#{∞} ...... 80

5.1.4 {∞, 4}4#{∞} ...... 83 5.2 Blended apeirohedra related to the hexagonal tiling of the plane ...... 86 5.2.1 {6, 3}#{} ...... 86

5.2.2 {∞, 3}6#{} ...... 88

vi 5.2.3 {6, 3}#{∞} ...... 89

5.2.4 {∞, 3}6#{∞} ...... 93 5.3 Blended apeirohedra related to the triangular tiling of the plane ...... 95 5.3.1 {3, 6}#{} ...... 95

5.3.2 {∞, 6}3#{} ...... 98 5.3.3 {3, 6}#{∞} ...... 99

5.3.4 {∞, 6}3#{∞} ...... 102 5.4 Remarks on blended apeirohedra ...... 104

6 Pure regular apeirohedra and their Wythoﬃans 105 6.1 {4, 6|4} ...... 106 6.2 {6, 4|4} ...... 108

6.3 {∞, 6}4,4 ...... 110

6.4 {∞, 4}6,4 ...... 112

6.5 {∞, 4}·,∗3 ...... 114

6.6 {6, 6}4 ...... 116

6.7 {4, 6}6 ...... 118

6.8 {6, 4}6 ...... 120 6.9 {6, 6|3} ...... 122

6.10 {∞, 6}6,3 ...... 123 6.11 {∞, 3}(a) ...... 125 6.12 {∞, 3}(b) ...... 127

7 Uniform polyhedra 130 7.1 New uniform polyhedra resulting from Wythoﬀ’s construction ...... 131 7.1.1 Uniform polyhedra derived from ﬁnite regular polyhedra ...... 132 7.1.2 Uniform apeirohedra derived from planar regular apeirohedra ...... 133 7.1.3 Uniform apeirohedra derived from blended regular apeirohedra ...... 134 7.1.4 Uniform apeirohedra derived from pure regular apeirohedra ...... 139 7.2 New construction for uniform polyhedra ...... 146 7.3 Uniform polyhedra generated through alternate construction ...... 148 7.3.1 Tetrahedral family ...... 148 7.3.2 Octahedral family ...... 148 7.3.3 Icosahedral family ...... 151 7.4 Summary of results ...... 157

Appendix 159

Bibliography 208

vii Chapter 1

Background

1.1 Introduction

Since ancient times, mathematicians have been studying polyhedra. Beginning with the convex regular Platonic solids, the subject then progressed to the non-regular but still convex Archimedean solids, [41]. By the 1800s, the non-convex Kepler-Poinsot star polyhedra had been discovered. In the last century, mathematicians began taking a more combinatorial approach to the study. This opened the door for the Petrie-Coxeter infinite polyhedra which have convex faces and skew vertex figures. Then came Coxeter’s Regular Polytopes, which, in addition to examing the geometry of polyhedra, examined the underlying symmetries and combinatorial properties of polyhedra. By the 1970s Grünbaum had extended these ideas about symmetry to allow for new types of polygons. These polygons can be skew or even infinite, [19]. To make polyhedra with these new polygons as faces, Grünbaum introduced the idea of skeletal polyhedra. This allowed for the regular Grünbaum- Dress apeirohedra (see [16], [17]) with skew and infinite faces. At this point, the list of regular polyhedra was complete. Then McMullen and Schulte examined and fully classified the regular polyhedra by their combinatorial properties (see [30], [31]). This study gave life to the field of abstract regular polytopes, [32]. The study of abstract regular polyhedra naturally gave rise to the study of non-regular abstract polyhedra, in particular, the study of chiral polyhedra. The chiral polyhedra have been fully classified by Schulte (see [37], [39]) and continue to be studied (see, for example, [34], [33], [25], [11]). The primary aim of this thesis is to take one of the techniques used to examine chiral polyhedra and apply it to skeletal polyhedra to find analogues of the Archimedean solids. More specifically, we take abstract regular polyhedra and realize them in Euclidean space, then based on those realizations and their symmetry groups we generate more polyhedra. To do this, we find a symmetry group in ordinary Euclidean space corresponding to the automorphism group of an abstract polyhedron. We then use Wythoff’s construction on the symmetry group to generate a realization of the abstract regular polyhedron and to generate other polyhedra with the same symmetries. These realizations are based on abstract objects, so their geometric properties only need fulfill the most basic combinatorial properties of an abstract polyhedron. As such, we need to refine our

1 definitions of polygon and polyhedron to account for the realizations of the abstract polyhedra. Based on Grünbaum’s work, polygons are viewed as sequences of connected edges, and polyhedra are viewed as skeletons comprised of vertices and edges, [19]. This allows us to look at polyhedra with skew faces and infinite faces. While the regular polyhedra with finite, planar faces have been studied extensively, there has been relatively little examination of the regular polyhedra with skew faces, skew vertex figures, or infinite faces. Here we will look at the objects generated by applying Wythoff’s construction to regular polyhedra with these properties. We will also examine when these objects are uniform polyhedra. Coxeter, Longuet-Higgins, and Miller found a complete list of uniform finite polyhedra with planar faces, [9]. In this thesis we will extend this list to include uniform polyhedra with skew faces, skew vertex figures, and infinite faces. The first chapter of this thesis provides background information and definitions which will be referenced throughout. In Chapter 2, we explicitly describe the various forms of Wythoff’s construction when applied to the symmetry groups of regular polyhedra. Chapters 3-6 give detailed descriptions of the skeletal Wythoffians generated in this way from the symmetry groups of the regular polyhedra. In Chapter 3, we discuss finite Wythoffians; in Chapter 4, we discuss planar Wythoffians; in Chapter 5, we discuss blended Wythoffians; and in Chapter 6, we discuss pure Wythoffians. In Chapter 7, we discuss the uniform polyhedra generated through Wythoff’s construction. We also examine a new construction that yields new uniform polyhedra which cannot be obtained by Wythoff’s construction. The Appendix contains tables of images of the facets and vertex figure for Wythoffians examined in Chapters 3-6. Alongside the tables are images of the Wythoffians themselves.

1.2 Abstract polytopes

In this section we will introduce abstract polytopes and many important related definitions and results. Abstract polytopes give us a way of examining the purely combinatorial properties of the geometric figures that have been studied for millenia. Historically, geometers have studied objects in some ambient space. In Euclid’s Elements a point is that which has no dimensions, a line is that which has length but not width, and a polygon is a planar figure bound by lines. It is from these simple concepts that the study of convex geometry was built. To help give a more intuitive understanding of abstract polytopes, we will begin with some basic concepts from convex geometry. For a more in-depth examination of convex polytopes see [2], [6], [20].

n Deﬁnition 1.1. A convex n-polytope is an n-dimensional convex hull of a ﬁnite set in E .

Each convex n-polytope P consists of faces which are themselves polytopes. For 0 ≤ d ≤ n − 1, a d-dimensional face (or d-face) of P is deﬁned as Fd := P ∩ H where H is a supporting hyperplane of P and d is the dimension of Fd. We also deﬁne F−1 := ∅ and Fn := P and call them faces. A 0-face is called a vertex, a 1-face is called an edge, and an (n − 1)-face is called a facet. This

2 terminology, as with most of the combinatorial terminology of convex geometry, carries over to abstract polytopes. We say that two faces F,G of P are incident if one is a subface of the other. Note that the faces F−1 and Fn are incident to all other faces of P .

Deﬁnition 1.2. The partially ordered set of all faces of P is the face lattice of P . The partial order is induced by incidence. The face lattice has a least and a greatest element, F−1 and Fn, respectively.

To examine abstract polytopes we must generalize the face lattice. Instead of considering the geometric object that the face lattice is associated with, we will instead look at how the elements of the face lattice interact. We will look at the properties of convex polytopes which determine how the face lattice behaves and expand them to other posets which are not derived from geometric objects. These posets will be called abstract polytopes (for more on abstract polytopes see [32]). Before we can detail the defining properties of an abstract polytope we need to go through some terminology. Most of the terms we use come from convex polytope theory, but with slightly modified definitions. Let P be a poset, then the elements of the poset are referred to as faces and we say two faces F and G are incident if F ≤ G or G ≤ F . We are now ready for the first defining property of an abstract polytope of rank n.

(A1) P contains a least face, F−1, and a greatest face, Fn.

In terms of the classical polytope theory we have seen that we can think of F−1 as the empty set which is contained in all faces, and we can think of Fn as the entire polytope which contains all faces. These are the improper faces of P. All faces, F , such that F−1 < F < Fn are proper faces of P. Define a flag as a maximal, totally ordered set of faces. Using this definition we arrive at the second defining property.

(A2) Each ﬂag of P contains exactly n + 2 faces.

As a result of (A2), P becomes a ranked poset. If F is a face, we define the rank of F as the st number i such that F occurs at the (i+2) position in every flag F−1 < F0 < ··· < Fn−1 < Fn that contains F . As with convex geometry, a 0-face is a vertex, a 1-face is an edge, and a (n − 1)-face is a facet. Property (A2) also tells us that there is exactly one face of every rank from −1 through n in each flag. This property should be fairly self-evident for classical polytopes. For example, every edge is incident to a vertex so any flag containing an edge must contain a vertex as well. Define a section of the poset as G/F := {H ∈ P | F ≤ H ≤ G} for faces F ≤ G of P. The sections then become polytopes in their own right. A face, F , is identified with the section F/F−1, which is a polytope of rank i if F has rank i. If F0 is a vertex, the section Fn/F0 is the vertex figure at F0.

3 The poset P is connected if for any two proper faces F,G ∈ P there exists a finite sequence of proper faces F = H0,H1, ..., Hk = G such that Hi and Hi+1 are incident for 0 ≤ i < k. If, in addition, each section of P is connected we call P strongly connected. Two flags of P are adjacent if they differ in exactly one face; if that face is the i-face then the flags are i-adjacent. If for any two flags of the poset, there exists a finite sequence of successively adjacent flags from one to the other, then it is flag-connected. We call a poset, P, strongly flag- connected if every section, including P, is flag-connected. We are now ready to look at the third property.

(A3) P is strongly ﬂag-connected.

Property (A3) can also be stated as “P is strongly connected”. Flag-connectedness is generally a more useful property which is why we prefer it to ordinary connectedness. In terms of classical polytopes both conditions simply mean that there can never be a polytope with two distinct pieces. This brings us to the ﬁnal property, the diamond condition.

(A4) P fulﬁlls the following diamond condition:

For two faces Fi−1 < Fi+1 of ranks i − 1 and i + 1, respectively, in P with 0 ≤ i ≤ n − 1

there are exactly two i-faces, H ∈ P, such that Fi−1 < H < Fi+1. In other words, every flag has exactly one i-adjacent flag for i = 0, ..., n − 1. If Φ is a flag of P and P fulfills the diamond quality, then we denote its unique i-adjacent flag as Φi.

For a simple example from classical polytopes consider a polygon. Each vertex (0-face) of the polygon (2-face) lies in exactly two edges (1-faces).

Deﬁnition 1.3. Let P be a poset as above. If P has properties (A1)-(A4) then it is an abstract n-polytope.

When P is infinite we also refer to it as an n-apeirotope. The lower rank polytopes borrow their names from convex geometry. Abstract 2-polytopes are polygons and 3-polytopes are polyhedra, respectively, or apeirogons and apeirohedra when infinite. More specifically, many polygons and polyhedra have actual names, e.g. the octagon and the cube. The names can get quite complicated as with the rhombicosidodecahedron. To simplify things we will use Schläflisymbols to denote polytopes. The Schläflisymbol for an abstract polytope is derived from its equivelar type, as described next.

A classical polytope is equivelar if there exist numbers p1, ..., pn−1 such that for each of its

ﬂags {F−1,F0, ..., Fn} and for each j = 1, ..., n − 1 the section Fj+1/Fj−2 is a polygon with pj edges. As with most of the geometric concepts, equivelarity has an analogue in abstract polytope theory. An abstract polytope is equivelar if there exist number p1, ..., pn−1 such that for each of its

ﬂags {F−1,F0, ..., Fn} and for each j = 1, ..., n − 1 there are exactly pj j-faces of P in the section

Fj+1/Fj−2 [32]. Then the Schl¨aﬂisymbol for the equivelar polytope is {p1, p2, ..., pn−1}, and we

4 say that P is of type {p1, p2, ..., pn−1}. In the case when P has rank 1, we write P = {}. Note that when n ≥ 2 the Schläflisymbol of a facet of P is {p1, p2, ..., pn−2} and the Schläflisymbol of a vertex figure at any vertex is {p2, p3, ..., pn−1}. Recall from before that the vertex figure at vertex

F0 is deﬁned as the section Fn/F0.

1.3 Regularity

For an abstract polytope, P, the automorphism group, Γ(P), is the group of all automorphisms ρ of P. Here an automorphism is an isomorphism from P to itself. An isomorphism φ : P → Q between abstract polytopes is an incidence preserving bijection between the face lattices of P and Q. For faces F,G in P and ρ ∈ Γ(P), the incidence preserving property means that F ≤ G if and only if ρ(F ) ≤ ρ(G). Thus, for ρ ∈ Γ(P), Φ = {F−1,F0, ..., Fn−1,Fn} is a ﬂag of P if and only if

ρ(Φ) = {ρ(F−1), ρ(F0), ..., ρ(Fn)} is a flag of P. We frequently use the fact that an automorphism of a polytope is uniquely determined by its effect on a single flag ([32], Proposition 2A4).

Definition 1.4. An abstract polytope is regular if Γ(P) is transitive on its flags. In other words, for any two flags Φ and Ψ of P there is some ρ ∈ Γ(P) such that ρ(Φ) = Ψ.

Note that the automorphism group of a regular polytope is simply transitive ([32], Proposition 2A5); that is, the automorphism ρ with ρ(Φ) = Ψ is unique. Before we can fully examine the automorphism groups of regular polytopes we will need some prelimary results about regular polytopes. The following results come from [32].

Lemma 1.1. Let P be an n-polytope with automorphism group Γ(P) and let Φ be a ﬂag of P.

(a) The i-adjacent ﬂag of the i-adjacent ﬂag to Φ is Φ itself, i.e. Φii = Φ. ([32], Proposition 2A2(a))

(b) For any ρ ∈ Γ(P), ρ preserves adjacency. That is, (ρΦ)i = ρΦi. ([32], Proposition 2A3(a))

(c) For j, k ∈ {0, 1, ..., n − 1} with |j − k| ≥ 2 the j-adjacent flag of the k-adjacent flag to Φ is the k-adjacent flag of the j-adjacent flag to Φ, ie Φkj = Φjk. ([32], Proposition 2A2(b))

Proof. For part (a), let G be a (i − 1)-face of P and let H be a (i + 1)-face of P which is incident to G. Then by the diamond condition there exist exactly two i-faces, F and F 0, which are incident to both G and H. Let Φ be a flag containing G < F < H, then Φi must contain G < F 0 < H. Since F,F 0 are the only i-faces incident to both G and H the i-adjacent flag to Φi must contain F . But the only flag which contains F and the j-faces of Φi with j 6= i is Φ. Thus Φii = Φ.

For part (b), recall that automorphisms preserve incidence. Deﬁne a ﬂag Φ := {F−1,F0,F1, ..., Fn}, then

ρΦ = {ρ(F−1), ρ(F0), ..., ρ(Fi−1), ρ(Fi), ρ(Fi+1), ..., ρ(Fn)}.

5 0 0 i 0 Let Fi 6= Fi be an i-face such that Fi−1 < Fi < Fi+1. Then Φ = {F−1,F0, ..., Fi−1,Fi ,Fi+1, ..., Fn} and i 0 ρΦ = {ρ(F−1), ..., ρ(Fi−1), ρ(Fi ), ρ(Fi+1), ..., ρ(Fn)}.

i 0 0 If ρΦ and ρΦ were not i-adjacent ﬂags then ρ(Fi) = ρ(Fi ) and hence Fi = Fi as ρ is a bijection. Thus (ρΦ)i = ρΦi. For part (c), without loss of generality assume j < k. Let

Φ = {F−1, ..., Fj−1,Fj,Fj+1, ..., Fk−1,Fk,Fk+1, ..., Fn} be a ﬂag of P where we allow j + 1 = k − 1. By the diamond condition there is exactly one other j- face, G, such that Fj−1 < G < Fj+1 and similarly only one other k-face such that Fk−1 < H < Fk+1. Then k Φ = {F−1, ..., Fj−1,Fj,Fj+1, ..., Fk−1,H,Fk+1, ..., Fn},

kj k j Φ = (Φ ) = {F−1, ..., Fj−1, G, Fj+1, ..., Fk−1,H,Fk+1, ..., Fn},

j Φ = {F−1, ..., Fj−1, G, Fj+1, ..., Fk−1,Fk,Fk+1, ..., Fn}, and jk j k Φ = (Φ ) = {F−1, ..., Fj−1, G, Fj+1, ..., Fk−1,H,Fk+1, ..., Fn}.

Thus Φkj = Φjk.

Theorem 1.1. ([32], Proposition 2B4) An n-polytope P is regular if and only if, for some ﬂag Φ of P and for all i ∈ {0, 1, ..., n −1}, there exists a (unique) involutory automorphism ρ ∈ Γ(P) such that ρΦ = Φi.

Proof. Assuming regularity of P we know that Γ(P) is transitive on the flags of P. This means that for the i-adjacent flags Φi of Φ there exists some ρ ∈ Γ(P) such that ρΦ = Φi. Then ρ is unique since Γ(P) is simply flag-transitive. Moreover,

ρ2Φ = ρ(ρΦ) = ρΦi = (ρΦ)i = (Φi)i = Φii = Φ, so ρ2 = ε, again by the simple flag-transitivity of Γ(P). This proves one direction. Now assume that for Φ, a flag of P, and all i ∈ {0, 1, ..., n−1}, there exists a (unique) involutory automorphism ρ ∈ Γ(P) such that ρΦ = Φi. To show regularity we need only show that for flags Φ and Ψ there exists some ρ ∈ Γ(P) such that ρΦ = Ψ. By the strong flag-connectedness property of polytopes there exists a sequence of successive adjacent flags Φ = Φ0, Φ1, ..., Φn = Ψ. We will induct on n. When n = 0 it is trivially true, and when n = 1 we see that Ψ = Φj for some j then let ρj be the involution from the condition and Ψ = ρjΦ. Now assume the hypothesis holds true for smaller values than n. Then there exists some ρ ∈ Γ(P) such that ρΦ = Φn−1. Then for some i i i i, Ψ and Φn−1 are i-adjacent. Thus Ψ = Φn−1 = (ρΦ) = ρΦ . Drawing on our base case there is

6 i i some ρi such that ρiΦ = Φ and so ρΦ = ρ(ρiΦ). Thus ρρiΦ = Ψ where ρρi ∈ Γ(P) as desired.

With regular polytopes, since Γ(P) is transitive on the flags, we can choose one fixed flag to be the base flag of P. The faces of this flag, {F−1,F0,F1, ..., Fn−1,Fn}, are called base faces. Let Φ be the base flag of P, an n-polytope. For each i = 0, ..., n − 1 the unique involution that takes Φ i to Φ is denoted ρi. Collectively ρ0, ρ1, ..., ρn−1 are the distinguished generators of Γ(P). [32] As a consequence of Lemma 1.1, 2 ρi = ε for all i and 2 (ρiρj) = ε for all i, j ∈ {0, 1, ..., n − 1} such that |i − j| ≥ 2. For any j-face Fj ∈ Φ with j 6= i, ρi(Fj) = Fj.

Additionally, if P is of type {p1, p2, ..., pn−1} then

pij (ρiρj) = ε with  1, if i = j  pij = pj, if j = i + 1  2, if |j − i| ≥ 2

We will constantly refer back to this last fact when performing Wythoff’s construction as it determines the size of the orbit of a face under a subgroup of Γ(P). Our last preliminary note about the automorphism groups of regular polytopes pertains to their relationship with string C-groups. A C-group is a group with a finite number of fixed involutory generators, denoted ρ0, ρ1, ..., ρn−1, such that the following “intersection property” holds. Let N := {0, 1, ..., n − 1} and let J, K ⊆ N then

hρj | j ∈ Ji ∩ hρk | k ∈ Ki = hρi | i ∈ J ∩ Ki.

2 A C-group is a string C-group if for generators ρi, ρj with |i − j| ≥ 2, (ρiρj) = ε. We have already seen this second property with the automorphism groups of regular polytopes. Automorphism groups of regular polytopes also possess the intersection property, making them string C-groups. In fact, there is a one-to-one correspondence between string C-groups and the automorphism groups of regular polytopes. This allows us to identify polytopes with their automorphism groups. For the proofs related to this fact and for more information on C-groups see [32], Section 2E.

7 1.4 Operations on regular polyhedra

Now we will examine some of the operations that can transform a group of one polytope into the group of another. Since we are only interested in polyhedra we will look at how the operations aﬀect a regular polyhedron, P, with automorphism group Γ(P) = hτ0, τ1, τ2i.

1.4.1 Duality - δ

A duality is a bijection between two polytopes δ : R → Q such that δ and δ−1 are both incidence reversing. That is, if F ≤ G in R then δ(F ) ≥ δ(G), and if H ≤ I in Q then δ−1(H) ≥ δ−1(I). When such a bijection exists, R and Q are said to be duals of one another. Generally, given two regular polytopes there will not be a duality between them. On the other hand, every polytope has precisely one dual obtained by reversing the order relation. If there exists a duality from a polytope to itself, that polytope is said to be self-dual. From a geometric standpoint we can think of this as letting the vertices of R be the centers of the facets of Q and the facets of R become the vertex ﬁgures of Q [2]. Within the Platonic solids the cube and octahedron are duals as are the icosahedron and dodecahedron while the tetrahedron is self-dual.

Figure 1.1: An octahedron whose vertices are the centers of the faces of a cube. Here it is clear that the octahedron is the dual of the cube.

Exchanging vertices and facets provides an easy method for finding the dual of a classical polytope. For abstract regular polytopes we can examine the underlying automorphism group to determine the dual polytope. Since an abstract regular polytope can be explicitly identified with its automorphism group we can now define the duality operation, which before acted on the faces of polytopes, as acting on the distinguished generators of the polytope. This thesis is primarily concerned with polyhedra so we will only explicitly describe the duality operation for polyhedra. By slight abuse of notation we again use δ to denote the duality operation. For P as above we define the duality operation as

δ :(τ0, τ1, τ2) → (τ2, τ1, τ0)

The dual of P is the regular polyhedron whose automorphism group has distinguished generators δ ρ0, ρ1, ρ2 where ρ0 := τ2, ρ1 := τ1, and ρ2 := τ0 [32]. Denote the dual of P as P . Note that

8 (Pδ)δ = P for all regular polytopes P.

1.4.2 Petrie duality - π

Petrie duality is the next operation we will examine. Geometrically, this operation is built on the idea of deﬁning the faces of a polyhedron as the Petrie polygons of a given regular polyhedron. A Petrie polygon’s edges and vertices are in the edge and vertex sets of the original polyhedron. Every two adjacent edges in a Petrie polygon are incident to the same face of P while no set of three sequential edges are incident to the same face of P. If it exists, the polyhedron, Pπ, whose faces are all of the Petrie polygons of P is the Petrie dual, or Petrial, of P. Petrie polygons are also referred to as the 1-zigzags of a polyhedron. A k-zigzag is formed by following the edge-path generated by alternately leaving each vertex by the kth edge to the left or right of the edge that was used to enter that vertex.

Figure 1.2: A Petrie polygon of the cube, one of the faces of the {4, 3}π.

As with duality, to ﬁnd Petrials of abstract polyhedra we can transform the distinguished generators of the polyhedra. Letting P be as before we deﬁne π as

π :(τ0, τ1, τ2) → (τ0τ2, τ1, τ2).

The distinguished generators of the Petrial of P are ρ0 := τ0τ2, ρ1 := τ1, and ρ2 := τ2 [32]. We denote the Petrial of P as Pπ. When Pπ is isomorphic to P, we say that P is self-petrial. Similar to duality, (Pπ)π = P for all regular polytopes. Petrie duality differs in that not all polyhedra have Petrie duals which are abstract polyhedra. In some cases two distinct polyhedra may have the same equivelar type. When this happens the Schläflisymbol needs to be altered to distinguish between the polyhedra. When the length of a polyhedron’s Petrie polygons determines the polyhedron up to isomorphism, that length is appended to the Schläflisymbol. For Q of equivelar type {p, q} with Petrie polygon of length r we 1 write Q = {p, q}r. If Q is uniquely determined by the length of the k-zigzags for 1 ≤ k ≤ j := b 2 qc, then we write

Q = {p, q}r1,r2,...,rj

If for some k the k-zigzag is unnecessary for specifying the polyhedron then we replace rj with “·”.

9 Moreover, a “∗” in a symbol like {p, q}r,∗s indicates that s is the length of the corresponding zigzag (here 2-zigzag) of the dual polyhedron, not of the polyhedron itself.

1.4.3 Facetting - ϕk

From a geometric standpoint, the kth facetting operation looks at the k-holes of a polyhedron of 1 type {p, q} for 2 ≤ k ≤ b 2 qc.A k-hole of a polyhedron is the polygon formed by following the edge-path generated by leaving each vertex by the kth edge to the right of the edge that was used to enter that vertex. The actual direction, right or left, does not matter so long as the same direction is used consistently throughout. For example, a 2-hole could be found by starting with an edge and then going from vertex to vertex by always leaving each vertex along the 2nd edge from the right. The facetting operation defines the k-holes of a polyhedron as the faces of a new object which may or may not be polytopal. The simplest example of facetting comes by finding a new regular polyhedron by defining the faces as the 2-holes of the icosahedron.

Figure 1.3: The ﬁgure in white is an icosahedron, {3, 5}, which has triangular faces. Outlined in red is a 2-hole of {3, 5}. The 2-hole is a pentagonal face of {3, 5}ϕ2 which is isomorphic to the 5 Kepler-Poinsot polyhedron {5, 2 }.

To translate the process to abstract regular polyhedra we again need to define the facetting operation on the distinguished generators of P. Thus far, duality and Petrie duality have only th altered τ0 and τ2. The k facetting operation affects τ1. We define it as

k−1 ϕk :(τ0, τ1, τ2) → (τ0, τ1(τ2τ1) , τ2).

For our purposes we are only interested in the case when k = 2. The distinguished generators of ϕ Γ(P 2 ) are then ρ0 := τ0, ρ1 := τ1τ2τ1, and ρ2 := τ2 [32]. Similarly to the k-zigzags, if two polyhedra have the same equivelar type but are distinctly determined by the length of their k-holes we can append that information to the Schläfli symbol to differentiate between the two polyhedra. Let hk be the length of a k-hole of a polyhedron, Q, of 1 type {p, q} for 2 ≤ k ≤ j := b 2 qc. Then we can write

Q = {p, q|h2, ..., hj}

If the length of a k-hole is unnecessary for identifying the polyhedron then we write a “·” in its

10 place. As before, if a “∗” occurs in the symbol, then the subsequent number gives the length of the k-hole of the dual polyhedron.

1.4.4 Halving - η

The halving operation only works for polyhedra of type {4, q} with q ≥ 3. The halved polyhedron, Pη, is self-dual and is of type {q, q}. The vertex set of Pη is contained in the vertex set of P as can be seen in the following example. When halving the cube one gets a tetrahedron with vertices given by alternate vertices of the cube.

Figure 1.4: The red tetrahedron is the result of performing the halving operation on the cube.

Let P be of this type with Γ(P) as before. Then

η :(τ0, τ1, τ2) → (τ0τ1τ0, τ2, τ1)

The new polyhedron has distinguished generators ρ0 := τ0τ1τ0, ρ1 := τ2, and ρ2 := τ1 [32].

1.4.5 Skewing - σ

The ﬁnal operation, skewing, applies only to regular polyhedra of type {p, 4}. This operation is deﬁned by 2 σ :(τ0, τ1, τ2) → (τ1, τ0τ2, (τ1τ2) )

σ 2 and the distinguished generators of P are ρ0 := τ1, ρ1 := τ0τ2, and ρ2 := (τ1τ2) [32]. In this thesis there is only one time when we will use skewing to understand the automorphism group of a regular polyhedron. We use it in the discussion of the family of pure regular apeiro- σ hedra, where {6, 4|4} −→{∞, 4}·,∗3 indicates that the polyhedron {∞, 4}·,∗3 is obtained from the polyhedron {6, 4|4} as the result of applying σ.

1.4.6 Mixing

We have now seen how to transform an automorphism group of a regular polyhedron into a new group. We can also combine the automorphism groups of two regular polyhedra and transform them into a new group. This process is called taking the mix of the two string C-groups. Let

11 Λ = hµ0, µ1, ..., µmi and ∆ = hν0, ν1, ..., νni be string C-groups with m ≤ n. Then we deﬁne the subgroup Λ♦∆ of the direct product Λ × ∆ by

Λ♦∆ := h(µ0, ν0), (µ1, ν1), ..., (µm−1, νm−1), (µm, νm), (ε, νm+1), ..., (ε, νn)i.

This group is often not a C-group, and consequently not the group of a regular polytope. In this thesis we will see a few examples of when the resulting mix is a polytope, which is denoted by P♦Q. In all of the cases we will examine, one of the initial C-groups is the group of a regular planar apeirohedron and the other initial C-group is the group of either the line segment or the linear apeirogon. In the next section we describe the concept of geometric blending which is closely related to mixing. For more on mixes and blended polytopes see [32], Sections 7A and 7E.

1.5 Realizations

We are interested in examining how objects which have the same face lattice structure as abstract polytopes can be manipulated in Euclidean space. To achieve this we will look at realizations of abstract polytopes in Euclidean space.

Definition 1.5. Following the definitions in [32], [29], and [28] a realization is a map from the d vertex set of an abstract n-polytope, P, to some Euclidean space, E . Call this mapping β0 and let the vertex set of P be denoted P0. Then V0 = β0(P0) is the vertex set of the realization. For V j = 1, ..., n we recursively define βj : Pj → Vj where Pj is the set of j-faces of P and Vj ⊆ 2 j−1 . V Here we use the notation 2 j−1 to denote the family of subsets of Vj−1 and say that Vj consists of elements

βj(F ) = {βj−1(G) | G ∈ Pj−1 and G ≤ F }.

Each Vj is the set of j-faces of the realization of P. Deﬁne β−1(F−1) := ∅ and deﬁne P := Vn as a geometric polytope which is the realization of the abstract regular P. Thus the realization is given by β := {βj}j=−1,0,...,n [32].

In other words, generating a realization of an abstract polytope is a recursive process where we deﬁne the realization of each face individually. The realization of an i-face F in P is deﬁned as the collection of the realizations of the facets of F . This collection of these (i − 1)-faces are the facets of βi(F ). Essentially we are determining the faces of the realization entirely by their facets. When this determination is unique the realization is faithful. More formally, the realization is faithful if

βi is a bijection for all i = 0, ..., n. We call a polytope P realizable if it admits a realization. Some polytopes are not realizable.

An example is the torus map {3, 6}(2,0). d Now let P be a regular polytope, and let P be a realization of P in E with vertex set V (P ) := V0. Recall that the distinguished generator ρi ∈ Γ(P) takes the base ﬂag, Φ, of P to its i-adjacent ﬂag,

12 i Φ . We can also look at the realization of Φ = {F−1,F0,F1, ..., Fn}, which will be

βΦ := {β−1(F−1), β0(F0), ..., βn(Fn)}.

d i If there exists an isometry of E mapping V (P ) to V (P ) such that it transforms βΦ into βΦ , deﬁne this isometry as ri. If such an isometry exists for all of the distinguished generators of Γ(P), we say that the realization is symmetric. The collection of the corresponding ri form the symmetry group of the realization P . Denote this symmetry group by

G(P ) := hr0, r1, ..., rn−1i.

The group G(P ) is a Euclidean representation of Γ(P) such that each isometry of G(P ) corresponds to an automorphism of Γ(P). The relations which hold for the automorphisms of Γ(P) carry over to G(P ). Let us explicitly describe this for regular polyhedra. Let P be a regular polyhedron of type {p, q} with automorphism group Γ(P) = hρ0, ρ1, ρ2i. Then we have at least the following relations: p q 2 (ρ0ρ1) = ε , (ρ1ρ2) = ε , (ρ0ρ2) = ε, where ε is the identity. For the corresponding symmetric realization P with symmetry group

G(P ) = hr0, r1, r2i the following relations will also be true:

p q 2 (r0r1) = 1 , (r1r2) = 1 , (r0r2) = 1, where 1 is the identity isometry of the ambient Euclidean space. The operations of section 1.4 all behave very similarly with the Euclidean representations of automorphism groups. For P as above with G(P ) = hr0, r1, r2i, we have

δ :(r0, r1, r2) → (r2, r1, r0)

π :(r0, r1, r2) → (r0r2, r1, r2) k−1 ϕk :(r0, r1, r2) → (r0, r1(r2r1) , r2)

η :(r0, r1, r2) → (r0r1r0, r2, r1) 2 σ :(r0, r1, r2) → (r1, r0r2, (r1r2) ).

We will use this fact extensively in determining the isometries of the polyhedra we will examine. Note that we can use the same Schläflisymbols to refer to the realizations of abstract regular regular polytopes as we use to describe the abstract polytopes. For the realization with the symbol {p1, p2, ..., pn−1, pn−1}, its facets are given by {p1, ...., pn−2} and its vertex figure is given by

{p2, ..., pn−1}. However, in the geometric context the “combinatorial” Schläflisymbol is refined by padding it with additional informational symbols representing the finer geometry. The result is a “geometric” Schläflisymbol. In the next section we discuss the various possibilities for a polygonal Schläflisymbol in rank 2. For the line segment, the only rank 1 polytope, we use the symbol {}.

13 From here on out when we use the terms polytope, polyhedron, or polygon we are referring to the symmetric realization of an abstract polytope, polyhedron, or polygon, respectively. We are justiﬁed in using these terms as a geometric n-polytope fulﬁlls the same conditions (A1)-(A4) as an abstract n-polytope which we shall repeat here for a geometric polytope P .

(P1) P has a least and a greatest face, F−1 and Fn.

(P2) Each ﬂag has n + 2 elements.

(P3) P is strongly ﬂag connected.

(P4) P fulﬁlls the diamond condition.

1.5.1 Blends of realizations

In Section 1.4.6 we discussed mixing the groups of two abstract polytopes. Here we will discuss a similar operation on the realizations of polytopes, the blending of realizations. In particular we will only examine the blend of a planar apeirohedra, P , with a line segment, {}, or a linear apeirogon, {∞} [32]. Suppose the symmetry groups of P , {}, and {∞} are, respectively,

G(P ) = hr0, r1, r2i, G({}) = hs0i, and G({∞}) = hs0, s1i. For our purposes, G(P ) acts on a plane 3 in E while G({}) and G({∞}) act on a line orthogonal to that plane. The blending process re- quires us ﬁrst to take the direct product of the groups, G(P ) × G({}) and G(P ) × G({∞}). The subgroups h(r0, s0), (r1, 1), (r2, 1)i of G(P ) × G({}) and h(r0, s0), (r1, s1), (r2, 1)i of G(P ) × G({∞}) are symmetry groups of regular apeirohedra. We say that the blend, P #{}, of P and {} has symmetry group h(r0, s0), (r1, 1), (r2, 1)i while the blend, P #{∞}, of P and {∞} has symmetry group h(r0, s0), (r1, s1), (r2, 1)i. For the rest of this thesis we will simplify the notation from (r, s) to rs for an element of the symmetry group of a blended apeirohedron. Also, note that rs = sr for any element r ∈ G(P ) and any s in either G({}) or G({∞}). These apeirohedra can be realized through Wythoﬀ’s construction (see Chapter 2) using the initial point (u, w) where u is the initial vertex of P and w is the initial vertex of {} or {∞}. Using these initial vertices, the blends P #{} and P #{∞} are combinatorially isomorphic to the mixes P ♦{} and P ♦{∞}, respectively. Note that while in this case the blending process behaves quite similarly to the abstract mixing operation, for the blends of arbitrary realizations of polytopes the result will not always be isomorphic to the mixes of their corresponding abstract polytopes. For a full discussion of blends and mixes see [32] Sections 7A and 7E.

1.6 Uniform polyhedra

When we perform Wythoﬀ’s construction in this thesis the objects which we generate are highly symmetric. Some of the more symmetric objects are uniform polyhedra.

Deﬁnition 1.6. A uniform polyhedron is a rank 3 geometric polytope such that the 2-faces are all regular polygons and the symmetry group is transitive on the vertices [40].

14 Cuboctahedron Truncated Octahedron Truncated Cube

Rhombicuboctahedron Truncated Cuboctahedron Snub Cube

Icosidodecahedron Truncated Dodecahedron Truncated Icosihedron

Rhombicosidodecahedron Truncated Icosidodecahedron Snub Dodecahedron

Truncated Tetrahedron

Figure 1.5: The Archimedean solids [6], [10], [44], [41].

15 The thirteen Archimedean solids are examples of uniform polyhedra derived from the Platonic solids through Wythoff’s construction. Along with the Platonic solids, these are the only convex uniform polyhedra ([10], [44]). The above definition does not require that the faces be convex polygons, and there are many other uniform polyhedra which do not have convex polygons as faces. Allowing for both convex and non-convex faces Coxeter, Longuet-Higgins, and Miller enumer- ated a complete list of finite uniform polyhedra with planar faces [9]. The completeness of the list was later proved, independently, by Skilling [42] and Har’El [24]. In the Coxeter, Longuet-Higgins, and Miller paper the authors used Wythoff’s construction and Schwarz triangles to create their list of uniform polyhedra. Here we will include a simplified explanation of their process, for a more complete description of Schwarz triangles see [9], [7], and [45]. Schwarz triangles lie on the surface of a sphere and are an extension of Möbiustriangles [5]. Their edges are the intersection of the surface of the sphere and a plane which bisects the sphere. Most importantly, repeatedly reflecting the triangle through its edges will cover the sphere a finite number of times. To construct a uniform polyhedron, first the authors chose a point on the sphere which was either a vertex of the Schwarz triangle, in the interior of an edge of the Schwarz triangle, or in the inerior of the Schwarz triangle. The point was chosen so that the non-zero distances between the point and its reflections across the edges of the triangle are all equal. The triangle and initial point are then repeatedly reflected across the edges of the triangle. Then an edge is defined as connected a point and its reflection across an edge of its triangle. This method generates 75 finite uniform polyhedra. The construction lies on the surface of a sphere so it only generates finite polyhedra. Furthermore, reflecting across the edges of Schwarz triangles only generates planar faces. In this thesis we will consider Wythoff’s construction a little differently which will allow us to examine uniform apeirohedra and uniform polyhedra with skew faces. In Chapters 4, 5, and 6 we will discuss apeirohedra with planar and non-planar faces. Note that a complete list of uniform planar apeirohedra with convex faces can be found in [23]. In Chapters 3 and 7 we will look at finite uniform polyhedra whose faces can be any type of finite polygon. Now we will classify the different types of regular polygons along with their Schläflisymbols. See [6], [19], and [38] for more on regular polygons. First we will look at the finite, planar polygons; then the finite, non-planar polygons; and finally we will look at the infinite polygons. The first type of regular polygons are the convex n-gons for n ≥ 3. This is simply the convex hull of n points evenly distributed on the circumference of a circle. We denote a convex n-gon as n {n}. The other finite, planar polygons are the star n-gons of density d where 1 ≤ d < 2 and n n and d are coprime. Denoted { d }, the star polygon is formed by again looking at a set of n points evenly distributed on the circumference of a circle. Moving in one direction, say counterclockwise, about the circle an edge is placed between a vertex and the dth vertex ahead of it. When d = 1, n { d } = {n}. The finite, non-planar polygons are the prismatic and antiprismatic polygons. A prismatic 2n-

16 (44) (4.82) (32.4.3.4) (33.42) (36) (63)

(3.6.3.6) (3.122) (3.4.6.4) (4, 6, 12) (34.6)

Figure 1.6: Uniform tilings with convex faces. The labels refer to the arrangement of faces at each vertex and are those used in [23].

Figure 1.7: Convex n-gons and star n-gons.

n n gon is formed from the vertices of a right prism with base { d } such that n is odd, 1 ≤ d < 2 , and n and d are coprime. The vertices coincide with those of the prism and edges are the diagonals of n n α n−2d the rectangular faces perpendicular to { d }. The Schläflisymbol is {2 · ( d ) } where 0 < α < n π is the angle between adjacent edges. An antiprismatic n-gon is found by alternately raising and lowering the vertices of a planar polygon perpendicularly to the plane of the planar polygon. The n α base planar polygon must have an even number, n, of vertices. The Schläflisymbol is {( d ) } n n−2d where 1 ≤ d < 2 , n and d are coprime, and 0 < α < n π is the angle between adjacent edges. n The vertices coincide with the vertices of an antiprism with a k-gonal base where k = 2 . When a prismatic or antiprismatic polygon is a facet or a vertex figure of a regular polyhedron, it is simply n written as d in the Schläflisymbol of the polyhedron. Both of these types of polygons can also be n viewed as { d }#{}. Due to their similarity to one another, in thesis they will often both be referred to as skew polygons. The infinite polygons are apeirogons, zigzag apeirogons, and helical apeirogons. The apeirogon lies on a straight line. It has an infinite number of vertices and edges (of the same length), leading to a Schläfli symbol of {∞}. The zigzag apeirogons, {∞α}, lie in a plane and have an angle of α between edges where 0 < α < π. They are of the form {∞}#{}. The helical apeirogons spiral above and below an n-gon and are of the form {n}#{∞}. The Schläflisymbol is {∞α,β} where

17 Figure 1.8: Prismatic and antiprismatic polygons.

0 < α, β < π, α + β > π, α is the angle between adjacent edges, and the helix is parametrized by (a cos βt, a sin βt, bt). With both the helical and zigzag apeirogons the angle measures are dropped from their notations when they are the facets of a regular apeirohedron.

Figure 1.9: Apeirogon, zigzag apeirogon, and helical apeirogon about square base.

While the facets of a uniform polyhedron are all regular polygons, they do not need to be congruent. Since the polyhedron is vertex transitive there will be the same arrangement of facets at each vertex. We call this arrangement the vertex configuration. The vertex figure is connected so we can list the faces at each vertex in an order given by adjacency, here adjacency means that the faces share an edge. At each vertex we can make such a list f1, f2, ..., fk where fi−1 and fi are both incident to a common edge. We then rewrite that list as (f1.f2.....fk). This symbol is how we denote the vertex configuration. This is similar to the vertex symbols used for the Archimedean plane tilings [36], [23]. The symbol is unique up to cyclic permutation and reversal of order. Here we will extend the notation to allow for infinite and skew faces. For a finite planar n-gon we use the notation n, for a skew n-gon we use the notation ns, for a truncated skew n-gon we use the notation tns, for a zigzag we use the notation ∞2, and for a helix over an n-gon we use the

18 notation ∞n. While this vertex configuration notation has most often been used for uniform tilings, it has also been used to describe how vertices are surrounded in non-uniform polyhedra. In describing the vertices of the Johnson solids Grünbaum and Johnson use this same vertex symbol notation, [22]. The Johnson solids are convex solids with regular faces but the vertices are surrounded in different ways. Thus in a Johnson solid different vertices are described by different vertex symbols. In this thesis all of the objects generated through Wythoff’s construction are vertex transitive so one vertex symbol will describe every vertex in the object. We will use this notation to describe every object we generate, even those with non-regular faces, [27].

1.7 Fundamental regions

Sometimes rather than examing a polytope in its entirety, it is enough to look at a typical region which models the behavior of the entire polytope. Such a typical region is called a fundamental region.

n n Deﬁnition 1.7. Let G be a discrete group of isometries of E . An open subset D of E is called a fundamental region for G if the following criteria are met:

(R1) For all r ∈ G such that r 6= 1 we have r(D) ∩ D = ∅.

n [ (R2) E = r(cl(D)). r∈G The groups G will typically be the symmetry groups of (discrete) regular polyhedra. If P is a regular polyhedron, then often we are only concerned with the action of G(P ) on the underlying n point set of P . In this case, if D is a fundamental region for G(P ) in its action on E , we call D0 := D ∩ P a fundamental region for G(P ) in P . Then the criteria in Deﬁnition 1.7 become

0 0 (R1’) For all r ∈ G(P ) such that r 6= 1 we have r(D ) ∩ D = ∅. [ (R2’) P = r(cl(D0)). r∈G(P ) The localized picture of P within the fundamental region shows exactly how P will behave in all of the transforms of the fundamental region. While the definition makes it clear that the fundamental region is typical, it does not show how to find a fundamental region. To find them we will utilize Fricke and Klein’s construction for fundamental regions of discrete groups G of n isometries in E as described in [1]. n Let v ∈ E be a point that is not held invariant under any non-identity transformation in G. For r ∈ G define H[r(v)] as the open half space containing v bounded by the hyperplane which perpendicularly bisects the line segment between v and r(v). The fundamental region of P centered at v is \ R := H[r(v)] r∈G

19 For a more in depth discussion of fundamental regions see [1] and [3]. We will use fundamental regions extensively when choosing our initial vertices for Wythoﬀ’s construction, see Chapter 2.

1.8 Polygonal complexes

We conclude this chapter with a brief discussion of polygonal complexes, or simply complexes.

n Deﬁnition 1.8. Following the deﬁnition in [35], a polygonal complex, C, in Euclidean space E is a collection of connected polygons, called faces, such that

(C1) The edge graph of C is connected.

(C2) For all vertices v in C, the vertex figure at v is connected. Here the term vertex figure refers to the graph whose vertices are the adjacent vertices to v in C and whose edges are defined as connecting two neighbors of v which are incident to a common face of C [40].

n (C3) C is discrete, that is, any compact set in E meets only ﬁnitely many faces of C.

(C4) Each edge of C is incident to a speciﬁed number of faces of C.

In the criterion (C4) when the speciﬁed number of faces is 2 the condition is essentially the diamond condition, and then C is a polyhedron.

3 Definition 1.9. In this thesis we use finite edge complex to refer to a figure in E composed of a sequence of points v1, v2, ..., vn with a specified finite number r ≥ 1 of line segments between each pair of points (vi, vi+1) and between (vn, v1). Similarly, an infinite edge complex is a sequence of points ..., v−1, v0, v1, ... with a specified finite number r ≥ 1 of identical line segments between each pair of points (vi, vi+1) for all i ∈ N such that the edge complex is discrete in the sense of (C3) from above.

Note that if the line segments joining the same pair of vertices are identified with each other, then a finite polygon or infinite polygon is obtained [12].

Deﬁnition 1.10. In this thesis we use the term semi-complex to refer to a connected collection of edge complexes, the faces of the semi-complex, which meets (C1), (C2), and (C3) from above. Additionally, each edge is incident to k faces where k can be at most three distinct numbers.

Almost all of the objects we will look at in this thesis are semi-complexes. Most of those are complexes and most of the complexes are polyhedra.

20 Chapter 2

Wythoﬀ’s construction

Wythoff’s construction as described in ([46], [6], [4]) provides a method for constucting polytopes in Euclidean space when given an initial vertex and the hyperplane reflection mirrors associated with the symmetry group of a regular polytope. The initial vertex and its transforms under the reflection group provide the vertex set of the constructed figure. [32] extends the construction to realizations of abstract regular polytopes as follows: Let Γ(P) := hρ0, ρ1, ..., ρn−1i be the automorphism group of an abstract regular n-polytope P, and let G(P ) := hr0, r1, ..., rn−1i be the corresponding Euclidean d representation of Γ(P) where P is a realization of P in Euclidean space E . Here, the reflections d r0, r1, ..., rn−1 are reflections in subspaces of E . Choose an initial vertex, v, such that r0(v) 6= v and ri(v) = v for all i ∈ {1, 2, ..., n − 1}. Define F0 := v. Recursively define each base t-face as

Ft := {r(Ft−1) | r ∈ hr0, ..., rt−1i} for t = 1, ..., n. Taking these faces along with F−1 := ∅ gives the base ﬂag {F−1,F0, ..., Fn} of P . The set of t-faces of the realization is then {r(Ft) | r ∈ G(P )}. The union of all of these sets gives the full realization.

2.1 Expanding Wythoﬀ’s construction

Here we will provide an extension of Wythoﬀ’s construction which generates a broader class of objects derived from regular polytopes. In this thesis we are primarily concerned with polyhedra so the construction will only be explicity described for the symmetry groups of regular polyhedra. As in [46], [6], [15], and [32] we will be examining the orbit of the initial vertex under the symmetry group. Although we may not always generate a polyhedron using this process we will always generate a poset of faces. d Continue to let P be an abstract regular polyhedron with realization P in E . Let Γ(P) = hρ0, ρ1, ρ2i be the automorphism group of P and let the corresponding Euclidean representation of the symmetry group of P be G(P ) = hr0, r1, r2i. Choose a non-empty subset E ⊆ {0, 1, 2}, and keep E ﬁxed. Each choice of E corresponds to a class of initial vertex choices all of which

21 are transient under rj for j ∈ E and invariant under all ri with i ∈ {0, 1, 2}\E. For a given E d choose some point v in E from its corresponding class of initial vertex choices. Every initial vertex corresponding to the same set E will yield combinatorially isomorphic figures. We use the fundamental region of P to limit our choices of initial vertex to make the choosing of the vertex a more feasible task. Choosing an initial vertex from within the fundamental region of P will result in a figure which is geometrically similar to the base polyhedron. For example if the faces of P are convex polygons and the vertex figures are skew polygons, choosing v from the fundamental region will generally result in a Wythoffian with convex faces corresponding to the faces of P and skew faces corresponding to the vertices of P . For each Wythoffian we will only examine initial vertices from within the fundamental region. We now impose the two conditions on v that it corresponds in the above way to a set E and that it is in the fundamental region of P . Once the initial vertex is chosen we can begin to generate base faces. Here we need to broaden the term base t-face to include any t-face incident with v whose vertex set is the orbit of v under precisely t distinguished generators of G(P ). Now that we are allowing more freedom in the choice of v it is possible that we can have up to three different kinds of base t-face for t = 1, 2. For instance, there is potentially a unique base 1-face corresponding to each I distinguished generator in G(P ). To differentiate between base faces we use the notation Ft to denote a base t-face where I is the set of the indices of the generating symmetries used to generate I Ft . To fully define the poset of faces generated by Wythoff’s construction we will give explicit definitions of each type of base t-face for t = 0, 1, 2, 3. Define the base 0-face ∅ F0 := v.

∅ Since there is only one base 0-face we will simplify the notation to F0 := F0 . For each i ∈ E we can deﬁne a base 1-face

{i} F1 := {r(v) | r ∈ hrii} = {v, ri(v)} and we deﬁne the set of base 1-faces as

{i} F1 := {F1 | i ∈ E}.

To define the base 2-faces we first need to determine which pairs of generating reflections of

G(P ) will generate a 2-face. The generators ri, rj will generate a base 2-face if two conditions are met. First we can have two distinct types of base edge, one generated by each generating reflection in the pair; or exactly one of the generating reflections corresponds to a base edge in the face. Second, the base edge generated by one of the reflections in the pair must not be invariant under the other reflection in the pair, else the orbit of the edge under the subgroup generated by the pair of reflections would simply be the edge itself and not a 2-face. We will call these conditions (W1) and (W2).

22 (W1) Both i, j ∈ E; or i ∈ E, j ∈ {0, 1, 2}\E.

{i} {i} (W2) If i ∈ E, then rj(F1 ) 6= F1 . {i} {j} {i} {i} Note that if F1 = F1 for i, j ∈ E then rj(F1 ) = F1 . Thus as a consequence of (W2), if {i} {j} i, j ∈ E then F1 6= F1 in order for ri and rj to be used in generating a 2-face of the Wythoffian. In the next section we will see some examples of when a particular initial vertex choice leads to two base edges being equal, thus violating (W2) and so they do not generate a 2-face. Now for any i, j which fulfill (W1) and (W2) we can define a base 2-face

{i,j} {k} {k} F2 := {r(F1 ) | r ∈ hri, rji, k ∈ {i, j},F1 ∈ F1}

and the set of all base 2-faces

{i,j} F2 := {F2 | i, j fulﬁll (W1) and (W2)}.

Now we can deﬁne the full Wythoﬃan

{0,1,2} {i,j} {i,j} WE(v) := F3 := {r(F2 ) | r ∈ G(P ),F2 ∈ F2}.

Finally we will deﬁne the base face of rank −1 as F−1 := ∅.

Essentially we start with a collection, E, of indices of generators from G(P ) and find a point, v, which is transient under all generators with indices in E. We then build up the faces starting with the edges. Each base k-face is built by taking the orbit of a specified set of base (k − 1)-faces under a specified set of generators. Then defining incidence follows naturally. I J If Ft−1 is used in the generation of Ft then

I J r(Ft−1) ≤ Ft

for all r ∈ hrj | j ∈ Ji. Consequently, for s ∈ G(P )

I J sr(Ft−1) ≤ s(Ft )

0 00 0 0 00 again with r ∈ hrj | j ∈ Ji. Additionally, for faces H,H ,H of WE(v) if H ≤ H and H ≤ H then H ≤ H00. Once the entire Wythoffian is constructed we can define its base flags. In each base flag the

first two faces are F−1 := ∅ and F0 = v. The next face is any one of the elements of F1. Our {i} {i} partial flag is now {F−1,F0,F1 }. Each base 2-face which uses F1 in its generation is an option {i} for the next face, that is, every base 2-face which is incident to F1 is an option. The final base face of the flag is the Wythoffian. A base flag will look like

{i} {i,j} {0,1,2} {F−1,F0,F1 ,F2 ,F3 }.

23 For a given choice of E there are generally going to be several distinct base flags which are not equivalent under G(P ). Each flag of WE(v) can be written as rΦ for r ∈ G(P ) and Φ a base flag of WE(v).

d Lemma 2.1. Let P be a regular polyhedron in E with symmetry group G(P ) = hr0, r1, r2i. Let {j} {j} j ∈ E, k 6∈ E and (j, k) = (0, 2) or (2, 0). Then rk(F1 ) = F1 .

Proof. Without loss of generality assume j = 0, k = 2 so rk(v) = v. Then

{0} r2(F1 ) = r2 ({v, r0(v)})

= {r2(v), r2r0(v)}

= {v, r0r2(v)}

= {v, r0(v)} {0} = F1 .

A direct consequence of Lemma 2.1 is that if ri and rj are a pair of generating reﬂections with i ∈ E and j ∈ {0, 1, 2}\E which fulﬁll (W2) then |i − j| = 1.

Lemma 2.2. The Wythoﬃan, WE(v), of the polyhedron P is vertex transitive under the symmetry group of P .

{i} Proof. Each base 1-face is deﬁned as F1 = {v, ri(v)} for some i ∈ E so clearly each vertex in a base edge is equivalent to v. {i,j} The base 2-face F2 is deﬁned as the orbit of its base edges under hri, rji. Every edge {i,j} {k} of F2 can be written r(F1 ) for r ∈ hri, rji and k = i, j for whichever of i, j ∈ E. Then {k} {i,j} r(F1 ) = r({v, rk(v)}) = {r(v), rrk(v)}. Thus each vertex in every edge of F2 is equivalent to v under hri, rji.

The base 3-face WE(v) is deﬁned as the orbit of its base 2-faces under G(P ). We know that the base 2-faces have all their vertices equivalent to v. Thus any vertex in the orbit of any 2-face is equivalent to v under G(P ).

For more on vertex transitive polyhedra and edge transitive polyhedra see [18].

Theorem 2.1. The Wythoﬃan, WE(v), based on the symmetry group G(P ) = hr0, r1, r2i for regular polyhedron P fulﬁlls polytope conditions (P1), (P2), and (P3).

Proof. (P1) The least face is F−1 = ∅ and the greatest face is F3 = WE(v).

(P2) As defined, each base flag has five elements and every other flag is the transform of a base flag so it will contain the same number of elements.

24 (P3) We will show strong connectedness of the faces and use Proposition 2A1 of [32] which tells us that when a poset of faces fulﬁlls the previous two conditions and is strongly connected then it is also strongly ﬂag connected. We only need to consider sections, H/H0, of rank 2 and 3. This leaves us with three cases 0 0 to consider: H is a 2-face with H = F−1; H = WE(v) with H a vertex; and H = WE(v) with 0 H = F−1.

First consider the section H/F−1 where H is a 2-face of WE(v). Without loss of generality let {i,j} 0 0 H be the base face F2 . Let F,F be two proper faces of the section. If F and F are incident to one another they are connected and there is nothing to show, so assume they are not.

If i ∈ E and j ∈ {0, 1, 2}\E then we have ri(v) 6= v, rj(v) = v, {v, ri(v)} is the only base edge {i,j} {i,j} in F2 , and all edges in F2 can be written {r(v), rri(v)} for some r ∈ hri, rji. This leads us to

2 · · ·{rjrirj(v), (rjri) (v)} ≥ rjrirj(v) = rjri(v) ≤ {rj(v), rjri(v)} ≥ rj(v) = 2 v ≤ {v, ri(v)} ≥ ri(v) = rirj(v) ≤ {rirj(v), rirjri(v)} ≥ rirjri(v) = (rirj) n n n n+1 ≤ · · · ≤ {(rirj) (v), (rirj) ri(v) ≥ (rirj) ri(v) = (rirj) (v) ≤ · · ·

{i,j} which cycles through all vertices and edges of F2 since it goes through r(v) for all r ∈ hri, rji. Thus F,F 0 are somewhere in that list, so F is connected to F 0. {i,j} If i, j ∈ E then ri(v) 6= v, rj(v) 6= v, {v, ri(v)} and {v, rj(v)} are the two base edges of F2 , {i,j} and all edges incident to F2 can be written as {r(v), rri(v)} or {r(v), rrj(v)} for some r ∈ hri, rji. Then, similar to before,

n n−1 n−1 n−1 n−1 · · ·{(rirj) (v), (rirj) ri(v)} ≥ (rirj) ri(v) ≤ {(rirj) ri(v), (rirj) (v)} ≥ n−1 (rirj) (v) ≤ · · · ≤ {rirj(v), ri(v)} ≥ ri(v) ≤ {v, ri(v)} ≥ v ≤ {v, rj(v)} ≥ rj(v) ≤ · · · k k k k k+1 ≤ {(rjri) (v), (rjri) rj(v)} ≥ (rjri) rj(v) ≤ {(rjri) rj(v), (rjri) (v)} ≥ · · ·

{i,j} 0 which lists all vertices and edges of F2 . Thus F,F are in the list and are connected to each other.

0 Next consider the case when H = WE(v) and H is a vertex. Since G(P ) acts vertex-transitively we may assume without loss of generality that H0 = v. Let F,F 0 be two proper faces in the section. {i,j} Similar to the section F2 /F−1 we will show how to connect a path through all proper faces of WE(v)/v. {i,j} {i} If there is only one base face, F2 , then there is only one base edge F1 and i = 0 or 2 and j = 1. Without loss of generality let i = 0 then j = 1. Since r1(v) = r2(v) = v and r0(v) 6= v, every {0} {0,1} edge (or face) incident to v must be of the form r(F1 ) (or r(F2 )) for some r ∈ hr1, r2i. Also {0} {0} note that by Lemma 2.1, r2(F1 ) = F1 . Now we can look at

{0} {0,1} {0} {0} {0,1} {0} 2 {0} F1 ≤ F2 ≥ r1(F1 ) = r1r2(F1 ) ≤ r1r2(F2 ) ≥ r1r2r1(F1 ) = (r1r2) (F1 ) ≤ · · ·

25 which cycles through all edges and faces which are incident to v. Since that is the complete set of proper faces of the section, F,F 0 must belong to it and are connected to each other. {i,j} {j,k} If there are exactly two base faces, F2 and F2 , then there are several diﬀerent ways for the base edges to be arranged. In each case we will proceed as before by constructing a path through all edges and faces incident to v. {j} Suppose F1 is the only base edge, then j = 1. Then

{1} {0,1} {1} {1,2} {1} {0,1} F1 ≤ F2 ≥ r0(F1 ) ≤ r0(F2 ) ≥ r0r2(F1 ) ≤ r0r2(F2 ) {1} {1,2} 2 {1} {1} ≥ r0r2r0(F1 ) ≤ r0r2r0(F1 ) ≥ (r0r2) (F1 ) = F1 showing all edges and faces incident to v are connected to each other. {i} {j} {k} {i} Suppose F1 and F1 are both base edges, then F1 is either equal F1 or it does not exist {k} {i} {j} {j} (that is, k 6∈ E). When F1 = F1 then rirk(F1 ) = rkri(F1 ) ≥ v and so

{i,j} {j} {j,k} {k} {k} {j,k} {j} {j} {i,j} F2 ≥ F1 ≤ F2 ≥ F1 = ri(F1 ) ≤ ri(F2 ) ≥ rirk(F1 ) = rkri(F1 ) ≤ rk(F2 )

{k} {i} which is a path through all proper faces of the section. When F1 does not exist rk(F1 ) and {j} rk(F1 ) are both incident to v, and thus

{j} {j,k} {j} {i,j} {i} {i} {i,j} {j} F1 ≤ F2 ≥ rk(F1 ) ≤ rk(F2 ) ≥ rk(F1 ) = F1 ≤ F2 ≥ F1 completes a circuit through all proper faces of the section. In both cases the section is connected. {i} {k} {j} {i} {i,j} {j,k} {i,j} Suppose F1 = F1 and F1 does not exist. Now F1 is incident to F2 ,F2 , rk(F2 ), {j,k} {i} and ri(F2 ). The edge rjrirkrj(F1 ) is still incident to v and is incident to the remaining four {j,k} {i,j} {j,k} {i,j} faces of the section: rjri(F2 ), rjrk(F2 ), rirkrjri(F2 ), and rirkrjrk(F2 ). Since the edge {i} {i} {i,j} {i,j} rj(F1 ) = rjrk(F1 ) is incident to both F2 and rjrk(F2 ) the two sets of 2-faces are connected and thus all the 2-faces of the section are connected to each other. All edges of the section are connected to four of these faces. All proper faces of the section are thus connected to one another. {i,j} {j,k} {i,k} If there are three distinct base faces F2 , F2 , and F2 then there are either three distinct {0} {2} {1} base edges or F1 and F2 are distinct and F1 does not exist. In the ﬁrst case

{0} {0,1} {1} {1,2} {2} {0,2} {0} F1 ≤ F2 ≥ F1 ≤ F2 ≥ F1 ≤ F2 ≥ F1 tracing a path through all proper faces of the section. In the second case

{0} {0,1} {0} {0,2} {2} {1,2} {2} {0,2} {0} F1 ≤ F2 ≥ r1(F1 ) ≤ r1(F2 ) ≥ r1(F1 ) ≤ F2 ≥ F1 ≤ F2 ≥ F1 tracing a path through all proper faces of the section. In both cases the section is entirely connected.

0 Finally, consider the section WE(v)/F−1. Let F,F be two proper faces of the Wythoﬃan. Then

26 we can trace a connected path from F to F 0 using only vertices and edges as follows. Start with

{bp} F ≥ rb1 rb2 ...rbm−1 rbm (v) = rb1 rb2 ...rbp (v) ≤ rb1 ...rbp−1 (F1 )

{bq} ≥ rb1 ...rbp−1 (v) = rb1 ...rbq (v) ≤ rb1 ...rbq−1 (F1 ) where p ≤ m−1 is the largest number such that bp ∈ E, and q ≤ p−1 is the next to largest number with bq ∈ E. Continue this pattern until we eliminate all ri and reach v. We can do the same thing to ﬁnd a connected path of vertices and edges from F 0 to v. Thus F and F 0 are connected by a series of incident proper faces of WE(v).

And thus WE(v) fulﬁlls (P3).

We purposely excluded polytope condition (P4), the diamond condition, from Theorem 2.1. As will be brieﬂy examine in the next section and repeatedly demonstrate in Chapters 3-6, there are several instances when a Wythoﬃan has edges which are incident to more than two faces.

2.2 Properties of Wythoﬃans

d Let P be a regular polyhedron in E with underlying abstract regular polyhedron P. Let G(P ) = hr0, r1, r2i. If P is of type {p, q}, then

p q 2 (r0r1) = (r1r2) = (r0r2) = 1

n when p and q are both finite. If p (or q) is infinite then there is no n such that (r0r1) = 1 (or n (r1r2) = 1). From the symmetry group we can generate Wythoffians which will have certain general combinatorial characteristics dependent on E and independent of the particular symmetry group. With some symmetry groups, for a given choice of E, different initial vertices will produce geometric figures with slightly different properties. Before we discuss the combinatorial properties of the Wythoffians we will introduce some notation. Table 2.1 shows two common notations and the notation which we will use throughout this thesis. The left most column indicates the choice of E, the indicies of generating symmetries under which the initial vertex is transient. The next column illustrates the relative placement of the initial vertex within the fundamental region idealized as a triangle. With convex polyhedra the fundamental region intersects the polyhedron in an actual triangle and the vertices, relative interiors of edges, and interior of the triangle represent the different choices of initial vertex. We can still use the triangle to demonstrate the choice of initial vertex even with non-convex polyhedra. The next column lists the diagrams as discussed in Coxeter’s Regular Polytopes, [6], which again illustrate the choice of initial vertex with respect to the generators of the symmetry group. The final column contains the notation which we will use throughout. The other two notations have historically been used to refer to Wythoffians derived from groups generated by plane reflections.

27 Here we will consider all symmetry groups of the 48 regular polyhedra which include those with non-planar generating reflections. So here we will use a different more generic notation. Table 2.1 shows the relationship between the historic notations and the notation we will use. The notation we use comes from the choice of E in the construction. Recall that E ⊆ {0, 1, 2} and the initial vertex is transient under all ri ∈ G(P ) with i ∈ E and invariant under all rj ∈ G(P ) with j 6∈ E. When E = {i} for i ∈ {0, 1, 2} we use the symbol P i to denote the corresponding Wythoffian. If E = {i, j} for i, j ∈ {0, 1, 2} we use the symbol P ij to denote the corresponding Wythoffian. Finally, when E = {0, 1, 2} we use the symbol P 012 to denote the corresponding Wythoffian. Then we clearly see P ij = P ji and P ijk = P ikj = P jik = ... for i, j, k ∈ {0, 1, 2}. For the sake of simplicity, for i < j < k we will always write P ij and P ijk.

As before, let E ⊆ {0, 1, 2} and let v be the initial vertex such that v is transient under all ri with i ∈ E. For i, j ∈ E assume that no two base edges, {v, ri(v)} and {v, rj(v)}, are equivalent under any transformations of G(P ). Then every edge of WE(v) is a transform of exactly one base edge and can be said to be of the same type as its corresponding base edge. In this thesis we will J J use the phrase a face H is of type Fi to mean that H is a transform of Fi under the symmetry group of the Wythoffian. In the rare cases where a face is a transform of multiple base faces, say J I J I Fi and Fi , we would say that it is of type Fi and Fi . These rare cases will be discussed in detail later in this section. The Wythoffian P 0 is isomorphic to the original polyhedron P and hence is of equivelar type {0} {0,1} {p, q}. The only base edge is F1 and the only base 2-face is F2 which is a geometrically regular p-gon. The vertex figure is a geometrically regular q-gon. 1 {1} {0,1} {1,2} The Wythoffian P has only one base edge, F1 . There are two base 2-faces, F2 and F2 . {0,1} {1,2} The face F2 is a geometrically regular p-gon and the face F2 is a geometrically regular q-gon. The vertex figure is a tetragon with vertex configuration (p.q.p.q). In the case when p and q are both finite, P 1 is isomorphic to the map on a surface known as the medial of P ([8], [14], [13]). When the Wythoffian is P 2 it is the realization of the dual polyhedron of P . As such, it is {2} {1,2} of equivelar type {q, p}. The only base edge is F1 and the only base 2-face is F2 which is a geometrically regular q-gon. The vertex figure is a geometrically regular p-gon. 01 {0} {1} {0,1} The Wythoffian P has two base edges, F1 and F1 . It then has two base 2-faces: F2 {1,2} which is a 2p-gon appearing as a truncated p-gon and F2 which is a geometrically regular q-gon. The vertex figure is an isosceles triangle since there are two 2p-gons and one q-gon meeting at each vertex. The vertex configuration is then (q.2p.2p). In the context of maps on surfaces, P 01 is called the vertex truncation of P [6]. 02 {0} {2} The Wythoffian P has two base edges, F1 and F1 . This figure has three base 2-faces. The {0,1} {1,2} face F2 is a geometrically regular p-gon, the face F2 is a geometrically regular q-gon, and the {0,2} face F2 is a tetragon. The vertex configuration is (p.4.q.4) which corresponds to a trapezoidal vertex figure. 12 {1} {2} {0,1} The Wythoffian P has two base edges, F1 and F1 . Its two base 2-faces are F2 , a regular {1,2} p-gon, and F2 , a 2q-gon which appears as a truncated q-gon. There is one p-gon and two 2q-

28 E Initial vertex placement Coxeter’s notation Notation for this thesis

{0} P 0

{1} P 1

{2} P 2

{0, 1} P 01

{0, 2} P 02

{1, 2} P 12

012 {0, 1, 2} P

Table 2.1: Different notations for the Wythoffian, WE(v), based on choice of E using the symmetry group of P which is a regular polyhedron of type {p, q}. 29 gons at each vertex yielding an isosceles triangle as the vertex figure. The vertex configuration is (p.2q.2q). The final combinatorial type of Wythoffian we can examine is P 012. This figure has three base {0} {1} {2} {0,1} {1,2} edges: F1 , F1 , and F1 . It also has three base faces: F2 , a 2p-gon (truncated p-gon); F2 , {0,2} a 2q-gon (truncated q-gon); and F2 , a tetragon. The vertex configuration of the final figure has a 2p-gon, a 2q-gon, and a tetragon surrounding each vertex, so the symbol is (2p.2q.4). The vertex figure is a triangle. The above analysis gives a combinatorial description of the Wythoffians. To get a more geometric idea of what is going on we need to look a bit deeper. The 2-faces of the Wythoffian may be skew, planar, star, or some combination of those. The nature of the face can often be determined by looking at the original figure, P . The facets of P directly inform the nature of the 2-faces of {0,1} type F2 . Similarly the geometry of the vertex figure of P determines the structure of the 2-faces {1,2} {0,1} of type F2 . For instance, if the facets of the original figure are skew then F2 will generally {0,2} be skew. The nature of the face, F2 , will be determined by its adjacent faces.

If we remove our assumption and allow ri(v) = srj(v) for i, j ∈ E, i 6= j, and some s ∈ G(P ) then the resulting figures will behave differently. This occurs in a couple of situations. For example, Wythoffians whose symmetry groups are generated by performing the Petrial operation on the symmetry group of a Platonic solid or a Kepler-Poinsot polyhedron can have equivalent base faces.

In the case of the Petrial, when the problem occurs, the base edges of WE(v) are not merely equivalent, but they are in fact equal. Let Q be an abstract regular polyhedron with automorphism group Γ(Q) = hσ0, σ1, σ2i whose Euclidean realization, Q, has symmetry group G(Q) = hs0, s1, s2i. π π The abstract Petrial Q of Q then has automorphism group Γ(Q ) = hσ0σ2, σ1, σ2i and the geo- π metric Petrial P := Q has symmetry group G(P ) = hs0s2, s1, s2i =: hr0, r1, r2i. When we look 02 012 at P and P we can choose an initial vertex v which is stabilized by s0 but not by s2 and consequently v is not stabilized by r0. Consequently, for this choice of v, r0(v) = r2(v) 6= v. In {0} {2} these two cases the base edges, F1 and F1 , are actually the same edge. This single edge is then {0,1} {0,1} {1,2} {1,2} incident to F2 , r2(F2 ), F2 , and r0(F2 ). This violates the diamond condition and thus 02 the resulting figure is not a polyhedron. Specifically, with P the point v is invariant under s0 and {0} {2} r1 so the only possible base edges are F1 and F1 . Thus for this choice of v there is only one base 012 edge and the Wythoffian is a non-polyhedral complex. With P the point v is invariant under s0 {1} alone so the base edge F1 exists and is incident to exactly two 2-faces making the Wythoffian a semi-complex (as defined in Section 1.8). The Wythoffians, P 02 and P 012, based on P with initial vertex v as above can be linked to 0 0 the Wythoffian for P with initial vertex v such that v is not stabilized by s0, r0, or r2. Then in 0 {0,2} {0} {0} {2} {2} WE(v ) the 2-face F2 is a quadrilateral consisting of edges F1 , r2(F1 ),F1 , and r0(F1 ). In WE(v) where v is stabilized by s0 these edges are all equal, so essentially the quadrilateral collapses in on itself and becomes a single edge. Geometrically, to transform the Wythoffian for P based on 0 {0,2} {2} v to the Wythoffian based on v, collapse faces s(F2 ) for each s ∈ G into the edge s(F1 ) and

30 {0} collapse any tranforms of F1 into a single point. The regular polyhedra derived through the facetting operation from another regular polyhedron also provide cases where the base edges of the Wythoﬃan can be transformed into one another under the symmetry group. Again let Q be a regular polyhedron with automorphism group Γ(Q) = hσ0, σ1, σ2i, and let {p, q} be its type. Then if the facetting operation yields a regular polyhedron, ϕ ϕ Q 2 , it has automorphism group Γ(Q 2 ) = hσ0, σ1σ2σ1, σ2i. The Euclidean realization, P := ϕ ϕ Q 2 , of Q 2 has symmetry group G(P ) = hs0, s1s2s1, s2i =: hr0, r1, r2i. With regards to taking Wythoﬃans of P , this symmetry group and its generators only behave oddly when the initial vertex, v, is stabilized by s1 but not by r1.

Suppose v is stabilized by s1 alone and that q is odd. Then

(q−1)/2 (q−1)/2 (r1r2) r1{v, r2(v)} = (s1s2s1s2) s1s2s1{v, s2(v)} q−1 = (s1s2) s1s2s1{v, s2(v)} q = (s1s2) s1{v, s2(v)}

= s1{v, s2(v)}

= {s1(v), s1s2s1(v)}

= {v, r1(v)} where the additional s1 in the fifth line comes from the fact that by our choice of v, s1(v) = v {2} {1} and thus s1s2(v) = s1s2s1(v). Clearly, in this case, F1 is congruent to F1 under an element (n−1)/2 {1,2} of G(P ). Since (r1r2) r1 ∈ hr1, r2i we see that both edges belong to the face F2 . Base (n−1)/2 faces are defined as the orbit of their base edges, so both (r1r2) r1 {v, r2(v)} and {v, r1(v)} {1,2} are incident to F2 . The face now has double edges causing the resulting figure to be an edge complex but not a polygon (see Section 1.8). As a result, the geometric Wythoffian is not a faithful realization of the corresponding abstract Wythoffian. As with the Petrial case we can examine how a similar Wythoffian for P can transform into 0 WE(v). Let us look at a vertex which has similar properties as v. That is, let v be a point that is 0 not held invariant by r1 nor by r2, just the same as v, but unlike v let v be transient under s1. The {1,2} 0 face F2 of WE(v ) is a polygonal face with an even number of vertices and edges. To transform {1,2} it to the face F2 of WE(v) the edges collapse onto one another yielding a face with half as {0,1} 0 many vertices and edges. This transformation does not affect the structures of F2 in WE(v ) {0,2} 0 {0,1} {0,2} and F2 in WE(v ) which have the same structures as F2 in WE(v) and F2 in WE(v), {1} {2} {0,1} {1,2} respectively. Edges equivalent to both F1 and F1 are incident to F2 in WE(v), F2 in {0,2} WE(v), and F2 in WE(v). While the resulting figure from v is different from the figure made from v0 it is clear how they are related. We can then perform the different operations on the group of Qϕ2 . Each time the result is the group of a regular polyhedron the corresponding Wythoffians will have similar issues as those discussed above. For examples, see the final eight polyhedra in the icosahedral family (Chapter 3).

31 As a final note before we begin examining the Wythoffians, we will discuss the methods used to calculate the figures. All Wythoffians in this thesis were initially calculated by hand. The basic properties of the regular polyhedra and their groups as described in [32] were used to determine the behavior of the Wythoffians. The results were then verified using Mathematica [26]. For each regular polyhedron the generating reflections were programmed into Mathematica then those reflections were repeatedly applied to each initial vertex. This allowed us to view and interact with the orbit of each initial vertex under the symmetry group. Mathematica was also used to help calculate the initial vertices used for the general Wythoffians and for the uniform cases. All the images in this thesis were generated by the author using this software.

32 Chapter 3

Finite regular polyhedra and their Wythoﬃans

There are eighteen finite regular polyhedra. These can be organized according to their symmetry groups. By performing combinations of the Petrial, dual, and facetting operations on the symmetry group of the tetrahedron, the symmetry group of the octahedron, and the symmetry group of the icosahedron we can find the symmetry groups of all the finite regular polyhedra. Depending on which regular polyhedron its group is derived from, each finite polyhedron is then said to have tetrahedral symmetry, octahedral symmetry, or icosahedral symmetry. Listed below are the finite regular polyhedra grouped by symmetry family with the relationships between their groups. This list comes from McMullen and Schulte’s Abstract Regular Polytopes ([32], p. 218).

T etrahedral Symmetry π {3, 3} ←→ {4, 3}3

Octahedral Symmetry π δ π {6, 4}3 ←→ {3, 4} ←→ {4, 3} ←→ {6, 3}4

Icosahedral Symmetry

π δ π {10, 5}3 ←→ {3, 5} ←→ {5, 3} ←→ {10, 3}5

l ϕ2 l ϕ2

5 π 5 δ 5 π {6, 2 } ←→ {5, 2 } ←→ { 2 , 5} ←→ {6, 5}

l ϕ2 l ϕ2

10 π 5 δ 5 π 10 5 { 3 , 3} ←→ { 2 , 3} ←→ {3, 2 } ←→ { 3 , 2 }

To perform Wythoﬀ’s construction we will ﬁrst have to identify the symmetry group corresponding to the symmetry group of the regular polyhedron. Then we will need to choose an initial

33 vertex, v, based on how it is transformed under the symmetry group. The resulting figures will generally be polyhedra with a few exceptions. Many of the Wythoffians will share an edge graph or, at the very least, a vertex set with a polyhedron which is isomorphic to an Archimedean solid. Here, we refer to the Archimedean solids by their names. For a full listing of the Archimedean solids see Section 1.6 or [10]. This chapter is composed of written descriptions of each of the Wythoffians corresponding to the finite regular polyhedra. The Appendix at the end of the thesis contains tables with an image of each type of face and the vertex figure for each Wythoffian. Alongside the tables there are images of the full Wythoffians. In this chapter, some of the regular polyhedra we will examine have multiple Wythoffians for a given initial vertex class. When this happens there is a generic case immediately after the label for the initial vertex class and then any other possible Wythoffians follow it. The tables in the appendix only show the faces, vertex figures, and Wythoffians for the generic cases. In all of the finite Wythoffians with regular skew faces, the faces are antiprismatic polygons. We will simply refer to them as skew polygons.

3.1 Regular polyhedra with tetrahedral symmetry

First we examine the two polyhedra which have underlying tetrahedral symmetry: {3, 3} and

{4, 3}3.

3.1.1 {3, 3}

Here we will examine the regular polyhedron with Schl¨aﬂisymbol {3, 3}, the tetrahedron. The automorphism group of the regular tetrahedron is Γ({3, 3}) := hσ0, σ1, σ2i where

3 3 2 (σ0σ1) = (σ1σ2) = (σ0σ2) = ε.

Its Euclidean realization has symmetry group G({3, 3}) := hs0, s1, s2i where

3 3 2 (s0s1) = (s1s2) = (s0s2) = 1.

Now we will examine the Wythoffians generated from G({3, 3}). For images of the faces, vertex figures, and full Wythoffians see Table 7.1 in the Appendix. In each case the initial vertex, v, is chosen to be transient under a specific subset of generators of G({3, 3}) and to be in the fundamental region of {3, 3}. For this symmetry group, all the Wythoffians are convex polyhedra. Additionally, for specifically chosen intial vertices, each Wythoffian can be made into a uniform polyhedron.

0 P Here, WE(v) is the regular tetrahedron, {3, 3} itself, with three regular triangles meeting at {0} {0,1} each vertex. This figure has six edges of type F1 and four triangular faces of type F2 . The vertex figure is a regular triangle with vertex configuration (3.3.3).

34 P 1 This polyhedron is a regular octahedron where four regular triangles meet at each vertex, (3.3.3.3), resulting in a convex square as the vertex ﬁgure. This ﬁgure has twelve edges of {1} {0,1} type F1 . There are four faces of type F2 which are regular triangles, and there are four {1,2} faces of type F2 which are also regular triangles.

P 2 This Wythoffian is a regular polyhedron and is the dual figure to {3, 3}. The tetrahedron is {2} its own dual so this Wythoffian is a regular tetrahedron. There are six edges of type F1 {1,2} and four faces of type F2 which are triangles. Three such triangles meet at each vertex, (3.3.3), yielding a regular triangle as the vertex figure.

01 {0} P This polyhedron is isomorphic to a truncated tetrahedron. It has six edges of type F1 and {1} {0,1} twelve edges of type F1 . There are four faces of type F2 which are convex hexagons {1,2} and there are four faces of type F2 which are regular triangles. Two hexagons and one triangle meet at each vertex, (3.6.6), yielding an isosceles triangle as the vertex ﬁgure. When {0} the initial vertex is chosen so that the edges of type F1 are the same length as the edges {1} of type F1 the Wythoﬃan is a uniform truncated tetrahedron with regular hexagons and triangles for faces.

P 02 Here, the Wythoffian is a polyhedron isomorphic to a cuboctahedron. There are twelve {0} {2} edges of type F1 and twelve edges of type F1 . There are three distinct types of 2- {0,1} {1,2} faces. The faces of type F2 are regular triangles, the faces of type F2 are also regular {0,2} triangles, and the faces of type F2 are convex rectangles. The vertex configuration at each {0,1} {1,2} vertex is one triangle of type F2 , one rectangle, one triangle of type F2 , and another rectangle yielding a vertex symbol of (3.4.3.4). The vertex figure is then a convex trapezoid. For a specific choice of initial vertex the Wythoffian is a uniform cuboctahedron where the rectangular faces are squares. When the Wythoffian is a uniform polyhedron the vertex figure is a rectangle.

P 12 This polyhedron is isomorphic to a truncated tetrahedron. There are twelve edges of type {1} {2} {0,1} F1 and six edges of type F1 . This figure has four regular, triangular faces of type F2 {1,2} and four convex, hexagonal faces of type F2 . Two hexagons and one triangle meet at each vertex, (3.6.6), yielding an isosceles triangle as the vertex figure. As with P 01, the initial vertex can be chosen so as to make the Wythoffian a uniform truncated tetrahedron.

P 012 The final Wythoffian is isomorphic to a truncated octahedron. There are twelve edges of type {0} {1} {2} F1 , twelve edges of type F1 , and twelve edges of type F1 . The figure has four convex, {0,1} {1,2} hexagonal faces of type F2 ; four convex, hexagonal faces of type F2 ; and six convex {0,2} rectangles generated by F2 . At each vertex there is one face of each type which together give a triangular vertex figure. For a specifically chosen vertex the faces can be made into regular hexagons and squares resulting in the uniform polyhedron the truncated octahedron.

35 Notice in the above how the self-duality of the tetrahedron {3, 3} is reﬂected in the structure of the Wythoﬃans. In fact, the structure is invariant under interchanging the superscripts 0 and 2.

3.1.2 {4, 3}3

The other member of the tetrahedral family is {4, 3}3 which is the Petrial of {3, 3}. As before let

Γ({3, 3}) = hσ0, σ1, σ2i, then

Γ({4, 3}3) = hσ0σ2, σ1, σ2i.

Let the symmetry group of the Euclidean realization be

G({4, 3}3) := hs0s2, s1, s2i =: hr0, r1, r2i.

Throughout we let v be the initial vertex used to generate the new figure using Wythoff’s construction, for images see Table 7.2 in the Appendix. The initial vertex comes from the fundamental region for {4, 3}3 which is the same as the fundamental region of {3, 3}. The choice of initial vertex is not as straight forward in as it was with {3, 3} since we are dealing with a Petrial of a convex polyhedron. For instance, there is no non-trivial vertex which is transient under r2 alone. Addi- tionally, every vertex which is stabilized by r0 is also stabilized by r2 so there is no vertex which is stabilized by r0 alone. In fact if any two of s0, s2, s0s2 fix any point, so does the third.

0 P When v is transient under r0 alone the resulting ﬁgure is the original regular polyhedron

{4, 3}3, with three skew quadrilaterals meeting at each vertex, (4s.4s.4s). This ﬁgure has six {0} {0,1} edges of type F1 and three skew quadrilaterals of type F2 . The vertex ﬁgure is a regular triangle.

1 P When v is transformed by only r1 the resulting polyhedron shares its vertex and edge set with a regular octahedron, but the faces are defined differently. This figure has twelve edges of {1} {0,1} type F1 . There are three faces of type F2 which are mutually orthogonal, intersecting, {1,2} convex squares; and there are four faces of type F2 which are regular triangles. The square faces are given by the equatorial squares of the octahedron and the triangle faces by a set of alternating faces of the octahedron. At every vertex two squares and two triangles converge giving a vertex figure of a crossed quadrilateral and vertex symbol (4.3.4.3). This object is a uniform polyhedron with planar faces known as the tetrahemihexahedron ([43], [9]).

P 01 This polyhedron shares its edge graph with a polyhedron which is isomorphic to the truncated {0} {1} tetrahedron. It has twelve edges of type F1 and twelve edges of type F1 . This gives three {0,1} skew octagons generated by F2 (truncated skew quadrilaterals) and four regular triangles {1,2} of type F2 . At each vertex there are two octagons and one triangle, (3.t4s.t4s), with a resulting vertex ﬁgure of an isosceles triangle. Since the skew octagons are truncations of skew squares they can never be made regular, so there is no initial vertex which will make this Wythoﬃan a uniform polyhedron.

36 P 02 The generic case of this Wythoffian is a polyhedron which shares its vertex set with a polyhe- {0} dron which is isomorphic to the cuboctahedron. It has twelve edges of type F1 and twelve {2} {0,1} edges of type F1 . Together they make three skew quadrilaterals of type F2 , four regular {1,2} triangles of type F2 , and six planar crossed quadrilaterals (which appear as bow ties) of {0,2} type F2 . At each vertex, in cyclic order, there is a crossed quadrilateral, a triangle, a crossed quadrilateral, and a skew quadrilateral with vertex configuration (4.3.4.4s). The vertex figure is a planar quadrilateral. Crossed quadrilaterals are not regular so this polyhedron is not uniform. There is one interesting subcase of P 02 when the only distinguished generator which stabilizes

the initial vertex is r1 but the initial vertex is also stabilized under s0. In this case we get {0} a polygonal complex rather than a polyhedron. The figure has six edges of type F1 which {2} are the same as the edges of type F1 . This gives three skew quadrilateral faces of type {0,1} {1,2} {0,1} {1,2} F2 and four regular triangles of type F2 . The faces F2 and F2 are the same as 02 {0,2} in the generic P . The key difference is that now there are no faces of type F2 . The resulting complex looks as if we took the Wythoffian P 02 and collapsed all of its faces of type {0,2} F2 into single edges. As a result, two skew quadrilaterals and two triangles are incident to each edge, with three quadrilaterals and three triangles meeting at each vertex. The resulting vertex figure is a regular triangle with double edges. As a result of this subcase, we see that P 02 is not a polyhedron for all choices of initial vertex.

P 012 When v is transient under all of the generating symmetries the resulting Wythoffian shares its vertex set with a polyhedron which is isomorphic to a truncated octahedron. In this {0} generic case the Wythoffian is a polyhedron. There are twelve edges of type F1 , twelve {1} {2} edges of type F1 , and twelve edges of type F1 . The figure has four skew, octagonal faces {0,1} {1,2} of type F2 (truncated skew quadrilaterals); four convex, hexagonal faces of type F2 {0,2} (truncated triangles); and six crossed quadrilaterals of type F2 . At each vertex there is one hexagon, one octagon, and one quadrilateral, (4.t4s.6). The vertex figure is a triangle. The skew octagon is a truncated skew quadrilateral and is thus never regular. As such, there is no initial vertex which makes this Wythoffian a uniform polyhedron. Now consider an initial vertex which is not stabilized by any of the distinguished generators {0} of {4, 3}3, but which is stabilized by s0. This figure has six edges of type F1 which are the {2} {1} same as the edges of type F1 and it has twelve edges of type F1 . This yields three skew {0,1} {1,2} octagons of type F2 and four convex hexagons of type F2 . These base faces are the 012 {0,2} same as for the generic P , the key difference is that the base face F2 has been collapsed into a single edge. The figure changes accordingly and is a semi-complex, see Section 1.8. Note that this is the only situation in which P 012 is not a polyhedron. The vertex figure here consists of two double edges with one common vertex.

As in P 02 and P 012 above, throughout the rest of this thesis we will use the term crossed quadrilateral to describe a planar quadrilateral that appears as the outline of a bow tie, see Figure

37 3.1.

Figure 3.1: A crossed quadrilateral.

3.2 Regular polyhedra with octahedral symmetry

There are four regular polyhedra with octahedral symmetry. Two of these polyhedra are Platonic solids: the octahedron, {3, 4}, and its dual polyhedron the cube, {4, 3}. The other two polyhedra in this symmetry family are the Petrial of the octahedron, {6, 4}3, and the Petrial of the cube,

{6, 3}4. The tables for the Wythoﬃans derived from these polyhedra are, respectively, Tables ??, 7.4, 7.5, and 7.6 in the Appendix. All of the initial vertices used in this section are chosen from within the standard fundamental region of the octahedral group.

3.2.1 {3, 4}

The octahedron’s automorphism group is

Γ({3, 4}) := hσ0, σ1, σ2i where 3 4 2 (σ0σ1) = (σ1σ2) = (σ0σ2) = ε.

The Euclidean realization of the octahedron has symmetry group

G({3, 4}) = hs0, s1, s2i with 3 4 2 (s0s1) = (s1s2) = (s0s2) = 1.

For this symmetry group there are no extra restrictions on the choice of initial vertex.

P 0 The first Wythoffian is the original regular octahedron {3, 4}. It has eight regular triangular faces, four of which meet at each vertex, (3.3.3.3). The vertex figure is a square.

1 {1} P This Wythoffian is a uniform cuboctahedron. The polyhedron has one edge type: F1 . It {0,1} {1,2} has eight regular triangular faces of type F2 and six convex, square faces of type F2 . The vertex configuration at each vertex is (3.4.3.4) yielding a vertex figure which is a convex rectangle.

38 P 2 The Wythoﬃan here is the dual to the octahedron, the regular cube. It has six convex square {1,2} faces of type F2 with three meeting at each vertex, (4.4.4). The vertex ﬁgure is a regular triangle.

01 P When the initial vertex is invariant under s2 alone then the resulting polyhedron is isomorphic {0} to the truncated octahedron. It has twelve edges of type F1 and twenty-four edges of type {1} {0,1} F1 . Together they make eight convex hexagonal faces of type F2 (truncated triangles) {1,2} and six convex square faces of type F2 . The vertex configuration is (4.6.6) with an isosceles triangle as the vertex figure. For a certain choice of initial vertex the hexagons are regular and the Wythoffian is the uniform polyhedron the truncated octahedron.

P 02 This polyhedron is isomorphic to the rhombicuboctahedron. It has twenty-four edges of type {0} {2} F1 and twenty-four edges of type F1 . All together they yield eight regular triangular faces {0,1} {1,2} {0,2} of type F2 , six convex squares of type F2 , and twelve convex rectangles of type F2 . At each vertex there is a triangle, a square, a rectangle, and a second square, in that order. Thus the vertex configuration is (3.4.4.4) and the vertex figure is a convex trapezoid. The initial vertex can be chosen so that the rectangular faces are squares and the Wythoffian is the uniform polyhedron the rhombicuboctahedron.

P 12 Here, Wythoff’s construction yields a polyhedron isomorphic to the truncated cube. It has {1} {2} twenty-four edges of type F1 and twelve edges of type F1 . Together these make eight {0,1} {1,2} regular triangles of type F2 and six convex octagonal faces of type F2 (truncated squares). The vertex configuration is (3.8.8) with an isosceles triangle as the vertex figure. When the initial vertex is chosen so that the base edges have the same length, the Wythoffian is the truncated cube, a uniform polyhedron.

P 012 When v is transient under all of the generating symmetries the resulting polyhedron is iso- {0} morphic to the truncated cuboctahedron. It has twenty-four edges of type F1 , twenty-four {1} {2} edges of type F1 , and twenty-four edges of type F1 . There are eight convex, hexagonal {0,1} {1,2} faces of type F2 (truncated triangles); six convex, octagonal faces of type F2 (trun- {0,2} cated squares); and twelve convex, rectangular faces of type F2 . The vertex configuration is (4.6.8) with a triangular vertex figure. As with the three prior Wythoffians, the initial vertex can be chosen so that the hexagons and octagons are regular and so that the rectangles are squares. In this case the object is the uniform polyhedron the truncated cuboctahedron.

3.2.2 {4, 3}

The next regular polyhedron we will analyze is the cube, {4, 3}. The cube is the dual of the octahedron. Thus when Γ({3, 4}) = hσ0, σ1, σ2i we see that

Γ({4, 3}) = hσ2, σ1, σ0i.

39 Let the Euclidean realization of the automorphism group of the cube be the symmetry group

G({4, 3}) = hs2, s1, s0i =: hr0, r1, r2i.

As with the octahedron there are no extra restrictions on initial vertex choice. The duality between the octahedron and the cube leads to many similarities between their Wythoffians as can be seen in this section. In fact, interchanging 0 and 2 in the superscripts from the Wythoffians of the octahedron result in the Wythoffians of the cube, and vice versa.

P 0 For our first Wythoffian the resulting polyhedron is the original regular cube, {4, 3}. It has six square faces, three of which meet at each vertex, (4.4.4). Thus the vertex figure is a regular triangle.

1 {1} P This polyhedron is the uniform cuboctahedron. It has twenty-four edges of type F1 which {0,1} {1,2} together make six convex square faces of type F2 and eight regular triangles of type F2 . The vertex conﬁguration is (3.4.3.4) giving a vertex ﬁgure of a convex rectangle.

P 2 Here, Wythoﬀ’s construction yields the dual polyhedron to the cube, the regular octahedron. As we saw in the last section, this polyhedron has eight triangle faces such that four meet at each vertex, (3.3.3.3). The vertex ﬁgure is then a square.

01 {0} P This polyhedron is isomorphic to the truncated cube. There are twelve edges of type F1 {1} {0,1} and twenty-four edges of type F1 . The faces are convex octagons of type F2 (truncated {1,2} squares) and regular triangles of type F2 . In total there are six octagons and eight triangles with two octagons and one triangle meeting at each vertex, (3.8.8). The vertex ﬁgure in this case is a triangle. An initial vertex can be chosen such that the octagons are regular and the Wythoﬃan is the uniform polyhedron the truncated cube.

P 02 This polyhedron is isomorphic to the rhombicuboctahedron. There are twenty-four edges of {0} {2} type F1 and twenty-four edges of type F1 . Then there are six convex squares of type {0,1} {1,2} {0,2} F2 , eight regular triangles of type F2 , and twelve convex rectangles of type F2 . In cyclic order, there is a square, a rectangle, a square, and a triangle at each vertex, (3.4.4.4), giving a vertex ﬁgure which is a convex trapezoid. The initial vertex can be picked so that the rectangles are squares in which case the polyhedron is the uniform rhombicuboctahedron. In this case the vertex ﬁgure is still a trapezoid.

P 12 This Wythoﬃan is a polyhedron which is isomorphic to the truncated octahedron. There are {1} {2} twenty-four edges of type F1 and twelve edges of type F1 . Together they make six convex {0,1} {1,2} squares of type F2 and eight convex hexagons of type F2 (truncated triangles). Two hexagons and one rectangle meet at each vertex, (4.6.6), yielding a triangular vertex ﬁgure. Given a certain initial vertex the polyhedron is uniform with regular hexagons, squares, and regular triangles for faces and is the truncated octahedron.

40 P 012 The final Wythoffian of the cube is a polyhedron isomorphic to the truncated cuboctahe- {0} {1} dron. There are twenty-four edges of type F1 , twenty-four edges of type F1 , and twenty- {2} {0,1} four edges of type F1 . Together they make six convex octagons of type F2 (truncated {1,2} squares), eight convex hexagons of type F2 (truncated triangles), and twelve convex rect- {0,2} angles of type F2 . An octagon, a hexagon, and a rectangle meet at each vertex, (4.6.8), with a triangular vertex figure. As with the previous three Wythoffians, given the right initial vertex this Wythoffian is a uniform polyhedron, the truncated cuboctahedron.

3.2.3 {6, 4}3

The Petrial of the octahedron is the regular polyhedron {6, 4}3. As above, let Γ({3, 4}) = hσ0, σ1, σ2i then

Γ({6, 4}3) = hσ0σ2, σ1, σ2i.

This group has Euclidean representation

G({6, 4}3) = hs0s2, s1, s2i =: hr0, r1, r2i.

Given this group, we are limited a bit in our choice of initial vertex, v. Any point invariant under r0 is also invariant under r2. Thus there is no v that is invariant under only r0 or under only r0 and r1.

0 P To realize {6, 4}3 we need to choose as initial vertex the point which is transient under r0 alone. This regular polyhedron has four regular, skew hexagonal faces which all meet at each

vertex, (6s.6s.6s.6s). The vertex ﬁgure is a convex square.

P 1 This Wythoffian shares its edge graph with the cuboctahedron. It has twenty-four edges of {1} {0,1} type F1 . Together they form four convex, regular hexagons of type F2 (the equatorial {1,2} hexagons of the cuboctahedron) and six convex squares of type F2 . The vertex configuration is (4.6.4.6). The hexagons all intersect which leads to a vertex figure which is a crossed quadrilateral. This is a uniform polyhedron with planar faces, in the notation of [9] it is 4 3 4 | 3.

P 01 This Wythoffian shares an edge graph with a polyhedron which is isomorphic to a truncated {0} {1} octahedron. There are twelve edges of type F1 and twelve edges of type F1 . They lead {0,1} to four skew dodecagons of type F2 (truncations of skew hexagon faces of {6, 4}3) and {1,2} six convex squares of type F2 . The vertex configuration is (4.t6s.t6s) with an isosceles triangle as the vertex figure. The skew dodecagons are never regular so this polyhedron is not uniform.

P 02 This Wythoﬃan shares a vertex set with a polyhedron which is isomorphic to a rhombicuboctahedron. In this general case, the Wythoﬃan is a polyhedron. There are twenty-four edges {0} {2} of type F1 and twenty-four edges of F1 . For 2-faces, there are four skew hexagons of type

41 {0,1} {1,2} {0,2} F2 , six convex squares of type F2 , and twelve crossed quadrilaterals of type F2 . At each vertex a crossed quadrilateral, a square, a crossed quadrilateral, and a skew hexagon

occur in cyclic order yielding a quadrilateral vertex ﬁgure with vertex symbol (4.4.4.6s). The crossed quadrilaterals are not regular so this cannot be a uniform polyhedron.

If we look at the case when the initial vertex is stabilized under s0 (the ﬁrst distinguished

generator of the octahedron) and r1 we generate a polygonal complex where each edge is incident to four faces. So clearly, P 02 is not always a polyhedron. There are twelve edges of {0} {2} type F1 which are the same as the edges of type F1 . The complex has four skew hexagons {0,1} {1,2} {0,2} of type F2 and six convex squares of type F2 just as before. The base face F2 from {2} before has now been collapsed into a single edge, F1 , and the rest of the figure has been altered accordingly. Two hexagons and two squares are incident to each edge making this figure a complex. The vertex figure is a triangle with double edges.

P 012 If the initial vertex is transient under all of the distinguished generators then the resulting figure will share a vertex set with a polyhedron which is isomorphic to the truncated cuboctahedron. In this generic case the figure is a polyhedron. There are twenty-four edges of type {0} {1} {2} F1 , twenty-four edges of type F1 , and twenty-four edges of type F1 . The figure has four {0,1} skew dodecagon faces of type F2 (truncated skew hexagons) , six convex octagons of type {1,2} {0,2} F2 (truncated squares), and twelve crossed quadrilaterals of type F2 . The vertex configuration is (4.8.t6s) with a triangular vertex figure. As before, the truncated skew hexagons are not regular so the figure is not a uniform polyhedron.

When we look at an initial vertex which is invariant under s0 but not under any of the 012 distinguished generators of G({6, 4}3) then the figure is similar to the generic P , but in this case (and only in this case) the Wythoffian is a semi-complex, see Section 1.8. There are {0} {2} twelve edges of type F1 which are the same as the edges of type F1 , and twenty-four edges {1} {0,1} of type F1 . This yields four skew dodecagons of type F2 and six convex octagons of type {1,2} {0} {2} F2 just as before. Each edge of type F1 = F1 is incident to four faces while the edges {1} {0,2} of type F1 are incident to only two faces as a result of the base face F2 collapsing into {2} the single edge F1 . The vertex figure consists of three vertices with two edges connecting the middle vertex to each of the other two.

3.2.4 {6, 3}4

Now we will examine the Petrial of the cube, {6, 3}4. The automorphism group of the cube is

Γ({4, 3}) = hσ2, σ1, σ0i based on the distinguished generators, σ0, σ1, and σ2, of the octahedron, thus the automorphism group of {6, 3}4 is

Γ({6, 3}4) = hσ2σ0, σ1, σ0i.

42 Then the Euclidean realization of {6, 3}4 has symmetry group

G({6, 3}4) = hs2s0, s1, s0i =: hr0, r1, r2i.

The duality between the octahedron and the cube can again be seen here. The generators s2s0, s1, and s0 of G({6, 3}4) are obtained from the generators of G({6, 4}3) by interchanging s0 and s2.

The Wythoﬃans of {6, 3}4 also share many similarities with the Wythoﬃans of {6, 4}3. As with the previous Petrial cases, every vertex which is stabilized by r0 is also stabilized by r2. Thus there is no point which is stabilized by r0 alone, nor is there one which is stabilized by r0 and r1 alone.

0 P The ﬁrst Wythoﬃan is the regular polyhedron {6, 3}4 itself. It shares its edge graph with the {0} cube and thus has eight vertices and twelve edges of type F1 . The four faces are the Petrie polygons of the cube which are regular, skew hexagons. Three faces meet at each vertex,

(6s.6s.6s), with a regular triangle as the vertex ﬁgure.

P 1 This polyhedral Wythoffian has the same edge graph as the cuboctahedron. It has twenty-four {1} edges of type of type F1 . Together they form four intersecting, regular, convex hexagons of {0,1} type F2 (the equatorial hexagons of the cuboctahedron) and eight regular triangles of type {1,2} F2 . The vertex configuration is (3.6.3.6) with a vertex figure of a crossed quadrilateral. 3 This is a uniform polyhedron with planar faces, in the notation of [9] it is 2 3 | 3.

01 P When the initial vertex is stabilized by r2 alone then the resulting polyhedron shares its edge graph with a polyhedron which is isomorphic to the truncated cube. There are twelve edges {0} {1} of type F1 and twenty-four edges of type F1 . Then there are four skew dodecagons of {0,1} type F2 (truncations of the skew hexagonal faces of {6, 3}4) and eight regular triangles {1,2} of type F2 . The vertex conﬁguration is (3.t6s.t6s) and the polyhedron has an isosceles triangle as a vertex ﬁgure. Since the skew dodecagons are truncated skew hexagons, they are not regular and the polyhedron is not uniform.

P 02 This Wythoffian shares its vertex set with a polyhedron which is isomorphic to a rhombicuboctahedron. In this general case, P 02 is a polyhedron. There are twenty-four edges of {0} {2} type F1 and twenty-four edges of type F1 . They yield four regular hexagons (convex or {0,1} skew depending on the exact choice of initial vertex) of type F2 , eight regular triangles {1,2} {0,2} of type F2 , and twelve crossed quadrilaterals of type F2 . The vertex configuration is (3.4.6.4) and the vertex figure is a convex trapezoid. The crossed quadrilaterals are not regular so this polyhedron is not uniform.

If we choose an initial vertex which is invariant under s2 in addition to r1 then we get a polygonal complex where four faces are incident to each edge. This is the only subcase in 02 {0} which P is not a polyhedron. There are twelve edges of type F1 which are the same as {2} {0,1} the faces of type F1 . The complex then has four skew hexagons of type F2 and eight {1,2} {0,2} regular triangles of type F2 just as before. The face that should be of type F2 is now

43 {2} just the edge F1 . Thus the figure has two hexagons and two triangles sharing each edge. The vertex configuration is (4.6.4.6.4.6.4.6) with a convex, double edged square vertex figure.

P 012 Suppose the initial vertex is transient under all of the distinguished generators, then, in this case, the resulting polyhedron shares its vertex set with a polyhedron which is isomorphic to {0} the truncated cuboctahedron. There are twenty-four edges of type F1 , twenty-four edges of {1} {2} type F1 , and twenty-four edges of type F1 . They come together to form four dodecagons {0,1} (which may be skew or convex depending on the precise choice of initial vertex) of type F2 {1,2} (truncated hexagons), eight convex hexagons generated by F2 (truncated triangles), and {0,2} twelve crossed quadrilaterals generated by F2 . The vertex conﬁguration is (4.6.12) and the vertex ﬁgure is a triangle. Again, the crossed quadrilaterals prevent the polyhedron from being uniform.

Now if we instead let the initial vertex be stabilized by s2 but not by any of the distinguished generators the resulting figure is a semi-complex. This is the only subcase where P 012 is not a {0} {2} polyhedron. There are twelve edges of type F1 which are the same as the edges of type F1 {1} and there are twenty-four edges of type F1 . Together they form four skew, dodecagonal {0,1} {1,2} faces of type F2 and eight convex hexagons of type F2 just as in the previous case. {0,2} The difference between these figures is that where this one should have a face of type F2 {2} there is only the edge F1 . Thus two dodecagons and two hexagons are incident to each edge {0} {2} of type F1 = F1 . The resulting vertex configuration is (6.6.t6s.t6s) and the vertex figure consists of two digons which share a common vertex.

3.3 Regular polyhedra with icosahedral symmetry

The remaining twelve finite regular polyhedra we will look at all have an icosahedral symmetry group. Images of their faces, vertex figures, and full Wythoffians are in Tables 7.7-7.18.

3.3.1 {3, 5}

The icosahedron has automorphism group

Γ({3, 5}) = hσ0, σ1, σ2i such that 3 5 2 (σ0σ1) = (σ1σ2) = (σ0σ2) = ε, and its Euclidean realization has corresponding symmetry group

G({3, 5}) = hs0, s1, s2i with 3 5 2 (s0s1) = (s1s2) = (s0s2) = 1.

44 The only restriction on the choice of initial vertex is that it comes from the standard icosahedral fundamental region.

0 P When the initial vertex is only transient under s0 then the resulting polyhedron is the original regular icosahedron. It has twenty regular triangles for faces, ﬁve of which meet at each vertex, (3.3.3.3.3). The resulting vertex ﬁgure is a regular pentagon.

P 1 This Wythoﬃan is an icosidodecahedron, a uniform polyhedron. There are sixty edges of {1} {0,1} type F1 . Together they make twenty regular triangles of type F2 and twelve regular, {1,2} convex pentagons of type F2 . In cyclic order there is a triangle, a pentagon, a triangle, and a pentagon surrounding each vertex with symbol (3.5.3.5). The vertex ﬁgure is a convex rectangle.

P 2 Here, Wythoﬀ’s construction yields the dual polyhedron to the icosahedron, the regular {2} dodecahedron. There are thirty edges of type F1 which together make twelve regular, {1,2} convex pentagons of type F2 . Three pentagons meet at each vertex, (5.5.5), which lead to a regular triangle for the vertex ﬁgure.

01 {0} P This polyhedron is isomorphic to a truncated icosahedron. There are thirty edges of type F1 {1} {0,1} and sixty edges of type F1 . This yields twenty convex hexagons of type F2 (truncated {1,2} triangles) and twelve regular, convex pentagons of type F2 . There are two hexagons and one pentagon at each vertex, (5.6.6), with a vertex figure of an isosceles triangle. For a specifically chosen initial vertex this Wythoffian is the uniform truncated icosahedron with regular hexagons and pentagons for faces.

P 02 This polyhedron is isomorphic to a rhombicosidodecahedron. There are sixty edges of type {0} {2} {0,1} F1 and sixty edges of type F1 . Together they make twenty regular triangles of type F2 ; {1,2} {0,2} twelve regular, convex pentagons of type F2 ; and thirty convex rectangles of type F2 . In cyclic order there is a triangle, a rectangle, a pentagon, and a rectangle at each vertex with vertex symbol (3.4.5.4). Thus the vertex ﬁgure is a convex trapezoid. Choosing the initial vertex such that the base edges have the same length results in the Wythoﬃan being the uniform polyhedron the rhombicosidodecahedron.

P 12 This polyhedral Wythoffian is isomorphic to a truncated dodecahedron. There are sixty {1} {2} edges of type F1 and thirty edges of type F1 . This gives twenty regular triangular faces {0,1} {1,2} of type F2 and twelve convex decagons of type F2 (truncated pentagons). The vertex configuration is (3.10.10) with an isosceles triangle for a vertex figure. For a specific initial vertex choice, the Wythoffian is the uniform polyhedron the truncated dodecahedron.

P 012 In the ﬁnal case suppose that the initial vertex is transient under all of the distinguished generators. This Wythoﬃan is a polyhedron which is isomorphic to a truncated icosidodec- {0} {1} ahedron. There are sixty edges of type F1 , sixty edges of type F1 , and sixty edges of {2} {0,1} type F1 . Together they make twenty convex hexagons of type F2 (truncated triangles),

45 {1,2} twelve convex decagons of type F2 (truncated pentagons), and thirty convex rectangles of {0,2} type F2 . The vertex configuration is (4.6.10) with a triangular vertex figure. As with the previous three Wythoffians the initial vertex can be chosen so that the resulting figure is a uniform polyhedron, the truncated icosidodecahedron.

3.3.2 {5, 3}

The dodecahedron is the dual of the icosahedron. Thus the automorphism group of the dodecahedron is

Γ({5, 3}) = hσ2, σ1, σ0i.

The Euclidean realization then has symmetry group

G({5, 3}) = hs2, s1, s0i =: hr0, r1, r2i.

There are no extra restrictions of the choice of initial vertex beyond choosing it from the fundamental region of the icosahedral group. As we saw with the cube and octahedron, the duality of the icosahedron and dodecahedron make it such that swapping 0 and 2 in the superscripts of a Wythoﬃan of the icosahedron leads to a Wythoﬃan of the dodecahedron, and vice versa.

0 P The first Wythoffian where the initial vertex is transient under r0 alone is the original regular {0} dodecahedron. There are thirty edges of type F1 and twelve convex, regular pentagons for faces. Three pentagons surround each vertex, (5.5.5), and give a vertex figure of a regular triangle.

1 {1} P This polyhedron is a uniform icosidodecahedron. There are sixty edges of type F1 . This {0,1} gives twelve convex, regular pentagons of type F2 and twenty regular triangles of type {1,2} F2 . The vertex configuration is (3.5.3.5) with a convex rectangle for a vertex figure. P 2 The Wythoffian is the dual polyhedron to the dodecahedron, the regular icosahedron. There {2} {1,2} are thirty edges of type F1 and twenty regular triangles of type F2 . The vertex figure is a convex, regular pentagon with vertex symbol (5.5.5.5.5).

P 01 This polyhedron is isomorphic to the truncated dodecahedron. It has thirty edges of type {0} {1} {0,1} F1 and sixty edges of type F1 . Together they make twelve convex decagons of type F2 {1,2} (truncated pentagons) and twenty regular triangles of type F2 . The vertex configuration is (3.10.10) with an isosceles triangle as a vertex figure. If the initial vertex is chosen so that the base edges have the same length then the Wythoffian is a uniform polyhedron, the truncated dodecahedron.

02 {0} P This polyhedron is isomorphic to the rhombicosidodecahedron. It has sixty edges of type F1 {2} {0,1} and sixty edges of type F1 . They form twelve convex, regular pentagons of type F2 ; {1,2} {0,2} twenty regular triangles of type F2 ; and thirty convex rectangles of type F2 . The vertex conﬁguration is (3.4.5.4) and the vertex ﬁgure is a convex quadrilateral. When the initial

46 vertex is carefully chosen the Wythoﬃan is a uniform polyhedron, the rhombicosidodecahedron.

P 12 This Wythoffian is a polyhedron which is isomorphic to the truncated icosahedron. There {1} {2} are sixty edges of type F1 and thirty edges of type F1 . They make twelve convex, regular {0,1} {1,2} pentagons of type F2 and twenty convex hexagons of type F2 (truncated triangles). There are two hexagons and one pentagon at each vertex, (5.6.6), yielding an isosceles triangle for a vertex figure. For a specific initial vertex the hexagons and pentagons are both regular and the Wythoffian is a uniform polyhedron, the truncated icosahedron.

P 012 In the final case the polyhedral Wythoffian is isomorphic to a truncated icosidodecahedron. {0} {1} {2} There are sixty edges of type F1 , sixty edges of type F1 , and sixty edges of type F1 . {0,1} Together they make twelve convex decagons of type F2 (truncated pentagons), twenty {1,2} convex hexagons of type F2 (truncated triangles), and thirty convex rectangles of type {0,2} F2 . At each vertex there is a rectangle, a hexagon, and a decagon. The vertex figure is a triangle. As with the previous three Wythoffians, for a specific initial vertex the edge lengths can be made equal so the faces become regular decagons, hexagons, and squares. In this case the Wythoffian is a uniform polyhedron, the truncated icosidodecahedron.

3.3.3 {10, 5}3

The Petrial of the icosahedron is {10, 5}3 with automorphism group

Γ({10, 5}3) = hσ0σ2, σ1, σ2i.

Then the Euclidean realization of {10, 5}3 has symmetry group

G({10, 5}3) = hs0s2, s1, s2i =: hr0, r1, r2i.

The initial vertex comes from the fundamental region of the icosahedral group. When choosing the initial vertex we need to take into account the fact that any point stabilized by r0 is also stabilized by r2. As a result, there is no point which is stabilized by r0 alone, nor is there any point stabilized by exactly r0 and r1.

0 {0} P The first polyhedron is the regular {10, 5}3 itself. It has thirty edges of type F1 which {0,1} form six regular, skew decagons of type F2 . There are five skew decagons at each vertex yielding a convex, regular pentagon as the vertex figure. The vertex configuration symbol is

(10s.10s.10s.10s.10s).

P 1 This polyhedron shares an edge graph with the icosidodecahedron. It has sixty edges of {1} {0,1} type F1 . They form six regular, convex decagons of type F2 and twelve regular, convex {1,2} pentagons of type F2 . The vertex conﬁguration is (5.10.5.10) and the vertex ﬁgure is a

47 planar, crossed quadrilateral. This is a uniform polyhedron with planar faces, in the notation 5 of [9] it is 4 5 | 5.

P 01 This polyhedral Wythoﬃan has an edge graph which is isomorphic to the edge graph of the {0} {1} truncated icosahedron. It has thirty edges of type F1 and sixty edges of type F1 . The {0,1} {1,2} faces of type F2 are skew icosagons (truncated skew decagons) and the faces of type F2 are convex, regular pentagons. At each vertex there is one pentagon and two skew icosagons,

(5.t10s.t10s), giving a vertex ﬁgure of an isosceles triangle. The skew icosagons are truncated skew decagons and so they are not regular and thus P 01 is not a uniform polyhedron.

02 P When the initial vertex is invariant under r1 alone the resulting figure shares its vertex set with a polyhedron which is isomorphic to the rhombicosidodecahedron. In this first generic {0} {2} case the figure is a polyhedron. It has sixty edges of type F1 and sixty edges of type F1 . {0,1} They make six regular, skew decagons of type F2 ; twelve regular, convex pentagons of {1,2} {0,2} type F2 ; and thirty crossed quadrilaterals of type F2 . The vertex configuration is (4.10s.4.5) and the vertex figure is a convex quadrilateral. The crossed quadrilaterals can never be regular polygons so P 02 is not uniform.

Rather than stabilizing under r1 alone, look at an initial vertex which is stabilized by only

s0 and r1. The resulting ﬁgure is a polygonal complex, note that this is the only case where 02 {0} P is not a polyhedron. It has thirty edges of type F1 which are identical to the edges of {2} {0,1} type F1 . There are six regular, skew decagons of type F2 and twelve regular, convex {1,2} {0,2} pentagons of type F2 just as before. In this case, though, there is no face of type F2 , {2} instead there are just edges of type F1 . As a result, each edge is incident to two decagons and two pentagons. There are three pentagons and three decagons meeting at each vertex and the vertex ﬁgure is a regular triangle with double edges. The edge graph is the same as that of a dodecahedron.

P 012 Finally, look at an initial vertex which is not stabilized by any of the distinguished generators. In this general case the resulting polyhedron shares a vertex set with a polyhedron which is {0} isomorphic to the truncated icosidodecahedron. There are sixty edges of type F1 , sixty edges {1} {2} of type F1 , and sixty edges of type F1 . They come together to form six skew icosagons {0,1} {1,2} of type F2 (truncated skew decagons), twelve convex decagons of type F2 (truncated {0,2} pentagons), and thirty crossed quadrilaterals of type F2 . The vertex configuration is 02 (4.t5s.t10s) with a triangular vertex figure. As with the generic P this figure has crossed quadrilaterals which prevent it from being uniform.

If we instead let the initial vertex be invariant under s0 and under none of the distinguished generators the ﬁgure is neither a polyhedron nor a polygonal complex, it is a semi-complex (see Section 1.8). This is the only time when P 012 is not a polyhedron. It has thirty edges of {0} {2} {1} type F1 which are the same as the edges of type F1 , and it has sixty edges of type F1 . {0,1} {1,2} They make six skew icosagons of type F2 and twelve convex decagons of type F2 , the

48 {2} {0,2} same as before. Here there is only an edge, F1 , rather than a base 2-face of type F2 . {0} {2} Thus the edges of type F1 = F1 are incident to two icosagons and two decagons while the {1} edges of type F1 are incident to one icosagon and one decagon. The vertex ﬁgure consists of two digons which share a common vertex.

3.3.4 {10, 3}5

The Petrial of the dodecahedron is {10, 3}5. Thus the automorphism group of {10, 3}5 is

Γ({10, 3}5) = hσ2σ0, σ1, σ0i.

Then the symmety of the Euclidean realization is

G({10, 3}5) = hs2s0, s1, s0i =: hr0, r1, r2i.

The initial vertex comes from the fundamental region of the icosahedral group. This is another Petrial of a Platonic solid and is yet another case where we have limitations on our initial vertex choices. All points which are invariant under r0 are also invariant under r2, thus there is no vertex invariant under r0 alone nor are there any which are invariant under only r0 and r1.

0 P For the ﬁrst choice of initial vertex the resulting polyhedron is the regular {10, 3}5 itself. It

has six regular, skew decagons with three arranged at each vertex, (10s.10s.10s.10s.10s.10s). The vertex ﬁgure is a regular triangle.

P 1 This polyhedron has an edge graph which is isomorphic to the edge graph of the icosidodeca- {1} hedron. There are sixty edges of type F1 . These lead to six regular, convex decagons of type {0,1} {1,2} F2 and twenty regular triangles of type F2 . The decagons are equatorial relative to an icosidodecahedron. The resulting vertex configuration is (3.10.3.10) and the vertex figure is a planar, crossed quadrilateral. This is a planar faced uniform polyhedron, in the notation of 3 [9] it is 2 3 | 5. P 01 This polyhedron has an edge graph which is isomorphic to the edge graph of the truncated {0} {1} dodecahedron. It has thirty edges of type F1 and sixty edges of type F1 . There are then {0,1} six skew icosagons of type F2 (truncated skew decagons) and twenty regular triangles {1,2} of type F2 . The resulting vertex configuration is (3.t10s.t10s) with an isosceles triangle as the vertex figure. The truncated skew decagons are not regular so this is not a uniform polyhedron.

02 P When the initial vertex is stabilized by r1 alone the resulting figure has the same vertex set as a polyhedron which is isomorphic to the rhombicosidodecahedron. This first general case {0} {2} is a polyhedron. It has thirty edges of type F1 and sixty edges of type F1 . There are {0,1} {1,2} six regular, skew decagons of type F2 ; twenty regular triangles of type F2 ; and thirty {0,2} crossed quadrilaterals of type F2 . The vertex configuration is (3.4.10s.4) with a convex

49 trapezoid as the vertex ﬁgure. The crossed quadrilaterals are not regular polygons so P 02 is not a uniform polyhedron.

If we instead let the inital vertex be stabilized by s2 and r1 then the resulting ﬁgure is a polygonal complex; this is the only case when P 02 is not a polyhedron. There are thirty {0} {2} edges of type F1 which are the same as the edges of type F1 . Together they make six {0,1} {1,2} regular, skew decagons of type F2 and twenty regular triangles of type F2 just as {0,2} before. Now there are no faces of type F2 , this base face from before has been collapsed {2} {0} into the base edge F1 = F1 . Consequently each edge is incident to two triangles and two decagons. This gives a vertex ﬁgure which is a double-edged, regular, convex pentagon. The edge graph is the same as that of the icosahedron.

P 012 Finally, choose an initial vertex which is transient under all of the distinguished generators

of {10, 3}5. The vertex set of the resulting figure will be the same as the vertex set of a polyhedron which is isomorphic to the icosidodecahedron. In the general case, P 012 is a {0} {1} polyhedron. There are sixty edges of type F1 , sixty edges of type F1 , and sixty edges of {2} {0,1} type F1 . Together they form six skew icosagons made by F2 (truncated skew decagons), {1,2} twenty convex hexagons made by F2 (truncated triangles), and thirty crossed quadrilater- {0,2} als made by F2 . The vertex configuration is (3.4.t10s) and the vertex figure is a triangle. The crossed quadrilaterals and the truncated skew decagons are not regular polygons for any initial vertex choice so this polyhedron is never uniform.

Look at another initial vertex which is invariant under s2 but none of the distinguished generators. In this case, and this case only, the ﬁgure is not a polyhedron but rather a semi- {0} {1} complex (see Section 1.8). There are thirty edges of type F1 and sixty edges of type F1 . {2} {0} The edges of type F1 are the same as those of type F1 . There are six skew icosagons of type {0,1} {1,2} F2 and twenty convex hexagons of type F2 just as before. Unlike in the more generic 012 {0,2} {2} {0} {2} P , here, F2 collapses into the edge F1 . As a result each edge of type F1 = F1 {1} is incident to two icosagons and two hexagons while each edge of type F1 is incident to one icosagon and one hexagon. Its vertex ﬁgure consists of two digons with one vertex in common.

5 3.3.5 {5, 2 } 5 nd To ﬁnd the group of the great dodecahedron {5, 2 } we must perform the 2 facetting operation on the group of the icosahedron. Thus

5 Γ 5, = hσ , σ σ σ , σ i. 2 0 1 2 1 2

Then the symmetry group of the Euclidean realization is

5 G 5, = hs , s s s , s i =: hr , r , r i. 2 0 1 2 1 2 0 1 2

50 The initial vertices come from the fundamental region of the icosahedral group. For initial vertices {1} {2} stabilized by s1, the base edges of type F1 and F1 are equivalent under the group. This equivalence aﬀects how the Wythoﬃans corresponding to certain initial vertices behave.

P 0 The first Wythoffian is the regular great dodecahedron itself. The faces are twelve convex, regular pentagons with five intersecting at each vertex, (5.5.5.5.5). The resulting vertex figure 5 is the regular pentagram, { 2 }.

1 {1} P In this polyhedron there are sixty edges of type F1 which come together to form twelve {0,1} {1,2} regular, convex pentagons of type F2 and twelve regular, planar pentagrams of type F2 . 5 5 The final vertex configuration is ( 2 .5. 2 .5) while the vertex figure is a convex rectangle. This 5 is a uniform polyhedron with planar faces, in the notation of [9] it is 2 | 2 5.

2 5 5 P This Wythoffian is the dual of {5, 2 }, the small stellated dodecahedron, { 2 , 5}, which is a {2} regular polyhedron. There are thirty edges of type F1 which yield twelve regular, planar {1,2} pentagrams of type F2 . There are five pentagrams meeting at each vertex giving a vertex 5 5 5 5 5 figure of a regular pentagon with vertex symbol ( 2 . 2 . 2 . 2 . 2 ).

01 {0} {1} P For this polyhedron there are thirty edges of type F1 and sixty edges of type F1 . Then {0,1} there are twelve convex decagons of type F2 (truncated pentagons) and twelve planar, {1,2} 5 regular pentagrams of type F2 . The vertex configuration is ( 2 .10.10) and the vertex figure is an isosceles triangle. For a specific choice of initial vertex the decagons are regular and the 5 figure is the planar faced uniform polyhedron 2 2 | 5 in the notation of [9].

02 {0} {2} P In this polyhedron there are sixty edges of type F1 and sixty edges of type F1 . They come {0,1} together to form twelve regular, convex pentagons of type F2 ; twelve planar, regular pen- {1,2} {0,2} tagrams of type F2 ; and thirty convex rectangles of type F2 . The vertex configuration 5 is ( 2 .4.5.4) and the vertex figure is a convex quadrilateral. This Wytoffian is a uniform polyhedron with planar faces for a specific initial vertex which changes the rectangles to squares 5 and the polyhedron is 2 5 | 2 in the notation of [9].

12 {1} {2} P This Wythoffian has sixty edges of type F1 and thirty edges of type F1 . There are twelve {0,1} convex, regular pentagons of type F2 and twelve planar, truncated pentagrams of type {1,2} 12 F2 . This is the most generic case of P and the figure is a polyhedron here. There is a 5 5 pentagon and two truncated pentagrams at each vertex, (5.t 2 .t 2 ). The resulting vertex figure is an isosceles triangle. The truncated pentagrams are not regular for any choice of initial vertex so this is not a uniform polyhedron.

Now let the initial vertex be invariant under r0 and s1 but not under r1 or r2. In the {0,1} {1,2} resulting ﬁgure, F2 is a convex regular pentagon just as before but F2 is now a double {1} {2} edged convex pentagon since F1 and F1 are equivalent under hr1, r2i. All edges of a given regular pentagon are incident to one double edged pentagon. There are three of these pairings meeting at each vertex with a double edged triangular vertex ﬁgure. Additionally

51 four pentagons (two regular and two double edged) are incident to each edge. The realization appears as a dodecahedron. This is the only time when P 12 is not a polyhedron.

P 012 When the initial vertex is transient under all of the distinguished generators the resulting {0} {1} Wythoffian has sixty edges of type F1 , sixty edges of type F1 , and sixty edges of type {2} {0,1} F1 . There are twelve convex decagons of type F2 (truncated pentagons); twelve planar, {1,2} {0,2} truncated pentagrams of type F2 ; and thirty convex rectangles of type F2 . There is a truncated pentagram, a rectangle, and a decagon at each vertex with vertex symbol 5 (4.t 2 .10). The vertex figure is a triangle. In this case, the Wythoffian is a polyhedron but never a uniform polyhedron since the truncated pentagrams are not regular polygons. Consider a slightly different initial vertex which is transient under all distinguished generators 012 but let it be invariant under s1. This is the only time when P is not a polyhedron. The {0,1} {0,2} decagons and rectangles of type F2 and F2 , respectively, are the same as before. The truncated pentagrams from before are now double edged pentagons. There is one truncated pentagram, two rectangles, and two decagons meeting at each vertex. The vertex figure appears as a crossed quadrilateral with an extra edge incident to two non-adjacent vertices.

5 3.3.6 { 2 , 5} 5 The dual of the great dodecahedron is the small stellated dodecahedron, { 2 , 5}. Thus

5 Γ , 5 = hσ , σ σ σ , σ i. 2 2 1 2 1 0

The Euclidean realization then has symmetry group

5 G , 5 = hs , s s s , s i =: hr , r , r i. 2 2 1 2 1 0 0 1 2

As with the other polyhedra in this family the initial vertices come from the icosahedral fundamental {1} region. When an initial vertex is invariant under s1 and not r1 then the base edges of type F1 {0} and F1 are equivalent. This equivalence leads to problems with two of the Wythoffians. One final note before discussing the Wythoffians of the small stellated dodecahedron is to note that 5 5 the duality between { 2 , 5} and {5, 2 } allows us to interchange 0 and 2 in the superscripts of a Wythoffian of one to get a Wythoffian of the other.

P 0 The first Wythoffian is the small stellated dodecahedron itself. It has twelve regular pentagram faces, thirty edges, and twelve vertices. The vertices are in the same configuration as an icosahedron. The vertex figure is a convex, regular pentagon since five pentagrams meet 5 5 5 5 5 at each vertex with vertex symbol ( 2 . 2 . 2 . 2 . 2 ).

1 P When the initial vertex is invariant under r0 and r2 then the resulting ﬁgure has sixty edges {1} {0,1} of type F1 . There are twelve faces of type F2 which are planar, regular pentagrams and

52 {1,2} there are twelve faces of type F2 which are convex, regular pentagons. In cyclic order there is a pentagram, a pentagon, a pentagram, and a pentagon meeting at each vertex with 5 5 vertex symbol (5. 2 .5. 2 ). The resulting vertex ﬁgure is a convex rectangle. The faces are all 5 regular polygons so this is the planar faced uniform polyhedron 2 | 2 5 in the notation of ??.

2 5 5 P This Wythoffian is the dual polyhedron to { 2 , 5}, the regular great dodecahedron {5, 2 }. It {2} has thirty edges of type F1 which together make twelve convex, regular, pentagonal faces. The vertex figure is a planar, regular pentagram with vertex configuration (5.5.5.5.5).

01 {0} {1} P This figure has sixty edges of type F1 and sixty edges of type F1 . Together they make {0,1} twelve truncated planar pentagrams of type F2 and twelve convex, regular pentagons of {1,2} type F2 . In this generic case the Wythoffian is a polyhedron. There are two truncated 5 5 pentagrams and one pentagon at each vertex, (5.t 2 .t 2 ), with an isosceles triangle as a vertex figure. The truncated pentagrams are not regular so this is not a uniform polyhedron.

Now consider an initial vertex which is stabilized by r2 and s1. The resulting figure is similar to the generic P 01 except this Wythoffian is not a polyhedron. This is the only case when 01 {1,2} P is not a polyhedron. The base face of type F2 is still a regular, convex pentagon. {1} {0} {0,1} Since the edges of type F1 and F1 are equivalent, the faces of type F2 are double edged pentagons. For these faces it is as if the truncated pentagrams from the previous case have collapsed down into pentagons. The two types of base face share all of their edges. The resulting object looks like a dodecahedron where two pairs of pentagons are incident to each edge. The vertex figure is a double edged, regular triangle.

02 {0} {2} P In this polyhedron there are sixty edges of type F1 and sixty edges of type F1 . They {0,1} make twelve planar, regular pentagrams of type F2 ; twelve convex, regular pentagons of {1,2} {0,2} type F2 ; and thirty convex rectangles of type F2 . There is a rectangle, a pentagon, a 5 rectangle, and a pentagram surrounding each vertex giving a vertex symbol (4.5.4. 2 ). The vertex ﬁgure is a convex quadrilateral. When the initial vertex is chosen so that the faces of {0,2} 5 type F2 are squares, the Wythoﬃan is the planar faced uniform polyhedron 2 5 | 2 in the notation of [9].

12 {1} {2} P Here, there are sixty edges of type F1 and sixty edges of type F1 . There are twelve planar, {0,1} {1,2} regular pentagrams of type F2 and twelve convex decagons of type F2 (truncated 5 pentagons). At each vertex there are two decagons and one pentagram, ( 2 .10.10). The vertex figure is an isosceles triangle. The initial vertex can be chosen so that the decagons are regular, in this case the Wythoffian is a uniform polyhedron with planar faces. In the 5 notation of [9] the uniform polyhedron is 2 2 | 5. P 012 Finally, consider an initial vertex which is transient under all distinguished generators. In {0} this generic case the Wythoffian is a polyhedron with sixty edges of type F1 , sixty edges of {1} {2} type F1 , and sixty edges of type F1 . They make twelve truncated planar pentagrams of {0,1} {1,2} type F2 , twelve convex decagons of type F2 (truncated pentagons), and thirty convex

53 {0,2} rectangles of type F2 . There is a truncated pentagram, a decagon, and a rectangle at each vertex. The vertex figure is a triangle. Truncated pentagrams are not regular polygons so this is not a uniform polyhedron. Look at an initial vertex which is not invariant under any of the distinguished generators 012 but is invariant under s1. This is the only time P is not a polyhedron. The base faces of {1,2} {0,2} type F2 and F2 are the same as before, they are a convex decagon and a rectangle, {0,1} respectively. The base face of type F2 is different, the truncated pentagram from before {0} {1} collapses into a double edged pentagon since the edges of type F1 and F1 are equivalent. The final object has one decagon, one rectangle, and one pentagon meeting at each vertex. The vertex figure is a triangle.

5 3.3.7 {6, 2 }

5 5 5 ϕ2 The Petrial of the great dodecahedron, {5, 2 }, is the regular {6, 2 }. Recalling that {5, 2 } = {3, 5} then when Γ({3, 5}) = hσ0, σ1, σ2i as before, we see that

5 Γ 6, = hσ σ , σ σ σ , σ i. 2 0 2 1 2 1 2

Its Euclidean realization then has the symmetry group

5 G 6, = hs s , s s s , s i =: hr , r , r i. 2 0 2 1 2 1 2 0 1 2

There are some restrictions on the choice of initial vertex beyond choosing it from the fundamental region of the icosahedral group. There are no points which are invariant under r0 alone or under only r0 and r1 since any point invariant under r0 is additionally invariant under r2. Some issues will also arise when the initial vertex is stabilized by s1 but not by r1. For this vertex choice, the {1} {2} {1,2} base edge F1 is equivalent to the base edge F1 under hr1, r2i which makes the base face F2 have double edges.

0 5 P In the ﬁrst case the resulting ﬁgure is the regular {6, 2 } itself. Since this is the Petrial of the great dodecahedron it has the same thirty edges and the same twelve vertices. The faces are

now regular, skew hexagons. Five faces meet at each vertex, (6s.6s.6s.6s.6s), with a regular pentagram as the vertex ﬁgure.

1 {1} P In this polyhedron there are sixty edges of type F1 . Together they form ten convex, regular {0,1} {1,2} hexagons of type F2 and twelve planar, regular pentagrams of type F2 . There are 5 5 two pentagrams and two hexagons which alternate around each vertex, ( 2 .6. 2 .6). The vertex ﬁgure is a planar crossed quadrilateral. All of the faces are regular polygons so this Wythoﬃan is a uniform polyhedron with planar faces. In the notation of [9] the uniform polyhedron is 5 5 3 2 | 3.

54 01 {0} {1} P In this polyhedron there are thirty edges of type F1 and sixty edges of type F1 . They {0,1} make ten skew dodecagons of type F2 (truncated hexagons) and twelve planar, regular {1,2} 5 pentagrams of type F2 . The vertex configuration is ( 2 .t6s.t6s) and the vertex figure is an isosceles triangle. The truncated hexagons are not regular polygons so the Wythoffian is not uniform.

02 {0} {2} P This Wythoffian has sixty edges of type F1 and sixty edges of type F1 . Together they make {0,1} {1,2} ten regular, skew hexagons of type F2 ; twelve planar, regular pentagrams of type F2 ; {0,2} 5 and thirty planar crossed quadrilaterals of type F2 . The vertex configuration is (4.6.4. 2 ) and the vertex figure is a convex quadrilateral. In this generic case, P 02 is a polyhedron. It is not a uniform polyhedron, however, since the crossed quadrilaterals are not regular polygons.

Now let the initial vertex be stabilized by s0 and r1 but not by r0. This is the only case when P 02 is not a polyhedron, it is a polygonal complex with four faces incident to each edge. {0} {2} There are thirty edges of type F1 which are identical to the edges of type F1 . Together {0,1} they form ten regular, skew hexagons of type F2 and twelve planar, regular pentagrams {1,2} {0} {2} of type F2 just as before. This time, though, there is only an edge of type F1 = F1 rather than a 2-face. As a result the ﬁgure is a polygonal complex with two pentagrams and two hexagons incident to each edge. The vertex ﬁgure is a double edged, convex pentagon.

P 012 Finally, let the initial vertex be variant under all distinguished generators. In the generic 012 {0} {1} case, P is a polyhedron. There are sixty edges of type F1 , sixty edges of type F1 , and {2} {0,1} sixty edges of type F1 . They make ten skew dodecagons of type F2 (truncated skew {1,2} hexagons), twelve truncated pentagrams of type F2 , and thirty crossed quadrilaterals of {0,2} type F2 . The vertex conﬁguration is (4.10.t6s) with a triangle as a vertex ﬁgure. The crossed quadrilaterals are not regular so this is not a uniform polyhedron.

Consider some initial vertex which is stabilized by s0 alone and none of the distinguished generators. The resulting figure in this case is a semi-complex, see Section 1.8. There are {0} {2} thirty edges of type F1 which are the same as the edges of type F1 and there are sixty {1} {0,1} edges of type F1 . Then just as before there are ten skew dodecagons of type F2 and {1,2} twelve truncated pentagrams of type F2 , but in this case there is only an edge of type {0} {2} {0,2} {0} F1 = F1 rather than a 2-face of type F2 . This difference is why each edge of type F1 {1} is incident to four faces and each edge of type F1 is incident to only two edges. The vertex figure consists of two digons which share a vertex.

Now look at some point which is invariant under s0 and s1 but not invariant under either 012 r0 or r1. This case of P is not a polyhedron. As with the previous case there is no face {0,2} {0,1} of type F2 and there are still ten skew dodecagons of type F2 . The twelve truncated {1,2} {1,2} pentagrams of type F2 collapse down into twelve double edged pentagons of type F2 . {0} {2} {1} {2} Here we have F1 = F1 and F1 is equivalent to F1 under the group meaning that each edge is incident to two faces of each type. The vertex ﬁgure consists of two digons which share a common vertex.

55 In the last case we will consider, let the initial vertex be invariant under s1 and nothing else. This case is more similar to the generic P 012 which is not stabilized under anything, but {0,1} it is not a polyhedron. As before, there are ten skew dodecagons of type F2 and thirty {0,2} crossed quadrilaterals of type F2 . The twelve truncated pentagrams from before collapse {1,2} into twelve double edged pentagons of type F2 . The vertex ﬁgure is still a triangle. The three subcases of P 012 are the only times when P 012 is not a polyhedron.

3.3.8 {6, 5}

5 ϕ2 δ π The Petrial of the small stellated dodecahedron, { 2 , 5}, is {6, 5}. We know that {6, 5} = (({3, 5} ) ) . Thus when Γ({3, 5}) = hσ0, σ1, σ2i as before we get that

Γ({6, 5}) = hσ2σ0, σ1σ2σ1, σ0i.

The Euclidean realization will then have symmetry group

G({6, 5}) = hs2s0, s1s2s1, s0i =: hr0, r1, r2i.

The initial vertex is still chosen from the fundamental region of the icosahedral group. Just as with the other Petrials, all points held invariant by r0 are additionally held invariant by r2. Thus there is no point which is invariant under r0 alone nor is there a point which is invariant under only r0 5 and r2. This polyhedron is derived from {5, 2 } which is formed using the facetting operation on {1} {2} {3, 5}. As a result, when the initial vertex is stabilized by s1 the base edges F1 and F1 are equivalent. They are equivalent within the resulting figure but not within the same face so this choice of initial vertex does not generate any different Wythoffians.

P 0 The first Wythoffian we will look at is the regular {6, 5} itself which has ten regular, skew hexagons for faces. The vertex figure is a regular, convex pentagon since five skew hexagons

meet at each vertex, (6s.6s.6s.6s.6s).

1 {1} P This polyhedron has sixty edges of type F1 . Together they make ten convex, regular {0,1} {1,2} hexagons of type F2 and twelve convex, regular pentagons of type F2 . The vertex conﬁguration is (5.6.5.6). The resulting vertex ﬁgure is a planar crossed quadrilateral. Since all of the faces are regular polygons this is a uniform polyhedron with planar faces whose 5 notation is 4 5 | 3, [9].

01 {0} {1} P In this polyhedron there are thirty edges of type F1 and sixty edges of type F1 . They make {0,1} ten skew dodecagons of type F2 (truncated skew hexagons) and twelve convex, regular {1,2} pentagons of type F2 . The vertex conﬁguration is (5.t6s.t6s) and the vertex ﬁgure is an isosceles triangle. Truncated skew hexagons can never be made regular so the polyhedron is not uniform for any choice of initial vertex.

56 02 {0} {2} P This Wythoffian has sixty edges of type F1 and sixty edges of type F1 . Together they {0,1} make ten regular, skew hexagons of type F2 ; twelve regular, convex pentagons of type {1,2} {0,2} F2 ; and thirty crossed quadrilaterals of type F2 . In this case, the Wythoffian is a polyhedron. The vertex configuration is (4.5.4.6s) and the vertex figure is a convex trapezoid. The crossed quadrilateral faces are not regular so P 02 is not a uniform polyhedron.

Consider a slightly diﬀerent initial vertex which is invariant under s2 and r1 but still transient

under r0 or r2. The resulting ﬁgure is a polygonal complex where each edge is incident to four faces; this is the only case when P 02 is not a polyhedron. It has thirty edges of type {0} {2} F1 which are the same as the edges of type F1 . Simililarly to the last case, the face set {0,1} consists of ten skew, regular hexagons of type F2 and twelve convex, regular pentagons of {1,2} {0,2} type F2 . Unlike with the last case, this time there is no face of type F2 . The resulting vertex ﬁgure is a double edged regular pentagram.

P 012 In the final case we let the initial vertex be such that it is transformed under all distinguished 012 {0} {1} generators. In the generic case P is a polyhedron which has sixty edges of type F1 ,F1 , {2} {0,1} and F1 . They make ten skew dodecagons of type F2 (truncated skew hexagons), twelve {1,2} convex decagons of type F2 (truncated pentagons), and thirty crossed quadrilaterals of {0,2} type F2 . The vertex configuration is (4.t6s.10) with a triangular vertex figure. The crossed quadrilateral faces are not regular polygons so the polyhedron is not uniform.

Now look at an initial vertex which is invariant under s2 and none of the distinguished {0} generators. This figure has thirty edges of type F1 which are the same as the edges of type {2} {1} 012 F1 and sixty edges of type F1 . Simililarly to the generic P , this figure has ten skew {0,1} {1,2} dodecagons of type F2 and twelve convex decagons of type F2 but it does not have any {0,2} {0} {2} faces of type F2 . Each edge of type F1 = F1 is contained in four faces while the edges {1} of type F1 are contained in two faces each making this figure a semi-complex. This is the only time when P 012 is not a polyhedron. The resulting vertex figure consists of two digons with a common vertex.

5 3.3.9 {3, 2 } 5 5 By performing the facetting operation on { 2 , 5} we get the regular polyhedron {3, 2 }, the great icosahedron. It will then have automorphism group

5 Γ 3, = hσ , σ σ σ σ σ σ σ , σ i. 2 2 1 2 1 0 1 2 1 0

The Euclidean realization then has symmetry group

5 G 3, = hs , s s s s s s s , s i =: hr , r , r i. 2 2 1 2 1 0 1 2 1 0 0 1 2

57 A quick note about choices of initial vertex beyond the assumption that it comes from the standard fundamental region of the icosahedral group. If the point is invariant under s1s2s1 then the base {1} {2} edges F1 and F1 are equivalent. In this case the equivalence has an eﬀect on the faces of the Wythoﬃan.

0 5 P For the first choice of initial vertex the Wythoffian is the regular {3, 2 } itself. It has twenty regular triangles as the faces and thirty edges. Five triangles meet at each vertex, (3.3.3.3.3), with a pentagram as the vertex figure.

1 {1} P The polyhedron that is generated here has sixty edges of type F1 . Together they form {0,1} twenty regular triangular faces of type F2 and twelve regular, planar pentagrams of type {1,2} 5 5 F2 . The vertex configuration is (3. 2 .3. 2 ) and the vertex figure is a convex rectangle. This figure is a uniform polyhedron with planar faces. In the notation of [9] the uniform polyhedron 5 is 2 | 3 2 .

2 5 {2} P The Wythoﬃan is the dual polyhedron, the regular { 2 , 3}. It has thirty edges of type F1 . {1,2} Together they make twelve regular, planar pentagrams of type F2 . Three pentagrams 5 5 5 meet at each vertex, ( 2 . 2 . 2 ), with a regular triangle as the vertex ﬁgure.

01 {0} {1} P In this polyhedron there are thirty edges of type F1 and sixty edges of type F1 . Together {0,1} they make twenty convex hexagons of type F2 (truncated triangles) and twelve regular {1,2} 5 pentagrams of type F2 . The vertex configuration is ( 2 .6.6) with an isosceles triangle as a vertex figure. When the initial vertex is chosen so that the hexagons are regular polygons 5 then the Wythoffian is the planar faced uniform polyhedron 2 2 | 3 in the notation of [9].

02 {0} {2} P For this polyhedron there are sixty edges of type F1 and sixty edges of type F1 . They make {0,1} {1,2} twenty regular, triangular faces of type F2 ; twelve regular pentagrams of type F2 ; and {0,2} 5 thirty convex rectangles of type F2 . The vertex configuration is (3.4. 2 .4) and the vertex figure is a crossed quadrilateral. The initial vertex can be chosen so that the rectangles become squares and the Wythoffian is then a uniform polyhedron with planar faces. In the 5 notation of [9], the uniform polyhedron is 3 3 | 2.

12 12 {1} P In the generic case of P the Wythoﬃan is a polyhedron with sixty edges of type F1 {2} {0,1} and thirty edges of type F1 . They form twenty regular, triangular faces of type F2 {1,2} and twelve truncated pentagrams of type F2 . There are two truncated pentagrams and 5 5 one triangle around each vertex, (3.t 2 .t 2 ). The vertex ﬁgure is an isosceles triangle. The truncated pentagrams are not regular so the polyhedron is not uniform.

Now consider an initial vertex which is stabilized by r0 and s1s2s1 but not by r1. This is the only time when P 12 is not a polyhedron. There are still twenty regular triangles of type {0,1} F2 . Instead of truncated pentagrams there are now twelve double edged pentagons of {1,2} type F2 . The pentagons are collapsed versions of the truncated pentagrams as a result of

58 {1} {2} F1 and F1 being equivalent. The ﬁnal vertex ﬁgure has ten edges arranged as a regular pentagram inscribed in a regular pentagon.

P 012 Finally look at the case when the initial vertex is transient under all of the distinguished {0} generators. This generic case is a polyhedron with sixty edges of type F1 , sixty edges {1} {2} of type F1 , and sixty edges of type F1 . They make twenty convex hexagons of type {0,1} {1,2} F2 (truncated triangles), twelve truncated pentagrams of type F2 , and thirty convex {0,2} rectangles of type F2 . There is a truncated pentagram, a rectangle, and a hexagon meeting 5 at every vertex, (4.t 2 .6). The resulting vertex ﬁgure is a triangle. The truncated pentagram is not a regular polygon so the polyhedron is not uniform.

The last case we will look at is when the initial vertex is invariant under s1s2s1 but not invariant under any of the distinguished generators. This is the only case when P 012 is not a polyhedron. As in the previous case there are still twenty convex hexagons of type {0,1} {0,2} F2 and thirty convex rectangles of type F2 . The truncated pentagrams are now twelve {1,2} double edged pentagons of type F2 . The vertex ﬁgure is rather complicated. It has six vertices arranged around a hexagon where every other edge is missing. Sharing vertices with the “hexagon” are two positionally dual regular triangles. There are three additional edges connecting vertices on opposite sides of the “hexagon”.

5 3.3.10 { 2 , 3} 5 5 The dual of {3, 2 } is the regular polyhedron { 2 , 3}, the great stellated dodecahedron. Thus it has automorphism group 5 Γ , 3 = hσ , σ σ σ σ σ σ σ , σ i. 2 0 1 2 1 0 1 2 1 2 The Euclidean realization has symmetry group

5 G , 3 = hs , s s s s s s s , s i =: hr , r , r i. 2 0 1 2 1 0 1 2 1 2 0 1 2

The fundamental region here is still the standard fundamental region for the icosahedral group. As with its dual polyhedron, there are initial vertex choices which lead to equivalences between base {0} {1} edges. If the initial vertex is invariant under s1s2s1 but not r1 then the base edges F1 and F1 are equivalent. The Wythoffian resulting from this vertex choice is affected accordingly. As we have seen with all other dual regular polyhedra, we can interchange the 0 and 2 in the 5 5 superscripts of a Wythoffian of {3, 2 } and get a Wythoffian of { 2 , 3}, and vice versa.

0 5 P The first Wythoffian is the regular polyhedron { 2 , 3} itself. It has thirty edges and twelve 5 5 5 regular pentagrams as faces. At each vertex there are three pentagrams, ( 2 . 2 . 2 ), yielding a regular triangle for the vertex figure.

1 {1} P In this polyhedron there are sixty edges of type F1 . Together they form twelve regular {0,1} {1,2} pentagram faces of type F2 and twenty regular triangles of type F2 . The vertex con-

59 5 5 ﬁguration is ( 2 .3. 2 .3) and the polyhedron has a vertex ﬁgure of a convex rectangle. This is a uniform polyhedron with planar faces. In the notation of [9] this uniform polyhedron is 5 2 | 3 2 .

2 5 5 P This Wythoffian is the regular {3, 2 }, the dual polyhedron of { 2 , 3}. It has thirty edges of {2} {1,2} type F1 and twenty triangular faces of type F2 . The vertex figure is a regular pentagram with vertex configuration (3.3.3.3.3).

01 {0} {1} P This Wythoﬃan has thirty edges of type F1 and sixty edges of type F1 . Together they {0,1} {1,2} make twelve truncated pentagrams of type F2 and twenty regular triangles of type F2 . In this generic case, P 01 is a polyhedron. There are two truncated pentagrams and one triangle 5 5 at each vertex, (3.t 2 .t 2 ). The resulting vertex ﬁgure is an isosceles triangle. Truncated pentagrams are never regular polygons, so this polyhedron is not uniform.

If we instead consider an initial vertex stabilized by r2 and s1s2s1 but not by r1 the figure is slightly different, and in this one case the Wythoffian is not a polyhedron. As with the {1,2} previous choice of initial vertex, the figure has twenty regular triangles of type F2 , but {0,1} unlike before this figure has twelve double edged pentagons of type F2 . The pentagons {0} {1} are collapsed truncated pentagrams resulting from the equivalence between F1 and F1 . There are two pentagons and two triangles incident to each edge. The resulting vertex figure consists of ten edges which are arranged as a pentagram inscribed in a pentagon.

02 {0} {2} P This polyhedron has sixty edges of type F1 and sixty edges of type F1 . Together they {0,1} {1,2} make twelve regular pentagrams of type F2 , twenty regular triangles of type F2 , and {0,2} 5 thirty convex rectangles of type F2 . The vertex configuration is (3.4. 2 .4) with a crossed quadrilateral as the vertex figure. For a carefully chosen initial vertex the rectangles are squares and the Wythoffian is a uniform polyhedron with planar faces. The uniform polyhedra 5 is 3 3 | 2 in the notation of [9].

12 {1} {2} P The polyhedron has sixty edges of type F1 and thirty edges of type F1 . They form {0,1} {1,2} twelve pentagrams of type F2 and twenty convex hexagons of type F2 . The vertex 5 configuration is ( 2 .6.6) and the polyhedron has a vertex figure which is an isosceles triangle. As with the last Wythoffian we can choose a specific initial vertex so that the hexagons become regular hexagons and the Wythoffian is a uniform polyhedron. The uniform polyhedron is 5 2 2 | 3 in the notation of [9]. P 012 Finally, look at the case when the initial vertex is transient under all of the distinguished generators. In the generic case the Wythoffian is a polyhedron. There are sixty edges of type {0} {1} {2} F1 , sixty edges of type F1 , and sixty edges of type F1 . They make twelve truncated pen- {0,1} {1,2} tagrams of type F2 , twenty convex hexagons of type F2 , and thirty convex rectangles of {0,2} type F2 . There is one truncated pentagram, one rectangle, and one hexagon surrounding 5 each vertex, (4.t 2 .6), yielding a triangular vertex figure. The truncated pentagrams are not regular polygons so the polyhedron is not uniform.

60 Consider a slightly altered initial vertex which is transient under all of the distinguished 012 generators and invariant under s1s2s1. This is the only case when P is not a polyhedron. {1,2} As with the previous choice of initial vertex there are twenty convex hexagons of type F2 {0,2} and thirty convex rectangles of type F2 . The truncated pentagrams transform into double {0,1} {0} {1} edged pentagons of type F2 due to the equivalence between F1 and F1 . The vertex ﬁgure is still a triangle.

10 5 3.3.11 { 3 , 2 } 10 5 5 The regular polyhedron { 3 , 2 } is the Petrial of {3, 2 }. Its automorphism group can then be derived from that of the icosahedron giving

10 5 Γ , = hσ σ , σ σ σ σ σ σ σ , σ i. 3 2 2 0 1 2 1 0 1 2 1 0

The Euclidean realization is then

10 5 G , = hs s , s s s s s s s , s i =: hr , r , r i. 3 2 2 0 1 2 1 0 1 2 1 0 0 1 2

There are some limitations on the choice of initial vertex beyond choosing it from the standard fundamental region of the icosahedral group. Any point which is stabilized by r0 is also stabilized by r2. Thus there is no point which is stabilized by r0 alone nor is there any point which is stabilized by only r0 and r1. Another issue that will arise is when the initial vertex is stabilized by s1s2s1 {1} {2} {1,2} but not by r1. In this case, F1 and F1 are equivalent and collapse F2 in the Wythoﬃan.

0 10 5 P The first Wythoffian is the regular polyhedron { 3 , 2 } itself. It has thirty edges and six regular, antiprismatic decagrams as faces. The vertex figure is a regular, planar pentagram 10 10 10 10 10 with vertex symbol ( 3 . 3 . 3 . 3 . 3 ).

1 {1} P This polyhedron has sixty edges of type F1 . There are six planar, regular decagrams of {0,1} {1,2} type F2 and twelve planar, regular pentagrams of type F2 . At each vertex there is 5 10 5 10 a pentagram, a decagram, a pentagram, and a decagram in cyclic order, ( 2 . 3 . 2 . 3 ). The vertex ﬁgure is a crossed quadrilateral. All faces are regular polygons so this is the planar 5 5 5 faced uniform polyhedron 3 2 | 3 in the notation of [9].

01 {0} {1} P In this polyhedron there are thirty edges of type F1 and sixty edges of type F1 . Together {0,1} they form six truncated, skew decagrams of type F2 and twelve planar, regular pentagrams {1,2} 5 10 10 of type F2 . The vertex conﬁguration is ( 2 .t 3 .t 3 ) giving an isosceles triangle as the vertex ﬁgure. Truncated skew decagrams are not regular polygons so this polyhedron is not a uniform polyhedron.

02 {0} P This Wythoﬃan is a polyhedron in the generic case which has sixty edges of type F1 and {2} {0,2} sixty edges of type F1 . They form six skew, regular decagrams of type F2 ; twelve

61 {1,2} {0,2} planar, regular pentagrams of type F2 ; and thirty crossed quadrilaterals of type F2 . 5 10 The polyhedron has a vertex configuration of (4. 2 .4. 3 ) and a vertex figure of a convex quadrilateral. Crossed quadrilaterals are not regular polygons so the Wythoffian is not a uniform polyhedron.

Now consider an initial vertex which is stabilized by s2 and r1 but not by r0 or r2. The resulting figure is a polygonal complex with four faces incident to each edge, and in this 02 {0} case only P is not a polyhedron. There are sixty edges of type F1 which are the same {2} 02 as the edges of type F1 . Similar to the more generic P the figure has six regular, skew {0,1} {1,2} decagrams of type F2 and twelve regular, planar pentagrams of type F2 . Unlike the {0,2} previous case, this time there is no face of type F2 . Consequently, each edge is incident to two pentagrams and two decagrams. There are three pentagrams and three decagrams surrounding each vertex yielding a double edged regular triangle as a vertex figure.

P 012 Finally, let the initial vertex be transient under all of the distinguished generators. In the {0} {1} generic case the resulting polyhedron has sixty edges of type F1 , sixty edges of type F1 , {2} {0,1} and sixty edges of type F1 . Together they form six truncated skew decagrams of type F2 , {1,2} twelve truncated planar pentagrams of type F2 , and thirty crossed quadrilaterals of type {0,2} 5 10 F2 . The vertex conﬁguration is (4.t 2 .t 3 ) with a vertex ﬁgure of a triangle. Truncated pentagrams and truncated skew decagrams are not regular polygons so this polyhedron is not uniform.

Consider an initial vertex which is stabilized under s2 and no distinguished generators. The {0} resulting ﬁgure, which is not a polyhedron, has sixty edges of type F1 , which are the same as {2} {1} the edges of type F1 , and it has sixty edges of type F1 . As before there are six truncated {0,1} {1,2} skew decagrams of type F2 and twelve truncated planar pentagrams of type F2 , unlike {0,2} {0} {2} before this time there is no face of type F2 . As a result, each edge of type F1 = F1 is incident to two truncated decagrams and two truncated pentagrams while each edge of type {2} F1 is incident to one decagram and one pentagram. The vertex ﬁgure consists of two digons with a common vertex.

Let a different initial vertex be invariant under s2 and s1s2s1 but not under any distinguished generators. This case is also not a polyhedron. As with the previous case there are six {0,1} truncated skew decagrams of type F2 . The truncated pentagrams collapse into double {1,2} {0,2} {0} {2} edged pentagons of type F2 . There are now no edges of type F2 since here F1 = F1 . This figure appears as a skeletal dodecahedron and its vertex figure is a double edged regular triangle.

Now consider a vertex which is invariant under s1s2s1 but not under anything else. As 012 {0,1} with the generic P there are six skew truncated decagrams of type F2 and thirty {0,2} crossed quadrilaterals of type F2 . The truncated pentagrams collapse into double edged {1,2} {1} {2} pentagons of type F2 due to the congruence between F1 and F1 . This case is also not a polyhedron. At each vertex there is a crossed quadrilateral, truncated decagram, crossed

62 quadrilateral, truncated decagram, and double edged pentagon. The resulting vertex ﬁgure is a crossed quadrilateral with an additional edge connecting two non-adjacent vertices. These three subcases are the only times in which P 012 is not a polyhedron.

10 3.3.12 { 3 , 3} 10 5 The last ﬁnite polyhedron which we will examine is { 3 , 3}, the Petrial of { 2 , 3}. Derived from the 10 group of the icosahedron, the group of { 3 , 3} is

10 Γ , 3 = hσ σ , σ σ σ σ σ σ σ , σ i. 3 0 2 1 2 1 0 1 2 1 2

The Euclidean realization of this polyhedron has symmetry group

10 G , 3 = hs s , s s s s s s s , s i =: hr , r , r i. 3 0 2 1 2 1 0 1 2 1 2 0 1 2

The initial vertex again comes from the standard fundamental region of the icosahedral group. As with all previous Petrials, this polyhedron has a couple of initial vertex choices which do not work.

All points stabilized by r0 are also stabilized by r2. Thus there are no points which are stabilized by r0 alone nor are there any points stabilized by only r0 and r1.

0 10 P In this first case, Wythoff’s construction yields the regular polyhedron { 3 , 3} itself. This figure has six regular, skew decagrams as faces and thirty edges. The vertex figure is a regular 10 10 10 triangle with vertex configuration ( 3 . 3 . 3 ).

1 {1} P This polyhedron has sixty edges of type F1 which together form six regular planar decagrams {0,1} {1,2} of type F2 and twenty regular triangles of type F2 . The resulting vertex conﬁguration 10 10 is (3. 3 .3. 3 ) with a convex, rectangular vertex ﬁgure. This is a uniform polyhedron with 3 5 planar faces. In the notation of [9] this uniform polyhedron is 2 3 | 3 .

01 {0} {1} P The polyhedron has thirty edges of type F1 and sixty edges of type F1 . It has six {0,1} {1,2} truncated skew decagrams of type F2 and twenty regular triangles of type F2 . The 10 10 vertex conﬁguration is (3.t 3 .t 3 ) with a vertex ﬁgure of an isosceles triangle. The truncated decagrams are not regular polygons so the polyhedron is not uniform.

02 {0} {2} P This Wythoffian has sixty edges of type F1 and sixty edges of type F1 . They form six {0,1} {1,2} regular skew decagrams of type F2 ; twenty regular triangles of type F2 ; and thirty {0,2} crossed quadrilaterals of type F2 . In this generic case the Wythoffian is a polyhedron. Around each vertex there is a quadrilateral, a triangle, a quadrilateral, and a truncated 10 decagram, (3.4.t 3 .4). The resulting vertex figure is a convex quadrilateral. The crossed quadrilaterals are not regular so the polyhedron is not uniform.

Now consider an initial vertex which is stabilized by s0 and r1. The resulting polygonal {0} complex, which in this case alone is not a polyhedron, has sixty edges of type F1 which are

63 {2} the same as the edges of type F1 . They come together to form six regular, skew decagrams {0,1} {1,2} of type F2 and twenty regular triangles of type F2 just as before. The main diﬀerence {0,2} is that now there is no face of type F2 , instead there is only a single edge. Consequently, each edge is incident to two faces of each type. The vertex ﬁgure of this complex is a double edged pentagon.

012 10 P Let the initial vertex be transient under all distinguished generators of G({ 3 , 3}). In the {0} generic case the Wythoffian is a polyhedron. The Wythoffian has sixty edges of type F1 , {1} {2} {0,1} F1 , and F1 . They form six truncated skew decagrams of type F2 , twenty convex {1,2} {0,2} hexagons of type F2 , and thirty crossed quadrilaterals of type F2 . The vertex configu- 10 ration is (4.6.t 3 ) and the resulting vertex figure is a triangle. The crossed quadrilaterals and truncated skew decagrams are not regular polygons so the polyhedron is not uniform for any choice of initial vertex.

Consider an initial vertex which is stabilzed under s0 and transient under r0, r1, and r2. The resulting ﬁgure is a semi-complex, see Section 1.8. This is the only case when P 012 is not a {0} {2} polyhedron. It has sixty edges of type F1 which are identical to the edges of type F1 , and {1} it has sixty edges of type F1 . Similar to the previous case, there are six truncated skew {0,1} {1,2} decagrams of type F2 and twenty convex hexagons of type F2 . In this case, there is no {0,2} 2-face of type F2 , there is only an edge. The resulting vertex ﬁgure consists of two double edges with a common vertex.

64 Chapter 4

Planar regular apeirohedra and their Wythoﬃans

The planar apeirohedra are infinite polyhedra which can be realized in 2-space. There are six planar apeirohedra, three with finite faces and three with infinite faces. The three with finite faces are the regular planar tessellations {4, 4}, {3, 6}, and {6, 3} and the three with infinite faces are their

Petrials {∞, 4}4, {∞, 6}3, and {∞, 3}6, respectively. For each apeirohedron we will examine how different choices of initial vertex behave under the symmetry group. The initial vertices will all be chosen from the fundamental triangle of the underlying regular tessellation. In each case, the resulting figure is still planar. Images of the faces, vertex figures, and Wythoffians can be found in Tables 7.19-7.24 in the Appendix. The planar apeirohedra can be categorized into two families, one based on the square tiling of the plane and the other based on the triangular tiling of the plane ([32], p.221).

π {4, 4} ←→ {∞, 4}4

π δ π {∞, 6}3 ←− {3, 6} ←→ {6, 3} ←→ {∞, 3}6

4.1 Square tiling

4.1.1 {4, 4}

The square tiling of the plane is the regular apeirohedron {4, 4}. Its automorphism group is given by

Γ({4, 4}) := hσ0, σ1, σ2i

4 4 2 where (σ0σ1) = (σ1σ2) = (σ0σ2) = ε. The corresponding symmetry group is

G({4, 4}) = hs0, s1, s2i

65 4 4 2 where (s0s1) = (s1s2) = (s0s2) = 1. All initial vertices are chosen from the fundamental triangle of {4, 4}.

0 {0} P The first Wythoffian is the regular apeirohedron {4, 4} itself. All edges are of type F1 and {0,1} all faces are convex squares of type F2 . Four squares meet at each vertex, (4.4.4.4), giving a convex square vertex figure.

1 {1} P In this Wythoffian all edges are of type F1 . This figure has two types of face: convex {0,1} {1,2} squares of type F2 and congruent convex squares of type F2 . The vertex figures are convex squares since as with P 0 the vertex configuration is (4.4.4.4). This is again a regular apeirohedron, a similar copy of the original square tessellation.

2 {2} P Here we get the dual polyhedron which is a copy of {4, 4}. All edges are of type F1 yielding {1,2} convex square faces of type F2 . Four squares meet at each vertex, (4.4.4.4), giving a convex square vertex ﬁgure.

P 01 This apeirohedron has two distinct types of 2-faces. The first type of base face is a convex {0,1} {1,2} octagon of type F2 (truncated square). The second type is a convex square of type F2 . Two octagons and one square meet at each vertex yielding an isosceles triangle for a vertex figure. The initial vertex can be chosen so that the octagons are regular in which case the Wythoffian is a uniform apeirohedron. In fact, it is the Archimedean tessellation (4.8.8).

P 02 This apeirohedron has three different types of 2-faces. The first is a convex square face of type {0,1} {0} {1,2} F2 all of whose edges are of type F1 . The second is a convex square of type F2 all {2} {0,2} of whose edges are of type F1 . The final face type is a convex rectangle of type F2 with {0} {2} alternating edges of type F1 and F1 . At each vertex there is a square of the first kind, a rectangle, a square of the second kind, and a rectangle with vertex symbol (4.4.4.4). The resulting vertex figure is convex quadrilateral. When the initial vertex is chosen so that the base edges have the same length, the rectangles are squares and the Wythoffian is a congruent copy of the original tessellation.

P 12 This Wythoffian is an apeirohedron with square and octagonal faces. The convex squares are {0,1} {1,2} of type F2 . The convex octagons are of type F2 (truncated squares). Two octagons and one square meet at each vertex giving an isosceles triangle for the vertex figure. If the initial vertex is chosen so that the octagons are regular, then the Wythoffian is again the Archimedean tessellation (4.8.8).

P 012 Finally, let the initial vertex be such that it is not held invariant under any of the distinguished generators. The resulting apeirohedron has two diﬀerent octagonal faces and a rectangular {0} {1} {0,1} face. The edges of type F1 and of type F1 make up convex octagons of type F2 {1} {2} (truncated squares). The edges of type F1 and F1 make up convex octagons of type {1,2} {0} {2} F2 (truncated squares). The edges of type F1 and F1 make up the convex rectangles {0,2} of type F2 . One octagon of each type and a rectangle come together at each vertex to

66 make a triangular vertex figure. When the initial vertex is chosen to make the base edges have equal length then the faces are regular polygons and the Wythoffian is again the Archimedean tessellation (4.8.8) with an isosceles triangle as the vertex figure.

4.1.2 {∞, 4}4

This regular apeirohedron is the Petrial of {4, 4}, thus its automorphism group is

Γ({∞, 4}4) = hσ0σ2, σ1, σ2i.

Its Euclidean symmetry group is

G({∞, 4}4) = hs0s2, s1, s2i := hr0, r1, r2i.

Note that there is no point which is held invariant by r0 which is not also held invariant by r2. All initial vertices come from the fundamental triangle of {4, 4}.

0 P The first Wythoffian is the regular apeirohedron {∞, 4}4 itself. Its 2-faces are apeirogons π which appear as infinite zigzags whose consecutive edges meet at an angle of 2 . Four apeirogons meet at each vertex, (∞2.∞2.∞2.∞2), giving a square vertex figure.

1 {1} P This apeirohedron only has edges of type F1 . They form straight line apeirogons of type {0,1} {1,2} F2 and convex squares of type F2 . About each vertex there is an apeirogon, a square, an apeirogon, and a square with vertex symbol (∞.4.∞.4). The apeirogons dissect the plane {1,2} into squares, exactly half of which are the square faces of type F2 . The vertex ﬁgure is crossed quadrilateral. All faces of this Wythoﬃan are regular polygons so this shape is a uniform apeirohedron.

01 {0,1} P This apeirohedron has finite and infinite faces. The apeirogonal faces are of type F2 , each of which is a truncation of a face of {∞, 4}4. The finite faces of this apeirohedron are convex {1,2} squares of type F2 . The vertex configuration is (4.t∞2.t∞2) and the resulting vertex figure is an isoceles triangle. The truncated zigzags are not regular apeirogons so this Wythoffian is not a uniform apeirohedron.

02 {0,1} P For this Wythoffian the generic case is a polyhedron whose faces of type F2 are zigzags π where the angle between consecutive edges is greater than or equal to 2 , convex squares of {1,2} {0,2} type F2 , and crossed quadrilaterals of type F2 (see Table 7.20). The vertex figure is a convex quadrilateral with vertex configuration (4.4.4.∞2). The crossed quadrilaterals are not regular so this is not a uniform apeirohedron.

Consider some other initial vertex which is invariant under s0 and r1 but not under r0. The {0} Wythoﬃan in this case, and this case only, is not a polyhedron. It has edges of type F1 {2} {0,1} which are equal to the edges of type F1 . There are zigzags of type F2 and convex

67 {1,2} squares of type F2 as before. This ﬁgure is similar to the previous one except that the crossed quadrilaterals have now been compressed into a single line segment and are no longer 2-faces. At each vertex there are four apeirogons and four squares with two apeirogons and two squares incident to each edge. The vertex ﬁgure of the complex is a double edged square.

P 012 Consider an initial vertex which is not invariant under any of the distinguished generators. In the generic case of this Wythoﬃan the ﬁgure is a polyhedron. There are apeirogonal faces {0,1} of type F2 which are truncated zigzags. There are also convex octagonal faces of type {1,2} {0,2} F2 (truncated squares) and crossed quadrilaterals of type F2 (see Table 7.20). There is one apeirogon, one octagon, and one quadrilateral at each vertex yielding a triangular vertex

ﬁgure with vertex symbol (4.8.t∞2). The truncated zigzags and crossed quadrilaterals are not regular polygons so the polyhedron is not uniform. To better visualize this apeirohedron think

of each zigzag as corresponding to and following a similar path to a zigzag face of {∞, 4}4

while the octagons are centered over the vertices of {∞, 4}4 and the crossed quadrilaterals lie

along the edges of {∞, 4}4, see Table 7.20. Finally, consider an initial vertex which is not invariant under any of the distinguished gen- 012 erators but is invariant under s0. This is the only case in which P is not a polyhedron. 012 {0,1} As with the generic P this Wythoffian has truncated zigzags of type F2 and convex {1,2} {0,2} octagons of type F2 . The difference is that each of the crossed quadrilaterals of type F2 {0} collapses into a single edge. In the resulting semi-complex edges of type F1 are incident to two apeirogons and two octagons. The resulting vertex figure consists of two digons which share a vertex.

4.2 Triangular tiling

4.2.1 {3, 6}

The triangular tiling of the plane is {3, 6}. Let the automorphism group of {3, 6} be

Γ({3, 6}) := hσ0, σ1, σ2i such that 3 6 2 (σ0σ1) = (σ1σ2) = (σ0σ2) = ε.

The Euclidean symmetry group then is

G({3, 6}) = hs0, s1, s2i where 3 6 2 (s0s1) = (s1s2) = (s0s2) = 1.

68 There are no extra restrictions on the choice of initial vertex for this symmetry group beyond choosing it from within the fundamental triangle for this tessellation.

P 0 The first Wythoffian is the apeirohedron {3, 6} itself. It is a planar figure with six regular triangles meeting at each vertex, (3.3.3.3.3.3). The vertex figure is a regular, convex hexagon.

1 {1} P In this tessellation all edges are of type F1 . The resulting ﬁgure has regular triangular {0,1} {1,2} faces of type F2 and regular hexagonal faces of type F2 . There are two triangles and two hexagons alternating about each vertex. The vertex ﬁgure is a convex rectangle. This uniform apeirohedron is the Archimedean tessellation (3.6.3.6).

P 2 The Wythoﬃan is the dual to {3, 6} which is the regular tessellation {6, 3}. All edges are of {2} {1,2} type F1 . They form regular hexagons of type F2 three of which meet at each vertex, (6.6.6). The vertex ﬁgure is a regular triangle.

P 01 This planar apeirohedron is a tessellation with two types of hexagonal 2-faces. The first {0,1} type is a convex hexagon of type F2 (truncated triangle). The second type of 2-face is {1,2} a convex regular hexagon of type F2 . One regular and two non-regular hexagons meet at each vertex yielding a vertex figure of an isosceles triangle. For a carefully chosen initial vertex the non-regular hexagons can be made regular and then the Wythoffian is a copy of the tessellation {6, 3} with a regular triangle for a vertex figure and vertex symbol (6.6.6).

02 {0} {2} P This tessellation has edges of type F1 and of type F1 . Together they form three types of {0,1} {1,2} 2-face. There are regular triangles of type F2 , regular convex hexagons of type F2 , and {0,2} convex rectangles of type F2 . The vertex configuration is (3.4.6.3) and the vertex figure is a convex quadrilateral. Given a specific initial vertex the base edges all have the same length and the faces are all regular polygons making the Wythoffian the Archimedean tessellation (3.4.6.3).

12 {1} {2} P This tessellation has edges of type F1 and of type F1 . The 2-faces are regular triangular {0,1} {1,2} faces of type F2 and convex dodecagons of type F2 (truncated hexagons). There are two dodecagons and one triangle at each vertex yielding an isosceles triangle vertex figure. For a specific initial vertex the dodecagons are regular and the Wythoffian is an Archimedean tessellation with symbol (3.12.12).

P 012 When the initial vertex is transient under all of the distinguished generators the resulting {0} {1} {2} tessellation has edges of type F1 , F1 , and F1 . They come together to form three {0,1} different types of 2-face. There are convex hexagons of type F2 (truncated triangles), {1,2} {0,2} convex dodecagons of type F2 (truncated hexagons), and convex rectangles of type F2 . There is one of each type of face meeting at each vertex giving a triangular vertex figure. As with the previous Wythoffians, the initial vertex can be picked so that the faces are regular polygons and the figure is the Archimedean tessellation (4.6.12).

69 4.2.2 {∞, 6}3

This regular apeirohedron is the Petrial of {3, 6}. Thus its automorphism group is

Γ({∞, 6}3) = hσ0σ2, σ1, σ2i.

Its Euclidean symmetry group is

G({∞, 6}3) = hs0s2, s1, s2i =: hr0, r1, r2i.

As with the Petrial of {4, 4} all points which are held invariant under r0 are also invariant under r2. Consequently there are no points invariant under r0 alone nor are there any which are invariant under r0 and r1 alone. All initial vertex choices are chosen from the fundamental triangle of {3, 6}.

0 P This Wythoﬃan is {∞, 6}3 itself. Its 2-faces are all apeirogonal zigzags with an angle of π {0,1} 3 between consecutive edges. These faces are of type F2 . Six apeirogons meet at each vertex, (∞2.∞2.∞2.∞2.∞2.∞2), resulting in a regular, convex hexagon as the vertex ﬁgure.

1 {0,1} P This apeirohedron has apeirogons of type F2 which appear as straight lines. The ﬁnite {1,2} faces are regular, convex hexagons of type F2 . Two straight lines and two hexagons alternate at each vertex, (6.∞.6.∞), with a resulting vertex ﬁgure of a crossed quadrilateral. This is a uniform apeirohedron since the faces are regular polygons.

01 {0,1} P The apeirohedron has both infinite and finite faces. The infinite faces are of type F2 which are truncations of the zigzags of {∞, 6}3. The finite faces are regular, convex hexagons of {1,2} type F2 . There are two zigzags and one hexagon meeting at each vertex, (6.t∞2.t∞2). The vertex figure is an isosceles triangle. The truncated zigzags are not regular polygons so this Wythoffian is not a uniform apeirohedron.

02 {0,1} P In this Wythoffian the faces of type F2 are regular zigzags. The finite faces are those of type {1,2} {0,2} F2 which are regular, convex hexagons and those of type F2 which are planar, crossed quadrilaterals. In this generic case, the Wythoffian is a polyhedron. At each vertex, in cyclic

order, there is a zigzag, quadrilateral, hexagon, and another quadrilateral, (4.6.4.∞2). The vertex ﬁgure is a convex quadrilateral. The crossed quadrilaterals are not regular polygons so the Wythoﬃan is not a uniform apeirohedron.

Now suppose that in addition to being invariant under r1, the initial vertex is also invariant 02 under s0. The Wythoﬃan in this case behaves very similarly to the generic P except that the crossed quadrilaterals have been compressed into single edges. Thus there are two {0,1} {1,2} apeirogons of type F2 and two regular, convex hexagons of type F2 incident to each edge. Consequently, in this case (and only in this case) P 02 is not a polyhedron. The resulting vertex ﬁgure is a double edged regular triangle.

70 P 012 In this first generic case, the Wythoffian is an apeirohedron. The Wythoffian has apeirogonal {0,1} faces of type F2 which are truncated zigzags. In addition to the apeirogons there are also {1,2} convex dodecagonal faces of type F2 (truncated hexagons) and crossed quadrilaterals of {0,2} type F2 . There is one apeirogon, one dodecagon, and one quadrilateral at each vertex, (4.12.t∞2), yielding a triangular vertex figure. The crossed quadrilaterals and truncated zigzags are not regular polygons so this is not a uniform apeirohedron. To better understand this apeirohedron, think of the zigzags as corresponding to and following a similar path to

the zigzag faces of {∞, 6}3 while the dodecagons are centered over the vertices of {∞, 6}3

and the crossed quadrilaterals lie along the edges of {∞, 6}3, see Table 7.22. Finally, consider an initial vertex which is not invariant under any of the distinguished gen- 012 erators but is invariant under s0. In this case alone, P is not an apeirohedron it is a semi-complex, see Section 1.8. Comparing the figure with the previous case, it has truncated {0,1} {1,2} zigzags of type F2 and convex dodecagons of type F2 . The difference is that the crossed {0,2} quadrilaterals of type F2 collapse into single edges in this Wythoffian. In the resulting {0} semi-complex, edges of type F1 are incident to two apeirogons and two dodecagons. The resulting vertex figure consists of two digons which share a vertex.

4.3 Hexagonal tiling

4.3.1 {6, 3}

The hexagonal tiling of the plane is {6, 3}. Deﬁne the automorphism group of {6, 3} as

Γ({6, 3}) := hσ0, σ1, σ2i such that 6 3 2 (σ0σ1) = (σ1σ2) = (σ0σ2) = ε.

The Euclidean symmetry group is given by

G({6, 3}) = hs0, s1, s2i where 6 3 2 (s0s1) = (s1s2) = (s0s2) = 1.

There are no extra restrictions on the choice of initial vertex beyond choosing them from the fundamental triangle of the tessellation. As {6, 3} is the dual of {3, 6}, interchanging 0 and 2 in the superscript of a Wythoﬃan for {3, 6} gives a Wythoﬃan for {6, 3}, and vice versa.

P 0 The Wythoﬃan is the regular apeirohedron {6, 3} itself. It has three convex, regular hexagons around each vertex, (6.6.6). The vertex ﬁgure is a regular triangle.

71 1 {1} P This tessellation only has edges of type F1 which make regular, convex hexagons of type {0,1} {1,2} F2 and regular triangles of type F2 . Alternating about each vertex are two triangles and two hexagons. The vertex ﬁgure is a convex rectangle. This is an Archimedean tessellation with symbol (3.6.3.6).

P 2 The Wythoﬃan in this case is the dual apeirohedron to {6, 3} which is the regular tessellation {1,2} {3, 6}. It has six regular triangles of type F2 surrounding each vertex, (3.3.3.3.3.3), with a convex, regular hexagon as the vertex ﬁgure.

01 {0} {1} P In this tessellation there are edges of type F1 and F1 . They form convex dodecagons {0,1} {1,2} of type F2 (truncated hexagons) and regular triangles of type F2 . At each vertex, there is one triangle and two dodecagons forming an isosceles triangle as the vertex ﬁgure. For a carefully chosen initial vertex the dodecagons are regular and the Wythoﬃan is the Archimedean tessellation (3.12.12).

02 {0} {2} P In this tessellation there are edges of type F1 and of type F1 . They form three different {0,1} types of 2-face. There are regular, convex hexagons of type F2 ; regular triangles of type {1,2} {0,2} F2 ; and convex rectangles of type F2 . The vertex configuration is (3.4.6.4). The vertex figure is a convex quadrilateral. For a carefully chosen initial vertex the rectangles are squares and the Wythoffian is the Archimedean tessellation (3.4.6.4).

12 {1} {2} P This tessellation has edges of type F1 and of type F1 . There are two types of hexagonal {0,1} {1,2} faces. There are regular, convex hexagons of type F2 and convex hexagons of type F2 (truncated triangles). One regular hexagon and two non-regular hexagons meet at each vertex yielding an isosceles triangle for a vertex figure. Given a specific choice of initial vertex all of the hexagons are regular and the Wythoffian is the regular tessellation {6, 3} with symbol (6.6.6).

012 {0} {1} {2} P In the final tessellation there are edges of all types: F1 , F1 , and F1 . These make three {0,1} types of 2-face: convex dodecagons of type F2 (truncated hexagons), convex hexagons of {1,2} {0,2} type F2 (truncated triangles), and convex rectangles of type F2 . There is one face of each type at each vertex yielding a triangular vertex figure. As with the prior Wythoffians for a specific initial vertex the faces are regular polygons and the Wythoffian is the Archimedean tessellation (4.6.12).

4.3.2 {∞, 3}6

This apeirohedron is the Petrial of {6, 3} and thus it has automorphism group

Γ({∞, 3}6) = hσ0σ2, σ1, σ2i.

72 The Euclidean symmetry group is

G({∞, 3}6) = hs0s2, s1, s2i =: hr0, r1, r2i.

It is worth noting that any point which is held invariant by r0 is also held invariant by r2, thus there are no points which are held invariant by r0 alone nor are there any held invariant by r0 and r1 alone. All initial vertices are chosen from the fundamental triangle of {6, 3}.

0 {0,1} P This Wythoﬃan is {∞, 3}6 itself. The 2-faces are apeirogons of type F2 and they are 2π zigzags with an angle of 3 between consecutive edges. Three of these apeirogons meet at each vertex, (∞2.∞2.∞2), resulting in a regular triangle as the vertex ﬁgure.

1 {0,1} P This apeirohedron has apeirogons of type F2 which are straight lines and it has regular {1,2} triangles of type F2 . The apeirogons dissect the plane into triangles, half of which are the {1,2} triangles of type F2 . The vertex conﬁguration is (∞.3.∞.3) yielding a crossed quadrilateral for the vertex ﬁgure. This is a uniform apeirohedron.

01 {0,1} P This apeirohedron has two types of 2-face. There are apeirogons of type F2 , each of which {1,2} is a truncated zigzag. The other faces of this apeirohedron are regular triangles of type F2 . The resulting vertex ﬁgure is an isoceles triangle with vertex symbol (3.t∞2.t∞2).

P 02 In this first case, the Wythoffian is an apeirohedron. This Wythoffian has three types of {0,1} {1,2} 2-face. There are regular zigzags of type F2 , regular triangles of type F2 , and crossed {0,2} quadrilaterals of type F2 . If we compare this Wythoffian with {∞, 3}6 the zigzags follow the same path as the zigzags of {∞, 3}6, the triangles lie over the vertices of {∞, 3}6, and

the crossed quadrilaterals lie along the edges of {∞, 3}6 (see Table 7.24). Around each vertex

ﬁgure there is a zigzag, quadrilateral, triangle, and another quadrilateral, (3.4.∞2.4). The resulting vertex ﬁgure is a convex quadrilateral. The crossed quadrilaterals are not regular polygons so this is not a uniform apeirohedron.

Now suppose that in addition to being invariant under r1, the initial vertex is also invariant 02 under s0. This is the only time when P is not a polyhedron. The Wythoﬃan in this case behaves very similarly to the previous case except that the crossed quadrilaterals have been {0,1} compressed into single edges. Thus there are two zigzags of type F2 and two regular {1,2} triangles of type F2 incident to each edge. The vertex ﬁgure is a double edged regular hexagon.

012 {0,1} P The Wythoffian has apeirogonal faces of type F2 which are truncated zigzags. In addition {1,2} to the apeirogons there are also convex hexagonal faces of type F2 (truncated triangles) {0,2} and crossed quadrilaterals of type F2 . There is one apeirogon, one hexagon, and one quadrilateral at each vertex, (4.6.t∞2), yielding a triangular vertex figure. In this case the Wythoffian is an apeirohedron, but it is not a uniform apeirohedron. To better understand this apeirohedron, think of the zigzags as corresponding to and following a similar path to

73 the zigzag faces of {∞, 3}6 while the dodecagons are centered over the vertices of {∞, 3}6 and the crossed quadrilaterals lie along the edges of {∞, 3}6, see Table 7.24. Finally, consider an initial vertex which is not invariant under any of the distinguished gener- 012 ators but is invariant under s0. This is the only situation in which P is not a polyhedron. {0,1} Comparing it with the previous Wythoffian it similarly has truncated zigzags of type F2 {1,2} {0,2} and convex hexagons of type F2 . Here, each of the crossed quadrilaterals of type F2 {0} collapses into a single edge. In the resulting figure, edges of type F1 are incident to two apeirogons and two hexagons causing the Wythoffian to be a semi-complex. The resulting vertex figure consists of two digons which share a vertex.

74 Chapter 5

Blended regular apeirohedra and their Wythoﬃans

In this chapter we will examine Wythoff’s construction applied to the regular blended apeirohedra. The symmetry groups of the blends in this chapter were discussed in Section 1.5.1, here we will quickly reiterate how to find the symmetry group of a blended apeirohedron. Let hs0, s1, s2i, ht0i, and ht0, t1i be symmetry groups of a planar polyhedron, P ; a line segment, {}; and a linear 3 apeirogon, {∞}, respectively. In E the reflections t0 and t1 are both plane reflections. Throughout this chapter when we refer to either the reflections t0 and t1 or the reflection planes of t0 and t1 we will consider them as plane reflections such that the plane of t0 coincides with the plane of P and the plane of t1 lies parallel to the plane of P . Consequently, the line segment and the linear apeirogon are orthogonal to P . To find the symmetry groups of the blends we take the direct product of the two related symmetry groups in the obvious way. The symmetry group of the blend

P #{} is the subgroup hs0t0, s1, s2i of G(P ) × G({}) and the symmetry group of the blend P #{∞} is the subgroup hs0t0, s1t1, s2i of G(P ) × G({∞}). For a more in depth explanation, see [32]. To obtain the realizations of the blend of P and {}, P #{}, or of P and {∞}, P #{∞}, we use the blend’s symmetry group to perform Wythoff’s construction with initial vertex (u, w) where u is the initial vertex of P and w is the initial vertex of {} or {∞}, respectively. For the other Wythoffians we will choose the initial vertices from the fundamental regions of the blends. The fundamental regions of the blends of P with a line segment are infinite cylinders over triangles and the fundamental regions of the blends of P with a linear apeirogon are prisms over triangles. Pictures of the fundamental regions are included throughout the chapter. In all cases the initial vertices for a blend and its Petrie dual are chosen from the same fundamental region. Here is a list of the blends of a planar apeirohedron with a line segment ([32], p. 222).

π {4, 4}#{} ←→ {∞, 4}4#{} π {3, 6}#{} ←→ {∞, 6}3#{} π {6, 3}#{} ←→ {∞, 3}6#{}

75 Here is a list of the blends of a planar apreirohedron with a linear apeirogon ([32], p. 222).

π {4, 4}#{∞} ←→ {∞, 4}4#{∞} π {3, 6}#{∞} ←→ {∞, 6}3#{∞} π {6, 3}#{∞} ←→ {∞, 3}6#{∞}

5.1 Blended apeirohedra related to the square tiling of the plane

5.1.1 {4, 4}#{}

The ﬁrst apeirohedron we examine in this section is the blend of the square tiling of the plane with a line segment. This blend is isomorphic to both {4, 4} and the mix {4, 4}♦{}. The automorphism groups of {4, 4} and {} are, respectively, Γ({4, 4}) = hσ0, σ1, σ2i and Γ({}) = hτ0i. In the direct 4 4 2 2 product of these groups we have (σ0σ1) = (σ1σ2) = (σ0σ2) = (σiτ0) = ε for i = 0, 1, 2. The blended apeirohedron has automorphism group

Γ({4, 4}#{}) = hσ0τ0, σ1, σ2i =: hρ0, ρ1, ρ2i where 4 4 2 (ρ0ρ1) = (ρ1ρ2) = (ρ0ρ2) = ε.

Let the blend {4, 4}#{} have symmetry group

G({4, 4}#{}) = hs0t0, s1, s2i =: hr0, r1, r2i such that 4 4 2 (r0r1) = (r1r2) = (r0r2) = 1.

Here si realizes σi for i = 0, 1, 2 and t0 realizes τ0. As a result the generator r0 is a half-turn and 3 the generators r1 and r2 are plane reflections. We will consider all of these reflections in E so s0, s1, s2, and t0 are all plane reflections. Some care will have to be taken in our choice of initial vertex to ensure an interesting Wythoffian.

If a point, v, is invariant under t0 then the Wythoffian of {4, 4}#{} with initial vertex v is the same as the Wytoffian of {4, 4} with initial vertex v. For the following Wythoffians assume that none of the initial vertex choices are invariant under t0, and consequently we will not look at any initial 1 2 12 vertices which are invariant under r0. This excludes the Wythoffians P , P , and P . All initial vertices are chosen from the fundamental region corresponding to {4, 4}#{}. This fundamental region is an infinite cylinder over the fundamental triangle of {4, 4}.

76 Figure 5.1: The fundamental region of the blend {4, 4}#{}.

P 0 The first Wythoffian is {4, 4}#{} itself and is isomorphic to {4, 4}. It 2-faces are all antipris- {0,1} matic squares ({4}#{}) of type F2 . Four faces meet at each vertex, (4s.4s.4s.4s), yielding a convex square as the vertex figure. The projection of this Wythoffian onto the reflection

plane of t0 appears as {4, 4}.

01 {0,1} P In this apeirohedron the faces of type F2 are skew octagons (truncated antiprismatic {1,2} squares) and the faces of type F2 are convex squares. Two octagons and one convex square meet at each vertex giving an isosceles triangle as a vertex ﬁgure with vertex symbol

(4.t4s.t4s). The truncated antiprismatic squares are not regular so this is not a uniform

apeirohedron. The projection of this Wythoffian onto the reflection plane of t0 appears as the Wythoffian P 01 of {4, 4}.

Now consider an initial vertex which is stabilized by s0 and r2. The resulting apeirohedron {0,1} has two types of 2-face. The faces of type F2 are skew octagons where consecutive edges {0} are orthogonal to each other. Within this face the edges of type F1 are mutually parallel {1} and are all orthogonal to the reflection plane of t0, and the edges of type F1 are parallel to the reflection plane of t0. Furthermore, the projection of this face onto the reflection plane {1,2} of t0 is a convex square. The faces of type F2 are convex squares. Two octagons and one square meet at each vertex figure yielding an isosceles triangle as the vertex figure.

02 {0,1} {1,2} P In this apeirohedron the faces of type F2 are antiprismatic squares, the faces of type F2 {0,2} are convex squares, and the faces of type F2 are convex rectangles. Cyclically, about each vertex, there is an antiprismatic square, a rectangle, a square, and a rectangle, (4s.4.4.4). The resulting vertex figure is a convex quadrilateral. For a specifically chosen initial vertex the {0,2} faces of type F2 are squares and the Wythoffian is a uniform apeirohedron with one kind of non-planar face. The projection of this Wythoffian onto the reflection plane of t0 appears as the Wythoffian P 02 of {4, 4}.

Now consider an initial vertex which is stabilized by s0 and r1. The resulting ﬁgure is a

77 semi-complex and is the only case when P 02 is not an apeirohedron. The figure has convex {1,2} square faces of type F2 each of which lies in one of two planes both of which are parallel {0,2} to the reflection plane of t0. The other faces of the figure are convex rectangles of type F2 which lie in planes which are orthogonal to the reflection plane of t0. Consequently, adjacent rectangles and squares are orthogonal to one another with the rectangles lying between the {0} {0,1} planes of the squares. Since F1 is invariant under r1, the face F2 is compressed into a {0} single edge. These edges, those of type F1 , are each incident to four rectangles. There are four rectangles and two squares meeting at each vertex yielding a vertex figure consisting of two isosceles triangles with a common vertex.

P 012 Here, let the initial vertex be transient under all of the distinguished generators. In this {0,1} apeirohedron the faces of type F2 are skew octagons (truncated antiprismatic squares), the {1,2} {0,2} faces of type F2 are convex octagons (truncated squares), and the faces of type F2 are convex rectangles. At each vertex there is one face of each type, (4.t4s.8), yielding a triangular vertex ﬁgure. The truncated antirprismatic squares are not regular so the Wythoﬃan is not

a uniform apeirohedron. The projection of this Wythoffian onto the reflection plane of t0 appears as the Wythoffian P 012 of {4, 4}.

Finally, consider an initial vertex which is stabilized by s0 and none of the distinguished {0,1} generators. In this apeirohedron the faces of type F2 are skew octagons similar in shape 01 {1,2} to the skew octagons in the subcase of P . There are also convex octagons of type F2 {0,2} and convex rectangles of type F2 . There is one face of each type meeting at each vertex with a triangular vertex ﬁgure.

5.1.2 {∞, 4}4#{}

This regular apeirohedron is the Petrial of {4, 4}#{} and is isomorphic to both {∞, 4}4 and the mix {∞, 4}4♦{}, [32]. As before, let Γ({4, 4}) = hσ0, σ1, σ2i, Γ({}) = hτ0i, and Γ({4, 4}#{}) = hσ0τ0, σ1, σ2i. Then the automorphism group of {∞, 4}4#{} is

Γ({∞, 4}4#{}) = hσ0τ0σ2, σ1, σ2i.

The symmetry group of the blend is

G({∞, 4}4#{}) = hs0t0s2, s1, s2i =: hr0, r1, r2i.

Here r0 is a point reflection (through the midpoint of the base edge of the underlying plane tessellation {4, 4}) and r1 and r2 are plane reflections. Individually s0, s1, s2, and t0 are plane reflections 3 in E . The initial vertices we will use come from the same fundamental region as {4, 4}#{}. As with

{4, 4}#{}, any initial vertex left invariant by t0 will result in the Wythoﬃan being the same as the corresponding Wythoﬃan derived from {∞, 4}4. Assume all choices of initial vertex are transient

78 under t0, and consequently we will not look at any initial vertices which are invariant under r0. This excludes P 1, P 2, and P 12.

0 P The ﬁrst Wythoﬃan is the regular apeirohedron {∞, 4}4#{} itself whose faces are regular

zigzag apeirogons, {∞}#{}, such that each edge is bisected by the reﬂection plane of t0. Four of these apeirogons meet at each vertex resulting in a convex, square vertex ﬁgure with vertex

symbol (∞2.∞2.∞2.∞2). The projection of this Wythoﬃan onto the reﬂection plane of t0

appears as {∞, 4}4.

01 {0,1} P In this apeirohedron the faces of type F2 are apeirogons which appear as truncations of the {1,2} faces of {∞, 4}4#{}, while the faces of type F2 are convex squares which lie parallel to the reflection plane of t0. Two apeirogons and one square meet at each vertex, (4.t∞2.t∞2). The vertex figure is an isosceles triangle. The truncated zigzags are not regular so this Wythoffian is not a uniform apeirohedron. The projection of this Wythoffian onto the reflection plane of 01 t0 appears as the Wythoffian P of {∞, 4}4.

Now consider some initial vertex which is invariant under r2 and s0. In this apeirohedron {0,1} the faces of type F2 are apeirogons which are still truncated zigzags except in this case {1,2} they project onto the reﬂection plane of t0 as straight lines. The faces of type F2 are still convex squares. Again, two apeirogons and one square meet at each vertex yielding an isosceles triangle as the vertex ﬁgure.

02 {0,1} P With this apeirohedron the faces of type F2 are regular zigzag apeirogons which are {1,2} bisected by the reflection plane of t0, the faces of type F2 are convex squares parallel to {0,2} the reflection plane of t0, and the faces of type F2 are planar crossed quadrilaterals which intersect the reflection plane of t0. Cyclically at each vertex there is an apeirogon, a crossed

quadrilateral, a square, and a crossed quadrilateral, (4.4.4.∞2). The vertex ﬁgure is convex quadrilateral. The crossed quadrilaterals are not regular so this apeirohedron is not uniform.

The projection of this Wythoffian onto the reflection plane of t0 appears as the Wythoffian 02 P of {∞, 4}4.

Now consider an initial vertex which is stabilized by r1 and s0. The face types of this figure are the same as with the more generic P 02, but the arrangement of the faces around each vertex is different. There are four crossed quadrilaterals, two squares, and two apeirogons meeting at each vertex. The resulting vertex figure consists of two non-intersecting convex rectangles. Since the vertex figure is not connected, the Wythoffian in this case, and this case only, is not an apeirohedron.

012 {0,1} P For this apeirohedron the faces of type F2 are truncated zigzag apeirogons. Within this {0} face, each edge of type F1 is bisected by the reflection plane of t0 while each edge of type {1} {1,2} F1 lies parallel to the reflection plane of t0. The faces of type F2 are convex octagons {0,2} (truncated squares) which lie parallel to the reflection plane of t0. The faces of type F2 are crossed quadrilaterals which intersect the reflection plane of t0 at their centers. There is

79 one face of each type at each vertex, (4.8.t∞2), with the resulting vertex ﬁgure is a triangle. The truncated zigzags and crossed quadrilaterals are not regular so this Wythoﬃan is not

a uniform apeirohedron. The projection of this Wythoffian onto the reflection plane of t0 012 appears as the Wythoffian P of {∞, 4}4. Now choose some initial vertex which is transient under all distinguished generators but 012 invariant under s0. With this apeirohedron the faces are the same as with the generic P with the primary exception being that the crossed quadrilaterals are now orthogonal to the

reﬂection plane of t0. The vertex ﬁgure is still a triangle.

5.1.3 {4, 4}#{∞}

Now we will consider the blend of the square tessellation of the plane with a linear apeirogon. We will continue to let the automorphism group of {4, 4} be Γ({4, 4}) = hσ0, σ1, σ2i and that of {∞} 2 be Γ({∞}) = hτ0, τ1i. In the direct product of these automorphism groups we have, (σiτj) = ε for i = 0, 1, 2 and j = 0, 1. The automorphism group of the blended polyhedron is

Γ({4, 4}#{∞}) = hσ0τ0, σ1τ1, σ2i and its symmetry group is

G({4, 4}#{∞}) = hs0t0, s1t1, s2i =: hr0, r1, r2i.

Here r0 and r1 are half-turns and r2 is a plane reflection. 3 In E the reflection planes corresponding to s0, s1, and s2 are orthogonal to the reflection planes corresponding to t0 and t1 which are parallel to one another. There is no point which is invariant under t0 and t1 so this will limit the choice of initial vertex. In all cases we will consider, any edge {2} of type F1 lies parallel to the reflection planes of t0 and t1. All initial vertices are chosen from the fundamental region of {4, 4}#{∞}. This is a prism over the fundamental triangle of {4, 4}.

80 Figure 5.2: The fundamental region of the blend {4, 4}#{∞}.

P 0 The ﬁrst Wythoﬃan is the regular apeirohedron {4, 4}#{∞} itself. Its faces are helical apeirogons spiraling around a cylinder with a square base. Each edge is incident to two helices which spiral upward in opposite orientations. Four helices meet at each vertex resulting in

an antiprismatic square vertex ﬁgure with vertex symbol (∞4.∞4.∞4, ∞4). The projection

of this Wythoﬃan onto the reﬂection plane of t0 appears as {4, 4}.

1 {0,1} P In this apeirohedron the faces of type F2 are regular helices over a square base while the {1,2} faces of type F2 are antiprismatic squares. At each vertex in alternating order there are two helices and two antiprismatic squares, (4s.∞4.4s∞4), yielding a convex rectangle as a vertex ﬁgure. The faces are all regular polygons so the Wythoﬃan is a uniform apeirohedron.

The projection of this Wythoffian onto the reflection plane of t0 appears as the Wythoffian P 1 of {4, 4}.

01 {0,1} P In this apeirohedron the faces of type F2 are helices centered about an octagon (truncations {1,2} of the helical faces of {4, 4}#{∞}). The faces of type F2 are regular squares, for some initial vertices they are skew and for some initial vertices they are convex. There are two helices and one quadrilateral at each vertex resulting in an isosceles triangle vertex ﬁgure with

vertex symbol (4.∞8.∞8). For a carefully chosen initial vertex the helices are regular helices about octagonal bases and the Wythoffian is uniform. The projection of this Wythoffian onto 01 the reflection plane of t0 appears as the Wythoffian P of {4, 4}.

Look at an initial vertex which is still invariant under r2, but is additionally invariant under

s0, then the resulting ﬁgure is altered slightly but is still an apeirohedron. Each face of type {0,1} 01 F2 is similar to the helices of the generic P in that one full rotation around the helix consists of eight vertices, but the cylinder at the center of the helix has a square base so the vertices are doubly covered under projection onto the base. As such, the faces of this

81 {1,2} type are no longer truncations of the faces of {4, 4}#{∞}. The faces of type F2 are still antiprismatic squares. The arrangement of face types about each vertex and the vertex ﬁgure

are the same as in the generic case. If the initial vertex is additionally held invariant by t1 then the quadrilateral faces are convex squares but the other face types and vertex ﬁgure remain the same.

Now consider an initial vertex which is invariant under r2 and s1. As with the previous case the helices have an eight vertex rotation cycle and are centered over a square. In this case {1,2} {1} there are no faces of type F2 . The edges of type F1 are incident to four helices making this figure a semi-complex. This is the only case in which P 01 is not an apeirohedron. The vertex figure consists of two digons with a shared vertex. A similar figure results when the

initial vertex is additionally held invariant under t0.

02 {0,1} P In this apeirohedron the faces of type F2 are regular helices centered about squares, the {1,2} faces of type F2 are convex squares lying parallel to the plane of t1, and the faces of type {0,2} F2 are convex rectangles which are not parallel to either of the reﬂection planes t0 or t1. Cyclically, about each vertex there is a square, a rectangle, a helix, and a rectangle with

vertex symbol (4.4.∞4.4). The resulting vertex figure is a skew quadrilateral. For a carefully {0,2} chosen initial vertex the faces of type F2 are regular and the Wythoffian is a uniform apeirohedron. The projection of this Wythoffian onto the reflection plane of t0 appears as the Wythoffian P 02 of {4, 4}.

Now consider some initial vertex which is stabilized by r1 and s0. In this apeirohedron the helix from the previous case condenses into a straight line. The convex squares are still

parallel to t1 but the rectangles are now orthogonal to t1. As a result, the figure appears as infinitely many spiral staircases centered at each straight line. The vertex figure is still a skew quadrilateral.

12 {0,1} P In this apeirohedron the faces of type F2 are regular helices centered about squares. The {1,2} faces of type F2 are skew octagons (truncated antiprismatic squares). There are two octagons and one helix meeting at each vertex, (t4s.t4s.∞4), yielding an isosceles triangle as the vertex ﬁgure. The truncated antiprismatic squares are not regular so P 12 is not a uniform

apeirohedron. The projection of this Wythoffian onto the reflection plane of t0 appears as the Wythoffian P 12 of {4, 4}.

Now consider an initial vertex which is invariant under r0 and s1. In this apeirohedron the {1,2} helix from the previous case is condensed into a straight line. The faces of type F2 are {1} still skew octagons but now the edges of type F1 are orthogonal to the plane of t0 while the {2} edges of type F1 are parallel to it. As such, consecutive edges of the octagon are orthogonal to one another and the skew octagon is not the truncation of a antiprismatic square. The vertex ﬁgure is still an isosceles triangle.

P 012 Now we will look at the cases in which the intitial vertex is transient under all of the dis-

82 tinguished generators. In this first generic case the Wythoffian is an apeirohedron. The {0,1} apeirogonal faces are of type F2 and appear as helices based around octagons (truncated {1,2} helices about squares). The finite faces are skew octagons of type F2 (truncated antipris- {0,2} matic square) and convex rectangles of type F2 . One face of each type meets at each vertex yielding a triangular vertex figure with vertex symbol (4.t4s.∞8). The truncated antiprismatic squares are not regular so this Wythoffian is not a uniform apeirohedron. The 012 projection of this Wythoffian onto the reflection plane of t0 appears as the Wythoffian P of {4, 4}.

Consider an initial vertex which is stabilized by either s0 or s1. In this apeirohedron the apeirogonal faces appear as helices about squares but the vertices exist in an eight vertex

rotation cycle. In each helix every other edge is orthogonal to the reflection plane of t0. The finite faces are still skew octagons and convex rectangles. The vertex figure is still a triangle.

Now consider an initial vertex which is stabilized by s0 and s1. In this apeirohedron the {0} {1} edges of type F1 and F1 all lie orthogonal to the reflection planes of t0 and t1. As a result {0,1} the faces of type F2 are straight lines which are orthogonal to the reflection planes of t0 {1,2} and t1, the faces of type F2 are skew octagons which when projected onto the plane of t0 {0,2} appear as squares, and the faces of type F2 are convex rectangles which are orthogonal to the plane t0. The resulting vertex figure is still a triangle.

If, in addition, the initial vertex is invariant under t0 the rectangular faces are parallel to the

plane of t0, and if the initial vertex is invariant under t1 then the octagonal faces are convex. In both of those cases, the combinatorial structure remains unchanged from the generic case and the ﬁgure is still an apeirohedron.

5.1.4 {∞, 4}4#{∞}

The last blended apeirohedron based on the square tessellation of the plane is {∞, 4}4#{∞}. It is the Petrial of {4, 4}#{∞}. As before, let Γ({4, 4}) = hσ0, σ1, σ2i and Γ({∞}) = hτ0, τ1i. Then the automorphism group of {∞, 4}4#{∞} is

Γ({∞, 4}4#{∞}) = hσ0σ2τ0, σ1τ1, σ2i =: hρ0, ρ1, ρ2i.

2 4 2 For this automorphism group, the following relations hold: (ρ0ρ2) = (ρ1ρ2) = (σiτj) = ε for all i = 0, 1, 2 and j = 0, 1. The symmetry group is given by

G({∞, 4}4#{∞}) = hs0s2t0, s1t1, s2i =: hr0, r1, r2i.

As with the previous apeirohedron, the reflection planes corresponding to s0, s1, and s2 are orthogonal to the reflection planes corresponding to t0 and t1 which are parallel to one another. Thus r0 is a point reflection, r1 is a half-turn, and r2 is a plane reflection. Again there is no point which is invariant under t0 and t1 so this will limit the choice of initial vertex. In all cases we will consider,

83 {2} any edge of type F1 lies parallel to the reﬂection planes of t0 and t1. Finally, any point which is 2 12 invariant under r0 is also invariant under r2, so this excludes P and P . The initial vertices all come from the fundamental region of {4, 4}#{∞}.

0 P The first Wythoffian is the regular apeirohedron {∞, 4}4#{∞} itself. The faces of type {0,1} F2 are regular zigzag apeirogons. These apeirogons lie in planes which cross through the reflection planes of both t0 and t1. When projected onto the plane of t0 they appear as

planar zigzags. Four zigzags meet at each vertex, (∞2.∞2.∞2.∞2), making the vertex ﬁgure

an antiprismatic square. The projection of this Wythoﬃan onto the reﬂection plane of t0

appears as {∞, 4}4

1 {0,1} P In this apeirohedron the faces of type F2 are straight lines. Each line corresponds to a zigzag of {∞, 4}#{∞} and is the line connecting the midpoints of the edges of the zigzag. {1,2} The faces of type F2 are antiprismatic squares. There are two squares and two lines alternating about each vertex, (4s.∞.4s.∞). The vertex figure is a crossed quadrilateral. This is a uniform apeirohedron. The projection of this Wythoffian onto the reflection plane 1 of t0 appears as the Wythoffian P of {∞, 4}4.

01 {0,1} 0 P In this apeirohedron the faces of type F2 are truncations of the faces of P . The faces of {1,2} type F2 are antiprismatic squares. There are two apeirogons and one square at each vertex, (4s.t∞2.t∞2). The resulting vertex figure is an isosceles triangle. The truncated zigzags are not regular polygons so this Wythoffian is not a uniform apeirohedron. The projection of this 01 Wythoffian onto the reflection plane of t0 appears as the Wythoffian P of {∞, 4}4.

Now consider an initial vertex which is stabilized by r2 and s0. The apeirogonal faces of this apeirohedron are compressed versions of the apeirogonal faces of the generic case. The {0} primary difference is that the edges of type F1 are now orthogonal to the reflection plane of t0. When projected onto the reflection plane of t0 these faces become straight lines. The faces {1,2} of type F2 are still antiprismatic squares. The vertex figure is still an isosceles triangle. {0,1} Consider another initial vertex which is invariant under r2 and s1. The faces of type F2 are again compressed versions of the apeirogons of the generic case. When projected onto

the reflection plane of t0 the apeirogonal faces appear as regular zigzags. This time the edges {1} {1} {1} of type F1 are orthogonal to the plane of t0. Here r2(F1 ) = F1 so there are no longer {1,2} any faces of type F2 . These edges are each incident to four apeirogons, so in this case the Wythoffian is not an apeirohedron. The vertex figure is then composed of two digons which share a common vertex.

If any of the above initial vertices are additionally held invariant under t0 or t1 then the Wythoﬃan changes very little. In the apeirogonal faces alternate edges are now parallel to

the reﬂection planes of t0 and t1, and in the ﬁnite faces the planarity of the faces changes

when the initial vertex is invariant under t1.

84 02 {0,1} {1,2} P In this apeirohedron the faces of type F2 are regular zigzags, the faces of type F2 are {0,2} convex squares lying parallel to the plane of t0, and the faces of type F2 are crossed quadrilaterals. At each vertex in cyclic order there are a crossed quadrilateral, a square, a crossed

quadrilateral, and an apeirogon, (4.4.4.∞2). The resulting vertex ﬁgure is a skew quadrilateral. The crossed quadrilateral faces are not regular so this is not a uniform apeirohedron.

The projection of this Wythoffian onto the reflection plane of t0 appears as the Wythoffian 02 P of {∞, 4}4.

Consider an initial vertex which is invariant under r1 and s0. The resulting apeirohedron is a compressed version of the generic P 02. The primary diﬀerence is that the crossed quadri-

laterals lie in planes orthogonal to the plane of t0. The vertex ﬁgure is a skew quadrilateral.

P 012 Lastly, we will look at the case where the initial vertex is transient under all distinguished {0,1} generators. In this apeirohedron the faces of type F2 are truncated zigzags. The faces of {1,2} {0,2} type F2 are skew octagons (truncated antiprismatic squares). The faces of type F2 are planar crossed quadrilaterals. There is one face of each type meeting at each vertex yielding

a triangular vertex figure with vertex symbol (4.t4s.t∞2). None of the faces are regular so P 012 is not a uniform apeirohedron. The projection of this Wythoffian onto the reflection 012 plane of t0 appears as the Wythoffian P of {∞, 4}4.

Consider an initial vertex which is invariant under s1 alone. Here again we have an apeirohedron which is just a slightly compressed version of the one corresponding to the generic {1} case. The main difference is that now the edges of type F1 are orthogonal to the plane of t0. The relative shape and position of the faces remain the same. While the faces of type {1,2} F2 are still skew octagons, now their projections onto the reflection plane of t0 appears as squares so they are no longer truncated antiprismatic squares. At each vertex there is still only one face of each type, accordingly, the vertex figure is still a triangle. If the initial vertex

is additionally invariant under s0 the Wythoffian behaves similarly. {0,1} Now consider an initial vertex which is invariant under s0 and t0. The faces of types F2 {1,2} 012 {0} {0} and F2 are the same as in the generic P . Here, r2(F1 ) = F1 so the faces of type {0,2} {0} F2 are compressed into single edges. As such, each edge of type F1 is incident to four faces and the figure is a semi-complex. The vertex figure is composed of two digons with a

common vertex. If the initial vertex is additionally invariant under s1 the Wythoﬃan behaves similarly.

In any of these cases if the initial vertex is additionally held invariant under t1 then the faces {1,2} of type F2 become convex. The relative shape and positioning of the other faces remains the same.

85 5.2 Blended apeirohedra related to the hexagonal tiling of the plane

5.2.1 {6, 3}#{}

We will now examine the blended apeirohedron {6, 3}#{}. Let the automorphism group of {6, 3} 6 3 2 be Γ({6, 3}) = hσ0, σ1, σ2i such that (σ0σ1) = (σ1σ2) = (σ0σ2) = ε. As in the last section, let 2 the automorphism group of the line segment be Γ({}) = hτ0i. Then (σiτ0) = ε for i = 0, 1, 2 in the direct product of the two groups. Then the automorphism group of the blend {6, 3}#{} is given by

Γ({6, 3}#{}) = hσ0τ0, σ1, σ2i, and its symmetry group is given by

G({6, 3}#{}) = hs0t0, s1, s2i =: hr0, r1, r2i.

Here r0 is a half-turn while r1 and r2 are plane reﬂections. When viewed on their own, s0, s1, s2, and t0 are all plane reﬂections.

When choosing an initial vertex we can ignore those vertices held invariant under t0. In such cases the Wythoffian will be the same as the corresponding Wythoffian from {6, 3}. This excludes P 1, P 2, and P 12. The initial vertices are chosen from the fundamental region of {6, 3}#{} which is an infinite cylinder over the fundamental triangle of {6, 3}.

Figure 5.3: The fundamental region of the blend {6, 3}#{}.

P 0 The ﬁrst Wythoﬃan is the regular apeirohedron {6, 3}#{} itself. The 2-faces are regular,

antiprismatic hexagons (isomorphic to {6}#{}), three of which meet at each vertex, (6s.6s.6s). The vertex ﬁgure is then a regular triangle. The projection of this Wythoﬃan onto the

reﬂection plane of t0 appears as {6, 3}.

01 {0,1} P In this apeirohedron the faces of type F2 are skew dodecagons (truncations of the faces {1,2} of {6, 3}#{}). The faces of type F2 are regular triangles which are all parallel to the reﬂection plane of t0. Two dodecagons and one triangle meet at each vertex giving a vertex

symbol (3.t6s.t6s). The vertex ﬁgure is an isosceles triangle. The truncated skew hexagons

86 are not regular so this is not a uniform apeirohedron. The projection of this Wythoffian onto 01 the reflection plane of t0 appears as the Wythoffian P of {6, 3}.

Now consider an initial vertex which is held invariant under s0 and r2. In this apeirohedron {0,1} the faces of type F2 are still skew dodecagons although now when projected onto the reflection plane of t0 they appear as regular, convex hexagons. The reason for this is that {0} {1,2} now the edges of type F1 are orthogonal to the reflection plane of t0. The faces of type F2 are still regular triangles. One triangle and two dodecagons meet at each vertex yielding an isosceles triangle for a vertex figure.

02 {0,1} P In this apeirohedron the faces of type F2 are regular, antiprismatic hexagons; the faces {1,2} {0,2} of type F2 are regular triangles; and the faces of type F2 are convex rectangles. At each vertex, in cyclic order, there is a rectangle, a hexagon, a rectangle, and a triangle with

vertex symbol (4.6s.4.3). The resulting vertex figure is a convex quadrilateral. For a specific {0,2} choice of initial vertex the faces of type F2 are squares and the Wythoffian is a uniform apeirohedron. The projection of this Wythoffian onto the reflection plane of t0 appears as the Wythoffian P 02 of {6, 3}. {1,2} Now choose some initial vertex which is stabilized by s0 and r1. The faces of type F2 are {0,2} regular triangles lying parallel to the reflection plane of t0; the faces of type F2 are convex {0} {0} rectangles which lie orthogonal to the reflection plane of t0. Since r1(F1 ) = F1 there is no {0,1} 2-face of type F2 . The skew hexagons from the previous figure have been compressed into this single edge, and as a result it is incident to six distinct rectangles making the Wythoffian a semi-complex. This is the only case in which P 02 is not an apeirohedron. The vertex figure consists of three isosceles triangles which all share a common vertex.

012 {0,1} P The resulting apeirohedron has three distinct types of 2-face. The faces of type F2 are {1,2} skew dodecagons (truncated skew hexagons). The faces of type F2 are convex hexagons {0,2} (truncated triangles) lying parallel to the reﬂection plane of t0. The faces of type F2 are convex rectangles which intersect the reﬂection plane of t0. One face of each type meets

at each vertex, with a triangular vertex figure and vertex symbol (4.6.t6s). The truncated hexagons are not regular so this Wythoffian is not a uniform apeirohedron. The projection 012 of this Wythoffian onto the reflection plane of t0 appears as the Wythoffian P of {6, 3}.

Finally, consider an initial vertex which is invariant under s0 but not under any of the distinguished generators. This apeirohedron will be very similar to the previous case. The faces are still skew dodecagons, convex hexagons, and convex rectangles. The difference is that the {0} edges of type F1 are orthogonal to the reflection plane of t0 so when the skew dodecagons are projected onto the reflection plane of t0 they appear as hexagons. The vertex figure is still triangular.

87 5.2.2 {∞, 3}6#{}

The Petrial of {6, 3}#{} is the regular apeirohedron {∞, 3}6#{}. Continue to let Γ({6, 3}) = hσ0, σ1, σ2i and Γ({}) = hτ0i. Then the automorphism group of {∞, 3}6#{} is

Γ({∞, 3}6#{}) = hσ0τ0σ2, σ1, σ2i, and the symmetry group is

G({∞, 3}6#{}) = hs0t0s2, s1, s2i =: hr0, r1, r2i.

The reflections r1 and r2 are still plane reflections as they were with {6, 3}#{}, but here r0 is a point reflection.

As with the last case, any point held invariant under t0 will yield the same ﬁgure as it would under the symmetry group of {∞, 3}6. Accordingly, we will restrict ourselves to points which are 1 2 12 not invariant under t0. This excludes P , P , and P . The points we choose are all from the fundamental region of {6, 3}#{}.

0 P The first Wythoffian is the regular apeirohedron {∞, 3}6#{} itself and the faces are regular zigzags. Three such zigzags meet at every vertex yielding a regular, triangular vertex figure

with vertex symbol (∞2.∞2.∞2). The projection of this Wythoﬃan onto the reﬂection plane

of t0 appears as {∞, 3}6.

01 {0,1} P In this apeirohedron the apeirogonal faces are of type F2 and appear as truncations of the {1,2} faces of {∞, 3}6#{}. The finite faces are of type F2 which appear as regular triangles. There are two truncated zigzags and one triangle at each vertex, (3.t∞2.t∞2). The vertex figure is an isosceles triangle. The truncated zigzags are not regular so this Wythoffian is

not a uniform apeirohedron. The projection of this Wythoffian onto the reflection plane of t0 01 appears as the Wythoffian P of {∞, 3}6.

Now consider a vertex which is invariant under r2 and s0. As with the previous case, this {0,1} is an apeirohedron with apeirogonal faces of type F2 and regular, triangular faces of {1,2} type F2 . The apeirogons are more compressed versions of the previous ones. They are still planar zigzags but now consecutive edges are orthogonal and their projection onto the

reﬂection plane of t0 appears as a straight line. The vertex ﬁgure is still an isosceles triangle.

02 {1,2} P In this apeirohedron the finite faces are regular triangles of type F2 and crossed quadri- {0,2} laterals of type F2 . The triangles are all parallel to the reflection plane of t0 and the crossed quadrilaterals all intersect that reflection plane at their centers. The infinite faces of {0,1} type F2 are regular zigzags. There is one crossed quadrilateral, one triangle, one crossed quadrilateral, and one zigzag meeting at each vertex, (4.3.4.∞2). The vertex figure is convex quadrilateral. The crossed quadrilaterals are not regular so the Wythoffian is not a uniform

88 apeirohedron. The projection of this Wythoffian onto the reflection plane of t0 appears as 02 the Wythoffian P of {∞, 3}6.

Now consider some initial vertex which is stabilized by r1 and s0. This ﬁgure is a compression of the previous one and it has similar face types. The most notable diﬀerences are that the

crossed quadrilaterals are now orthogonal to the reflection plane of t0 and that there are now three triangles, three zigzags, and six crossed quadrilaterals meeting at each vertex. The vertex figure is also very different in that it is now a collection of three non-intersecting rectangles. The vertex figure is disconnected so this is not an apeirohedron. This is the only case in which P 02 is not an apeirohedron.

012 {0,1} 02 P In this apeirohedron the faces of type F2 are truncations of the zigzags of P . For {1,2} finite faces there are convex hexagons of type F2 which lie parallel to the reflection plane {0,2} of t0, and there are crossed quadrilaterals of type F2 which interesect that reflection plane at their centers. There is one apeirogon, one crossed quadrilateral, and one hexagon

meeting at each vertex, (4.6.t∞2), yielding a vertex ﬁgure which is a triangle. The crossed quadrilaterals and truncated zigzags are not regular polygons so this Wythoﬃan is not a

uniform apeirohedron. The projection of this Wythoffian onto the reflection plane of t0 012 appears as the Wythoffian P of {∞, 3}6. Finally, consider an intitial vertex that is transient under all distinguished generators but

invariant under s0. This apeirohedron is a compressed version of the previous case. The faces are still apeirogons, hexagons, and crossed quadrilaterals. The crossed quadrilaterals now lie

orthogonal to the reﬂection plane of t0. This results in each vertex having a corresponding

vertex reﬂected across t0. The vertex ﬁgure is still a triangle.

5.2.3 {6, 3}#{∞}

We will now look at the blend of the hexagonal tessellation of the plane with a linear apeirogon. As before let the automorphism group of {6, 3} be Γ({6, 3}) = hσ0, σ1, σ2i and let the automorphism group of the apeirogon be Γ({∞}) = hτ0, τ1i. In the direct product of these groups we have 2 (σiτj) = ε for all i = 0, 1, 2 and j = 0, 1. Then the automorphism group of the blend becomes

Γ({6, 3}#{∞}) = hσ0τ0, σ1τ1, σ2i

6 3 2 such that (σ0τ0σ1τ1) = (σ1τ1σ2) = (σ0τ0σ2) = ε. In the blend of the two components, the reﬂection planes of the distinguished generators of {6, 3} are all orthogonal to the reﬂection planes of the distinguished generators of {∞}. The blended apeirohedron has symmety group

G({6, 3}#{∞}) = hs0t0, s1t1, s2i =: hr0, r1, r2i, where r0 and r1 are half-turns and r2 is a plane reﬂection.

89 Since the reﬂection planes of t0 and t1 are parallel to one another they have no points in 2 common. As such there is no point which is only transient under r2 so there is no P . This is the only restriction on choice of initial vertex beyond the fundamental region. Here, the fundamental region is a prism over the fundamental triangle of {6, 3}.

Figure 5.4: The fundamental region of the blend {6, 3}#{∞}.

P 0 The ﬁrst Wythoﬃan is the regular apeirohedron {6, 3}#{∞} itself. Each face is a regular helix about a hexagonal base. There are six helices around each hexagon of {6, 3}. Three of these helices are left-handed and three are right-handed. The three left-handed helices never intersect, nor do the three right-handed helices. Of these six helices, any pair of enantiomorphic ones will share every third vertex. Six helices meet at each vertex and the resulting vertex

ﬁgure is {3}#{}, a regular, prismatic hexagon with vertex symbol (∞6.∞6.∞6.∞6.∞6.∞6).

The projection of this Wythoﬃan onto the reﬂection plane of t0 appears as {6, 3}.

1 {0,1} P In this apeirohedron the faces of type F2 are regular helices about hexagonal bases. There are again three left-handed and three right-handed helices about each hexagon. Each pair of enantiomorphic helices will cross at the center of every third edge without sharing any {1,2} vertices. The faces of type F2 are regular, prismatic hexagons. Cyclically at each vertex there is a helix, a hexagon, a helix, and a hexagon, (6s.∞6.6s.∞6). The resulting vertex figure is a convex rectangle. This is a uniform apeirohedron. The projection of this Wythoffian onto 1 the reflection plane of t0 appears as the Wythoffian P of {6, 3}.

01 {0,1} P For this apeirohedron, the faces of type F2 are helices about dodecagonal bases (truncated hexagonal helices) with six helices (three left-handed and three right-handed) centered at each dodecagon. In each pair of enantiomorphic helices, the faces cross at the center of every sixth {0} edge. Each edge of type F1 is incident to two enantiomorphic helices centered over diﬀerent

90 {1,2} dodecagons. The faces of type F2 are regular, prismatic hexagons. There is one hexagon and two helices meeting at each vertex, (6s.∞12.∞12). The resulting vertex figure is an isosceles triangle. For a certain initial vertex the helices are regular helices over dodecagons and the Wythoffian is uniform. The projection of this Wythoffian onto the reflection plane 01 of t0 appears as the Wythoffian P of {6, 3}.

Consider an initial vertex which is stabilized by r2 and s0. In this apeirohedron the helices are now centered about regular hexagons, but there is a twelve vertex cycle between vertices {0} in the same relative position. This twelve vertex cycle occurs because the edges of type F1 are now orthogonal to the reflection plane of t0. The helices still occur in enantiomorphic pairs about each base hexagon and still cross at the center of every sixth edge. The faces of {1,2} type F2 are the same as in the generic case, and so is the relative arrangement of all the faces. The vertex figure is still an isosceles triangle. {1} If either of the previous initial vertices is held invariant under t1 then the edges of type F1 are parallel to the reflection plane of t1. As a result, each pair of enantiomorphic helices {1,2} are now both incident to every sixth edge. Additionally, the faces of type F2 are convex triangles lying parallel to the reflection plane of t0 rather than the skew hexagons they were in {1} the previous cases. Thus there are three faces incident to each edge of type F1 making this a semi-complex rather than an apeirohedron. Then there are four helices and one triangle at each vertex yielding two triangles incident to a common edge as the vertex figure.

Now consider an initial vertex which is invariant under r2 and s1. This ﬁgure is a semi- complex. Similar to the last case the helices spiral around regular hexagons in a twelve vertex {1} cycle. In this case the edges orthogonal to the base hexagon are those of type F1 . The {1} {1} enantiomorphic pairs of helices are each incident to every sixth edge. Since r2(F1 ) = F1 , {1,2} there is now no face of type F2 . As a result there are six helices incident to each edge of {1} type F1 . The resulting vertex ﬁgure consists of three digons such that one vertex is incident to all three.

02 {0,1} P In this semi-complex the faces of type F2 are regular helices spiraling about a regular hexagonal base. There are six helices (three right-handed and three left-handed) about each hexagon, enantiomorphic pairs of which coincide at every third vertex. The faces of type {1,2} {0,2} F2 are convex triangles lying in planes parallel to t0. The faces of type F2 are convex rectangles. There are two rectangles incident to each edge of the triangle making this Wythoﬃan a semi-complex and not an apeirohedron. At each vertex there is a triangle, two

rectangles, two helices, and another two rectangles, (3.4.4.∞6.∞6.4.4). The vertex figure consists of two intersecting skew quadrilaterals which share an edge. The projection of this 02 Wythoffian onto the reflection plane of t0 appears as the Wythoffian P of {6, 3}. {0,1} Consider an initial vertex which is invariant under r1 and s0. The faces of type F2 are straight lines orthogonal to the reflection plane of t0. The other face types are still triangles

and rectangles, though the rectangles are now orthogonal to the plane of t0. Each edge of the

91 triangles is incident to two rectangles, both of which lie in the same plane and are orthogonal to the triangle. The resulting semi-complex is composed of inﬁnite stacks of rectangles with spirals of triangles about each apeirogon. The vertex ﬁgure is a bipyramid centered over a regular hexagon with alternating faces missing.

12 {1,2} P In this apeirohedron the faces of type F2 are regular helices over regular hexagonal bases. About each hexagonal base there are three left-handed and three right-handed helices. The {1,2} faces of type F2 are truncated prismatic hexagons. There are two truncated prismatic hexagons and one helix meeting at each vertex, (t6s.t6s.∞6), yielding an isosceles triangle for the vertex ﬁgure. The truncated prismatic hexagons are not regular so this Wythoﬃan is

not a uniform apeirohedron. The projection of this Wythoffian onto the reflection plane of t0 appears as the Wythoffian P 12 of {6, 3}.

When the initial vertex is invariant under r0 and s1 the resulting figure is not an apeirohedron. {0,1} The faces of type F2 are now straight lines which are orthogonal to the reflection plane of {1,2} t0. Each face of type F2 traces the edge graph of a prism over regular triangle base such {1,2} that each edge connecting the base triangles is included twice in F2 . Then each vertex is incident to four edges within the face, two from one of the base triangles and two coinciding edges connecting the triangles. None of these “prisms” meet face to face, but each edge of {2} {1,2} type F1 is incident to two faces of type F2 . The vertex figure is a bipyramid over a regular hexagon.

P 012 In this apeirohedron we let the initial vertex be transient under all symmetries. The faces of {0,1} type F2 are helices around dodecagons (truncated helices over hexagons). There are six such helices, three right-handed and three left-handed, about each dodecagon. Enantiomor- {1,2} phic pairings of these helices cross at the center of every sixth edge. The faces of type F2 {0,2} are truncated prismatic hexagons. The faces of type F2 are convex rectangles. There is one rectangle, one truncated prismatic hexagon, and one helix at each vertex, (4.t6s.∞12), resulting in a triangular vertex figure. The truncated prismatic hexagons are not regular so this Wythoffian is not a uniform apeirohedron. The projection of this Wythoffian onto the 012 reflection plane of t0 appears as the Wythoffian P of {6, 3}.

Consider an initial vertex which is stabilized by s0 alone. In this apeirohedron, the ﬁnite faces are similar to those of the generic case, but the helices are now centered about hexagons even though they each have a twelve vertex cycle to get back to the same relative position in the {0} helix. This comes from the fact that the edges of type F1 are orthogonal to the plane of t0. Consequently, the rectangular faces are also orthogonal to the plane of t0. The vertex ﬁgure is still a triangle.

If either of the previous initial vertices is additionally invariant under t1 then the edges of {1} type F1 are parallel to the reﬂection plane of t0. The helices and rectangles are similar but {1,2} the faces of type F2 are now convex hexagons with double edges. As a result two helices and two rectangles are incident to each edge of the hexagon, so this is a semi-complex and

92 not an apeirohedron. The vertex figure of this semi-complex is composed of two triangles which share a common edge. {1} Now consider an initial vertex which is stabilized by s1 alone. Here the edges of type F1 are orthogonal to t0. The helices consist of twelve vertex cycles about a regular hexagonal {1,2} 12 base. Each face of type F2 is similar to the “prism”-like faces of the subcase of P . {0,2} Consequently, this Wythoffian is not an apeirohedron. The faces of type F2 are still {1,2} convex rectangles. There are two helices, two rectangles, and one face of type F2 at each vertex. The vertex figure is then composed of three adjacent triangles with a common vertex.

The ﬁnal choice of initial vertex we examine here is stabilized by s0 and s1, and again does {0} {1} not produce an apeirohedron. Edges of types F1 and F1 are orthogonal to the plane of {0,1} t0. Thus the faces of type F2 are linear apeirogons rather than helices. The faces of type {1,2} F2 are similar to the “prism”-like faces of the last case. The vertex ﬁgure is the edge graph of a bipyramid over a hexagon.

5.2.4 {∞, 3}6#{∞}

The ﬁnal apeirohedron based on the hexagonal tessellation of the plane is the Petrial of {6, 3}#{∞}.

Continue to let the automorphism group of {6, 3} be Γ({6, 3}) = hσ0, σ1, σ2i and let the automorphism group of the apeirogon be Γ({∞}) = hτ0, τ1i as before. The automorphism group of this apeirohedron is then

Γ({∞, 3}6#{∞}) = hσ0τ0σ2, σ1τ1, σ2i.

The blended apeirohedron has symmety group

G({6, 3}#{∞}) = hs0t0s2, s1t1, s2i =: hr0, r1, r2i, were r0 is a point reflection, r1 is a half-turn, and r2 is a plane reflection. There are some restrictions as to the choice of initial vertex beyond the fundamental region which is the same as with {6, 3}#{∞}. Since the reflection planes of t0 and t1 are parallel to one another they have no points in common. Thus there will be no point held invariant by both t0 and t1. Additionally any point held invariant under r0 is also held invariant under r2 so there are no 2 12 points held invariant under r0 alone. Together these exclude P and P .

0 P The initial Wythoﬃan is the regular apeirohedron {∞, 3}6#{∞} itself. The faces are regular

zigzags lying in planes which intersect the reﬂection planes of t0 and t1. When projected

onto the reﬂection plane of t0 these apeirogons appear as the regular zigzag faces of a copy of

{∞, 3}6. At each vertex of {∞, 3}6#{∞} there are two zigzags which project onto the same

zigzag of {∞, 3}6 in the reﬂection plane of t0. The two zigzags are reﬂections of each other

through t1. Since three such pairs of zigzags meet at each vertex, the vertex ﬁgure is {3}#{},

a regular, prismatic hexagon, with vertex symbol (∞2.∞2.∞2.∞2.∞2.∞2). The projection

of this Wythoﬃan onto the reﬂection plane of t0 appears {∞, 3}6.

93 1 {0,1} P In this apeirohedron the faces of type F2 are linear apeirogons. Each straight line corresponds to a zigzag of {∞, 3}6#{∞} such that the straight line bisects every edge of the {1,2} zigzag. The faces of type F2 are regular, prismatic hexagons which appear as {3}#{}. There are two apeirogons and two prismatic hexagons at each vertex, (6s.∞2.6s.∞2). The resulting vertex figure is a planar crossed quadrilateral. The faces are all regular polygons so the Wythoffian is a uniform apeirohedron. The projection of this Wythoffian onto the 1 reflection plane of t0 appears as the Wythoffian P of {∞, 3}6.

01 {0,1} P In this apeirohedron the faces of type F2 are truncations of the zigzag faces of {∞, 3}6#{∞}. {1,2} The faces of type F2 are prismatic hexagons. There are two apeirogons and one prismatic hexagon meeting at each vertex, (6s.t∞2.t∞2), yielding an isosceles triangle as the vertex ﬁg- ure. The truncated zigzag faces are not regular polygons so this Wythoﬃan is not a uniform

apeirohedron. The projection of this Wythoffian onto the reflection plane of t0 appears as 01 the Wythoffian P of {∞, 3}6. {0,1} Consider an initial vertex which is invariant under r2 and s1. The faces of type F2 are non-regular zigzags which when projected onto the reflection plane of t0 appear as regular {1} zigzag faces of a copy of {∞, 3}6. This is because the edges of type F1 are orthogonal to {1} {1} {1,2} the reflection plane of t0. Since r2(F1 ) = F1 there is no face of type F2 . As such, six {1} faces are incident to each edge of type F1 , thus the figure is a semi-complex and not an apeirohedron. The vertex figure is composed of three digons which share a common vertex. {0,1} Now consider an initial vertex which is stabilized by r2 and t1. The faces of type F2 are {1,2} truncated zigzags. The faces of type F2 are regular triangles which lie in planes parallel {1} to the reflection plane of t1. Each edge of type F1 is incident to one triangle and two apeirogons. Consequently, the figure is a semi-complex and not an apeirohedron. There are four apeirogons and one triangle at each vertex yielding two isosceles triangles with a common edge as the vertex figure.

02 {0,1} P In this Wythoffian the faces of type F2 are regular zigzag apeirogons. The faces of type {1,2} {0,2} F2 are triangles lying parallel to the plane of t0. The faces of type F2 are planar, crossed quadrilaterals. There are two crossed quadrilaterals and one triangle incident to each {2} edge of type F1 making this Wythoffian a semi-complex and not an apeirohedron. The vertex figure consists of two intersecting skew quadrilaterals which share an edge and the

vertex symbol is (3.4.4.∞2.∞2.4.4). The crossed quadrilaterals are not regular so this is not

a uniform apeirohedron. The projection of this Wythoffian onto the reflection plane of t0 02 appears as the Wythoffian P of {∞, 3}6.

012 P In this apeirohedron the initial vertex is transient under all generating reﬂections of G({∞, 3}6#{∞}). {0,1} {1,2} The faces of type F2 are truncated zigzags. The faces of type F2 are skew dodecagons {0,2} which appear as truncations of {3}#{}. The faces of type F2 are planar, crossed quadrilaterals. There is one face of each type at each vertex, (4.t6s.t∞2), yielding a triangular vertex

94 ﬁgure. The crossed quadrilaterals and truncated zigzags are not regular polygons so this is

not a uniform apeirohedron. The projection of this Wythoffian onto the reflection plane of t0 012 appears as the Wythoffian P of {∞, 3}6. {0,1} Consider an initial vertex which is invariant under s1 alone. The faces of type F2 and {0,2} 012 {1} F2 are the same as for the generic P . Here the edges of type F1 are orthogonal to {1,2} the reflection plane of t0. This causes each face of type F2 to compress into an object which appears as the edge graph of a prism over a triangle. It is a dodecagon whose edges trace the edge graph of a prism such that each vertex is hit twice and each edge connecting the triangular bases is also traced twice. Accordingly, this figure is not an apeirohedron. The {1,2} figure now has one face of type F2 , two zigzags, and two crossed quadrilaterals at each vertex. The vertex figure now consists of two triangles with a common vertex. {0,1} Here consider an initial vertex which is invariant under s0 and t0. The faces of type F2 {1,2} 012 {0} {0} and F2 are similar to those of the generic P . Now r2(F1 ) = F1 so there is no face {0,2} {0} of type F2 . The edges of type F1 are incident to two zigzags and two skew dodecagons so this figure is a semi-complex and not an apeirohedron. The resulting vertex figure is a quadrilateral.

Now consider an initial vertex which is held invariant under s0, s1, and t0. This figure is a combination of the previous two cases, and as such it is not an apeirohedron. The faces of {0,1} {1,2} type F2 are zigzags, the faces of type F2 are dodecagons whose edges trace the edge {0,2} graph of triangular prisms, and there are no faces of type F2 . There are six faces of type {1,2} F2 and six zigzags incident to each vertex. {0,1} Finally, consider an initial vertex which is stabilized by t1. The faces of type F2 and {0,2} 012 F2 are similar to the generic P and as with the generic case this Wythoffian is also an {1} apeirohedron. The edges of type F1 are parallel to the reflection plane of t0 and consequently {1,2} the faces of type F2 are convex hexagons which lie parallel to the plane of t0. There are two zigzags, two crossed quadrilaterals, and one hexagon at each vertex yielding a vertex figure which consists of two triangles which share an edge.

5.3 Blended apeirohedra related to the triangular tiling of the plane

5.3.1 {3, 6}#{}

This regular apeirohedron is the blend of the triangular tessellation of the plane, {3, 6}, with a line segment, {}. Let the automorphism group of the tessellation be Γ({3, 6}) = hσ0, σ1, σ2i such 3 6 2 that (σ0σ1) = (σ1σ2) = (σ0σ2) = ε. As before, let the automorphism group of the line segment 2 be Γ({}) = hτ0i. In the direct product of these two groups, (τ0σi) = ε for i = 0, 1, 2. Then the

95 automorphism group of the blended apeirohedron is

Γ({3, 6}#{}) = hσ0τ0, σ1, σ2i, and its symmetry group is

G({3, 6}#{}) = hs0t0, s1, s2i =: hr0, r1, r2i.

Here r0 is a half-turn while r1 and r2 are both plane reflections. As with the other blends with line segments we will ignore all cases where the initial vertex is stabilized by t0 since the resulting Wythoffian will be the same as the corresponding Wythoffian derived from {3, 6}. Accordingly, we will not examine P 1, P 2, and P 12. The initial vertices which we will look at will all be chosen from the fundamental region of {3, 6}#{} which is a one-sided infinite cylinder over the fundamental triangle of {3, 6}.

Figure 5.5: The fundamental region of the blend {3, 6}#{}.

P 0 The ﬁrst Wythoﬃan is the regular apeirohedron {3, 6}#{} itself. The faces of this apeirohedron are regular prismatic hexagons ({3}#{}). Six of these polygons meet at each vertex

yielding a convex, regular hexagon as the vertex ﬁgure with vertex symbol (6s.6s.6s.6s.6s.6s).

The projection of this Wythoﬃan onto the reﬂection plane of t0 appears as {3, 6}.

01 {0,1} P In this apeirohedron the faces of type F2 are skew dodecagons which are truncations of {1,2} the faces of {3, 6}#{}. The faces of type F2 are convex, regular hexagons which exist in parallel pairs on opposite sides of the reﬂection plane of t0. There are two dodecagons

and one hexagon at each vertex, (6.t6s.t6s), yielding an isosceles triangle for a vertex figure. Since the dodecagons are truncated prismatic hexagons they are not regular polygons and this Wythoffian is not a uniform apeirohedron. The projection of this Wythoffian onto the

96 01 reﬂection plane of t0 appears as the Wythoﬃan P of {3, 6}.

Now consider some initial vertex which is stabilized under r2 and s0 but no other generating {0,1} symmetries. The Wythoffian in this case is not a polyhedron. A face of type F2 shares its edges with the edge graph of a triangular prism where three edges are incident to each vertex violating the diamond condition. This face is a compressed version of the skew dodecagon of the previous initial vertex and similar to the “prism”-like faces of the subcases of P 12 and 012 {1,2} P of {6, 3}#{∞} and {∞, 3}6#{∞}. The faces of type F2 are still convex hexagons existing in parallel pairs on either side of the reflection plane of t0. The vertex figure is the edge graph of a pyramid over a square base.

02 {0,1} P In this apeirohedron the faces of type F2 are regular, prismatic hexagons. The faces of {1,2} type F2 are convex, regular hexagons which again occur in parallel pairs one on each side {0,2} of the reﬂection plane of t0. The faces of type F2 are convex rectangles which intersect the reﬂection plane of t0. There is a rectangle, a convex hexagon, a rectangle, and a prismatic

hexagon meeting at each vertex, (4.6.4.6s). The resulting vertex figure is a convex quadri- {0,2} lateral. When the initial vertex is chosen so that the faces of type F2 are squares, the Wythoffian is a uniform apeirohedron. The projection of this Wythoffian onto the reflection 02 plane of t0 appears as the Wythoffian P of {3, 6}.

Consider an initial vertex which is invariant under r1 and s0. This Wythoﬃan has no face {0,1} {0} of type F2 , only an edge of type F1 . The edge can be viewed as a compression of {1,2} the skew hexagon from the previous case. As before the faces of type F2 are regular, {0,2} convex hexagons which occur in pairs, and the faces of type F2 are convex rectangles. The rectangles are now orthogonal to the reﬂection plane of t0 and to the hexagons. Each pair of

hexagons on either side of the reflection plane of t0 is incident to six rectangles. This set of eight faces makes a prism over a regular hexagon. Each rectangular face belongs to two such {0} sets. As such, each edge of type F1 is incident to three rectangles and each edge of type {2} F1 is incident to one rectangle and two hexagons. The Wythoffian is a semi-complex whose projection onto the reflection plane of t0 appears as a copy of {6, 3}. The vertex figure is the edge graph of a pyramid over a triangle.

012 {0,1} P In this apeirohedron the faces of type F2 are skew dodecagons (truncated prismatic {1,2} hexagons). The faces of type F2 are convex dodecagons (truncated hexagons) which exist {0,2} in parallel pairs on either side of the reﬂection plane of t0. The faces of type F2 are convex rectangles which are bisected by the reﬂection plane of t0. There is one face of each type

meeting at each vertex giving a triangular vertex ﬁgure with vertex symbol (4.t6s.12). The truncated prismatic hexagons are not regular polygons so this Wythoﬃan is not a uniform

apeirohedron. The projection of this Wythoffian onto the reflection plane of t0 appears as the Wythoffian P 012 of {3, 6}. Finally we will look at an initial vertex which is transient under the distinguished generators {0,1} 012 but is invariant under s0. The faces of type F2 are compressed versions of the generic P

97 {0,1} 01 skew dodecagons. They are similar to the faces of type F2 of the subcase of P and as such this Wythoﬃan is not an apeirohedron. The other face types are the same as with the generic P 012, convex dodecagons and convex rectangles though the rectangles are orthogonal

to the reﬂection plane of t0. The vertex ﬁgure is composed of two isosceles triangles which share their base edge.

5.3.2 {∞, 6}3#{}

Now we will examine the Petrial of {3, 6}#{}. As before let Γ({3, 6}) = hσ0, σ1, σ2i and Γ({}) = hτ0i. Then the automorphism group of {∞, 6}3#{} is

Γ({∞, 6}3#{}) = hσ0τ0σ2, σ1, σ2i, and the symmetry group is

G({∞, 6}3#{}) = hs0t0s2, s1, s2i =: hr0, r1, r2i.

The generator r0 is a point reﬂection while r1 and r2 are plane reﬂections. Again we will ignore 1 2 12 all initial vertices which are invariant under t0, excluding P , P , and P . The initial vertices we will look at all come from the fundamental region used for {3, 6}#{}.

0 P The first Wythoffian is the regular apeirohedron {∞, 6}3#{} itself. The faces of this regular apeirohedron are regular zigzags. Each of these zigzags occurs in a pair with its reflection

through t0. Six zigzags meet at each vertex resulting in a regular, convex hexagon as the

vertex ﬁgure with vertex symbol (∞2.∞2.∞2.∞2.∞2.∞2). The projection of this Wythoﬃan

onto the reﬂection plane of t0 appears as {∞, 6}3.

01 {0,1} P In this apeirohedron the faces of type F2 are truncations of the apeirogonal faces of {1,2} {∞, 6}3#{}. The faces of type F2 are convex, regular hexagons. These hexagons exist in parallel pairs on either side of the reﬂection plane of t0. Two apeirogons and one hexagon meet

at each vertex, (6.t∞2.t∞2), yielding an isosceles triangle for the vertex figure. The truncated zigzags are not regular polygons so this apeirohedron is not uniform. The projection of this 01 Wythoffian onto the reflection plane of t0 appears as the Wythoffian P of {∞, 6}3.

Now consider some initial vertex which is invariant under r2 and s0. The apeirogons from the 0 {0,1} generic P are compressed a bit to form this Wythoffian’s faces of type F2 . Now the edges {0} of type F1 are orthogonal to the reflection plane of t0 and are incident to four faces of type {0,1} {1} F2 , but the edges of type F1 are still parallel to the reflection plane of t0. The faces of {1,2} type F2 are still convex, regular hexagons. As a result this figure is a semi-complex and not an apeirohedron. There are two hexagons and four zigzags at each vertex making the vertex figure into two isosceles triangles which share a vertex.

02 {0,1} {1,2} P In this apeirohedron the faces of type F2 are regular zigzags. The faces of type F2 are

98 convex, regular hexagons which exist in parallel pairs on either side of the reﬂection plane {0,2} of t0. The faces of type F2 are crossed quadrilaterals which intersect the reﬂection plane of t0 at their centers. At each vertex there is a quadrilateral, a zigzag, a quadrilateral, and

a hexagon, (4.∞2.4.6). The resulting vertex ﬁgure is a convex quadrilateral. The crossed quadrilateral faces are not regular polygons so the Wythoﬃan is not a uniform apeirohedron.

The projection of this Wythoffian onto the reflection plane of t0 appears as the Wythoffian 02 P of {∞, 6}3.

Now consider some initial vertex which is stabilized by r1 and s0. The faces are all the same types as with the generic P 02. Now the crossed quadrilaterals lie orthogonal to the {2} reflection plane of t0. Here each edge of type F1 is incident to two hexagons and one crossed quadrilateral, so this figure is a semi-complex and not an apeirohedron. There are three hexagons, three crossed quadrilaterals, and three zigzags at each vertex. The vertex figure is the edge graph of a triangular prism.

012 {0,1} {1,2} P In this apeirohedron the faces of type F2 are truncated zigzags. The faces of type F2 are convex dodecagons (truncated hexagons) which occur in parallel pairs on either side of the {0,2} reflection plane of t0. The faces of type F2 are crossed quadrilaterals which intersect the reflection plane of t0 at their centers. One face of each type meets at each vertex, (4.12.t∞2), and the vertex figure is a triangle. The truncated zigzags and crossed quadrilaterals are not regular polygons to this Wythoffian is not a uniform apeirohedron. The projection of this 012 Wythoffian onto the reflection plane of t0 appears as the Wythoffian P of {∞, 6}3. Finally, consider some initial vertex which is transient under all distinguished generators and

invariant under s0. The faces are of the same types as with the previous initial vertex. Now

the crossed quadrilaterals are orthogonal to the reflection plane of t0 and as a result each {2} edge of type F1 is incident to two dodecagons and one crossed quadrilateral. This figure is a semi-complex and not an apeirohedron. There are two dodecagons, two zigzags, and one crossed quadrilateral at each vertex. The vertex figure is the edge graph of two similar triangles which share an edge.

5.3.3 {3, 6}#{∞}

We will continue to let the automorphism group of {3, 6} be Γ({3, 6}) = hσ0, σ1, σ2i. Let the automorphism group of the apeirogon be Γ({∞}) = hτ0, τ1i. In the direct product of their groups, 2 (σiτj) = ε for i = 0, 1, 2 and j = 0, 1. The automorphism group of {3, 6}#{∞} is

Γ({3, 6}#{∞}) = hσ0τ0, σ1τ1, σ2i, and the symmetry group is

G({3, 6}#{∞}) = hs0t0, s1t1, s2i =: hr0, r1, r2i.

99 Here r0 and r1 are half-turns and r2 is a plane reﬂection. The only condition this imposes on our choice of initial vertex is that there is no point which is invariant under both t0 and t1, excluding P 2. Additionally the initial vertex must come from the fundamental region of {3, 6}#{∞} which is a prism over the fundamental triangle of {3, 6}.

Figure 5.6: The fundamental region of the blend {3, 6}#{∞}.

P 0 The ﬁrst Wythoﬃan is the regular apeirohedron {3, 6}#{∞} itself. This apeirohedron has regular helical faces that spiral about a triangular base. Six such helices meet at each vertex alternating between “right” and “left” handed helices so that helices over adjacent triangles in the tessellation {3, 6} have opposite orientation and share every third edge. When projected

onto the reﬂection plane of t0 the apeirohedron appears as {3, 6}. The vertex ﬁgure is a

regular antiprismatic hexagon with vertex symbol (∞3.∞3.∞3.∞3.∞3.∞3). The projection

of this Wythoﬃan onto the reﬂection plane of t0 appears as {3, 6}.

1 {0,1} P In this apeirohedron the faces of type F2 are regular helices over regular triangle bases. Each of these helices corresponds to a face of {3, 6}#{∞} and has a similar orientation. {1,2} The faces of type F2 are regular antiprismatic hexagons. The vertex configuration then consists of a hexagon, a helix, a hexagon, and a helix, (6s.∞3.6s.∞3). The vertex figure is crossed quadrilateral. All faces are regular so this Wythoffian is a uniform apeirohedron. The 1 projection of this Wythoffian onto the reflection plane of t0 appears as the Wythoffian P of {3, 6}.

01 {0,1} P This apeirohedron has helical faces over hexagons of type F2 which are truncations of the helical faces of {3, 6}#{∞}. Helices around adjacent hexagonal cylinders share every sixth {1,2} edge. The faces of type F2 are regular antiprismatic hexagons. There is one hexagon and two helices at each vertex, (6s.∞6.∞6), yielding an isosceles triangle as a vertex ﬁgure.

100 For a speciﬁc choice of initial vertex the helices are regular and the Wythoﬃan is a uniform

apeirohedron. The projection of this Wythoffian onto the reflection plane of t0 appears as the Wythoffian P 01 of {3, 6}.

Consider an initial vertex which is stabilized under r2 and s0. The resulting apeirohedron is similar to the generic P 01 in that its helices traverse a six vertex cycle before returning to the same relative position. Unlike the generic case, here the projection of the helix onto

the reflection plane of t0 is just a triangle, so these helices are not truncations of the faces {1,2} of {3, 6}#{∞}. The faces of type F2 are still regular antiprismatic hexagons. The vertex figure and vertex configuration remain the same.

If either of the previous two initial vertex choices is additionally held invariant under t1 then {1,2} the only substantial change is that the faces of F2 are now convex hexagons. All other faces and conﬁgurations remain the same and the Wythoﬃans are still apeirohedra.

Now consider an initial vertex which is invariant under r2 and s1. The figure is a semi- {0,1} complex and not an apeirohedron. The faces generated by F2 are helices over triangles whose vertices exist in six vertex cycles as in the previous case. There are no finite faces. Six {1} helices are incident to each edge of type F1 . The resulting vertex figure consists of three digons with a common vertex.

02 {0,1} P In this apeirohedron the faces of type F2 are regular helices over triangles; the faces of {1,2} {0,2} type F2 are convex, regular hexagons; and the faces of type F2 are convex rectangles. At each vertex there is a rectangle, a helix, a rectangle, and a hexagon, (4.∞3.4.6), giving a {0,2} quadrilateral as a vertex figure. For a specific choice of initial vertex the faces of type F2 are squares and the Wythoffian is a uniform apeirohedron. The projection of this Wythoffian 02 onto the reflection plane of t0 appears as the Wythoffian P of {3, 6}.

Now consider an initial vertex which is invariant under r1 and s0. This apeirohedron is a compressed version of the generic P 02. The helices from the previous case are now linear apeirogons. The hexagons and rectangles are the same as before. The vertex ﬁgure is still a quadrilateral.

12 {0,1} P In this apeirohedron the faces of type F2 are regular helices over triangles and the faces {1,2} of type F2 are skew dodecagons (truncated antiprismatic hexagons). There are two dodecagons and one helix meeting at each vertex, (t6s.t6s.∞3), yielding a triangular vertex figure. The dodecagons are truncated antiprismatic hexagons which are not regular polygons so P 12 is not a uniform apeirohedron. The projection of this Wythoffian onto the reflection 12 plane of t0 appears as the Wythoffian P of {3, 6}.

Consider an initial vertex which is stabilized by r0 and s1. The resulting figure is an apeirohedron. The helix from the generic P 12 compresses into a linear apeirogon. The faces of type {1,2} F2 are still skew dodecagons. The vertex configuration and vertex figure the same as in the generic case.

101 P 012 In this apeirohedron we use an initial vertex which is transient under all symmetries of {0,1} G({3, 6}#{∞}). The faces of type F2 are helices about hexagons (truncations of he- {1,2} lices over triangles), the faces of type F2 are skew dodecagons (truncated antiprismatic {0,2} hexagons), and the faces of type F2 are convex rectangles. One face of each type meets at each vertex of this apeirohedron yielding a triangular vertex ﬁgure with vertex symbol 012 (4.t6s.∞6). The truncated antiprismatic hexagons are not regular polygons so P is not

a uniform apeirohedron. The projection of this Wythoffian onto the reflection plane of t0 appears as the Wythoffian P 012 of {3, 6}. {1,2} Consider an initial vertex which is stabilized by either s0 or s1. The faces of type F2 and {0,2} F2 are still skew dodecagons and convex rectangles, respectively. The helix now consists of cycles of six vertices about a triangular base. The vertex figure is still triangular. Additionally

holding the initial vertex invariant under t1 or t0, respectively, generates a similar figure. The Wythoffian is an apeirohedron in all of these cases. {0,1} Finally, consider an initial vertex which is invariant under s0 and s1. The faces of type F2 {1,2} {0,2} are linear apeirogons. The faces of type F2 and F2 are again skew dodecagons and convex rectangles, respectively. The key difference between this apeirohedron and the others {0} {1} is that edges of type F1 and F1 are orthogonal to the reflection plane of t0. The vertex configuration and figure is the same as in the generic P 012.

5.3.4 {∞, 6}3#{∞}

This apeirohedron is the Petrial of {3, 6}#{∞}. We will continue to let the automorphism group of {3, 6} be Γ({3, 6}) = hσ0, σ1, σ2i and the automorphism group of the apeirogon be Γ({∞}) = hτ0, τ1i. The automorphism group of {∞, 6}3#{∞} is

Γ({∞, 6}3#{∞}) = hσ0τ0σ2, σ1τ1, σ2i, and the symmetry group is

G({∞, 6}3#{∞}) = hs0t0s2, s1t1, s2i =: hr0, r1, r2i.

Here r0 is a point reflection, r1 is a half-turn, and r2 is a plane reflection. These symmetries impose some conditions on which initial vertices can be used. There is no vertex which is invariant under t0 and t1. All points held invariant by r0 are also held invariant by 2 12 r2, thus there is no point held invariant under r0 alone. These conditions exclude P and P . The final condition on the initial vertex choice is that it must come from the fundamental region which is the same fundamental region as for {6, 3}#{∞}.

0 P The first Wythoffian is the regular apeirohedron {∞, 6}3#{∞} itself. The faces of type {0,1} F2 are regular zigzags, six of which meet at each vertex, (∞2.∞2.∞2.∞2.∞2.∞2). The projection of the faces of this apeirohedron onto the reflection plane of t0 is a copy of {∞, 6}3.

102 The vertex ﬁgure is a regular antiprismatic hexagon. The projection of this Wythoﬃan onto

the reﬂection plane of t0 appears as {∞, 6}3.

1 {0,1} {1,2} P In this apeirohedron the faces of type F2 are linear apeirogons. The faces of type F2 are regular antiprismatic hexagons. There are two hexagons and two apeirogons meeting at each

vertex, (6s.∞.6s.∞), yielding a crossed quadrilateral as the vertex ﬁgure. This is a uniform

apeirohedron. The projection of this Wythoffian onto the reflection plane of t0 appears as 1 the Wythoffian P of {∞, 6}3.

01 {0,1} P In this apeirohedron the faces of type F2 are truncations of the zigzag faces of {∞, 6}3#{∞}. {1,2} The faces of type F2 are regular antiprismatic hexagons. There are two apeirogons and one hexagon meeting at each vertex, (6s.t∞2.t∞2). The resulting vertex ﬁgure is an isosceles triangle. This apeirohedron is not uniform since the truncated zigzags are not regular

polygons. The projection of this Wythoffian onto the reflection plane of t0 appears as the 01 Wythoffian P of {∞, 6}3.

Consider an initial vertex which is invariant under r2 and s0. In this apeirohedron the faces of {0,1} {0} type F2 are zigzag apeirogons where the edges of type F1 are orthogonal to the reﬂection {1,2} plane of t0. They are not truncations of the faces of {∞, 6}3#{∞}. The faces of type F2 are regular antiprismatic hexagons. The vertex ﬁgure is still an isosceles triangle.

If either of the previous initial vertex choices is additionally held invariant under t1 the faces {1,2} of type F2 are convex hexagons. All other faces and conﬁgurations remain the same and the Wythoﬃans are still apeirohedrons.

Now consider an initial vertex which is stabilized under r2 and s1. The apeirogons of type {0,1} F2 are zigzags, again they are not truncations of the faces of {∞, 6}3#{∞}. The skew hexagons from before are now compressed into a single edge. As a result, there are six apeirogons incident each to such edge and this Wythoﬃan is not an apeirohedron. The vertex ﬁgure consists of three digons with a vertex in common.

02 {0,1} {1,2} P In this apeirohedron the faces of type F2 are regular zigzags, the faces of type F2 are {0,2} convex regular hexagons, and the faces of type F2 are crossed quadrilaterals. At each vertex there is a quadrilateral, a hexagon, a quadrilateral, and an apeirogon, (4.6.4.∞2). The resulting vertex figure is quadrilateral. The crossed quadrilaterals are not regular polygons so this Wythoffian is not a uniform apeirohedron. The projection of this Wythoffian onto the 02 reflection plane of t0 appears as the Wythoffian P of {∞, 6}3.

Consider an initial vertex which is stabilized by r1 and s0. In this apeirohedron the apeirogons {0,1} {1,2} of type F2 are linear apeirogons. The faces of type F2 are again convex hexagons and {0,2} the faces of type F2 are still crossed quadrilaterals. The vertex ﬁgure is still a quadrilateral.

012 {0,1} {1,2} P In this apeirohedron the faces of type F2 are truncated zigzags, the faces of type F2 {0,2} are truncated antiprismatic hexagons, and the faces of type F2 are crossed quadrilaterals.

103 There is one face of each type meeting at each vertex yielding a triangle for a vertex ﬁgure

with vertex symbol (4.t6s.t∞2). The truncated antiprismatic hexagons and truncated zigzags are not regular polygons so this Wythoffian is not a uniform apeirohedron. The projection of 012 this Wythoffian onto the reflection plane of t0 appears as the Wythoffian P of {∞, 6}3. {0,1} Consider an initial vertex which is invariant under s0 and t0. The faces of types F2 and {1,2} 012 F2 are of similar type as the ones from the generic P . Unlike in that case, here the crossed quadrilateral is compressed into a single edge. As a result there are two apeirogons and two dodecagons incident to each of those edges making this a semi-complex and not an apeirohedron. The vertex figure is two digons with a common vertex.

Consider an initial vertex which is stabilized by s0 and s1. In this apeirohedron the faces of {0,1} type F2 are zigzags which are not truncated zigzags, nor are they regular zigzags. The other face types are skew dodecagons and crossed quadrilaterals as before. The vertex figure is a triangle. {1,2} When the initial vertex is additionally invariant under t1 the faces of type F2 are convex. All other face types and configurations remain the same and the figures will still be

apeirohedra. When it is additionally invariant under t0 there are no fundamental changes.

5.4 Remarks on blended apeirohedra

As has been seen throughout this chapter there is a relationship between a Wythoffian of a regular planar apeirohedron and the corresponding Wytoffian of a regular blended apeirohedron based on that planar polyhedron. For P a regular planar apeirohedron with symmetry group G(P ) and L either a line segment or a linear apeirogon with symmetry group G(L), set Q := P #L, the blend of P and L with symmetry group G(Q). Let u be a point in the fundamental triangle of P such that u is transient under all generators of G(P ) with subscripts in some non-empty set E ⊆ {0, 1, 2}, and let v be a point in the fundamental region of L. Then (u, v) is an initial vertex of the Wythoffian,

WE(u, v), of Q. For each of the twelve blended apeirohedra in this chapter we have seen that when (u, v) is transient under all generators of G(Q) with subscipts in the same E as above, then the projection of WE(u, v) onto the plane of P is the Wythoffian WE(u) of P . When L is a line segment, then projection of WE(u, v) onto L is the line segment connecting v and r(v) with r ∈ G(L). When L is a linear apeirogon, the projection of WE(u, v) onto L is L itself. Thus, in a sense, a Wythoffian of P “blended” with a line segment or a linear apeirogon is a Wythoffian of P #{} or P #{∞}, respectively.

104 Chapter 6

Pure regular apeirohedra and their Wythoﬃans

In this chapter we examine the Wythoﬃans of the pure apeirohedra. These twelve apeirohedra are those which are not the blends of any regular polytopes. The automorphism groups of all twelve can be derived from that of the cubical honeycomb {4, 3, 4}. The automorphism group of the cubical honeycomb is

Γ({4, 3, 4}) = hτ0, τ1, τ2, τ3i

4 3 4 2 2 with the relations (τ0τ1) = (τ1τ2) = (τ2τ3) = (τiτj) = τi = ε for i, j ∈ {0, 1, 2, 3} with |i−j| ≥ 2 ([32], p. 231). We let the honeycomb have symmetry group

G({4, 3, 4}) = ht0, t1, t2, t3i with the same relative relations. Each generating reﬂection of a pure apeirohedron is found by combining the generating reﬂections of {4, 3, 4}. As in [32] (p. 231) the automorphism group of

{4, 6|4} is Γ({4, 6|4}) = hτ0, τ1τ3, τ2i which can then be used to ﬁnd the groups of the other pure apeirohedra. Below we show the relationships between the groups of all the pure apeirohedra ([32], p.224).

π δ π {∞, 4}6,4 ←→ {6, 4|4} ←→ {4, 6|4} ←→ {∞, 6}4,4 ↓ σ ↓ η ϕ2 (a) {∞, 4}·,∗3 {6, 6}4 ←→ {∞, 3} l π l π

δ ϕ2 (b) {6, 4}6 ←→ {4, 6}6 ←→ {∞, 3} ↓ σδ ↓ η π {∞, 6}6,3 ←→ {6, 6|3}

105 Figure 6.1: The fundamental region of {4, 3, 4}.

In Figure 6.1, the dashed edges outline the base cubic facet of {4, 3, 4}. The fundamental region of {4, 3, 4} is a simplex which lies within the base facet. In the fundamental region the front wall lies in the reflection plane of t0, the back wall lies in the reflection plane of t1, the top wall lies in the reflection plane of t2, and the bottom wall lies in the reflection plane of t3. The generating symmetries of the pure apeirohedra are composed of combinations of the reflections t0, t1, t2, t3. As such, the fundamental regions of the pure apeirohedra often have similarities with the fundamental region of {4, 3, 4}. I utilized Fricke and Klein’s construction ([1]) described in Secton 1.7 to find the fundamental region for each of the pure apeirohedra. The fundamental regions only depend on the underlying symmetry groups, not the combinatorics of the polyhedra. Many of the fundamental regions for the pure apeirohedra coincide and share walls with one another and with the fundamental region of {4, 3, 4} depending on how their symmetry groups are related. For instance, all polyhedra related through duality or Petrie duality have the same fundamental region. Taking advantage of this fact expedited the construction of the fundamental regions. These fundamental regions are not as intuitive as those belonging to the finite regular polyhedra, planar apeirohedra, or blended apeirohedra. To help clarify what the regions look like I have included pictures of the fundamental regions throughout this chapter. Often we choose the initial vertex of a Wythoffian from within the fundamental region so that the Wythoffian is geometrically similar to the original regular apeirohedron. This is not necessary and other choices of initial vertex lead to combinatorially isomorphic Wythoffians even though the geometry may be changed. For pictures of the faces, vertex figures, and Wythoffians see Tables 7.37-7.48 in the Appendix.

6.1 {4, 6|4}

The apeirohedron {4, 6|4} is one of the Petrie-Coxeter skew polyhedra. The realization has finite, convex faces and a finite, skew vertex figure. Based on the cubical honeycomb ([32], p. 231), the

106 automorphism group is

Γ({4, 6|4}) = hτ0, τ1τ3, τ2i and the symmetry group is

G({4, 6|4}) = ht0, t1t3, t2i =: hr0, r1, r2i,

4 6 2 with the relations (r0r1) = (r1r2) = (r0r2) = 1. Note that r0 and r2 are plane reflections, and that r1 is a halfturn. For the Wythoffians the initial vertices have all been chosen so that they are within the convex hull of the base face of {4, 6|4}. This choice leads to the resulting figures being more geometrically similar to {4, 6|4}. Other points in the fundamental region which belong to the same Wythoffian class generate combinatorially isomorphic figures, but the planarity of the faces may be different.

In the cases when the initial vertex is transient under r1, choosing a point outside of the convex {0,1} hull of a face of {4, 6|4} alters the planarity of the faces of type F2 but neither the regularity nor the number of vertices are changed. For pictures of the Wythoﬃans, their faces, and their vertex ﬁgures refer to Table 7.37 in the Appendix.

(a) Fundamental region in honeycomb. (b) Fundamental region in {4, 6|4}.

Figure 6.2: Figure (a) shows the fundamental region of {4, 6|4} within {4, 3, 4}. Figure (b) shows the fundamental region and the generating reﬂections of the symmetry group of {4, 6|4} within {4, 6|4}. The red plane is r0, the blue line is r1, and the green plane is r2. The base face of {4, 6|4} is outlined with dashed edges.

P 0 The resulting apeirohedron is {4, 6|4} itself. It has convex square faces, six of which meet at each vertex, and so the vertex configuration is (4.4.4.4.4.4). The faces incident to the initial vertex lie in three mutually orthogonal planes such that each face and its point reflection through the initial vertex lie in the same plane. The vertex figure is a regular, antiprismatic

107 hexagon.

1 {0,1} P In this apeirohedral Wythoffian the faces of type F2 are convex squares while the faces of {1,2} type F2 are regular, antiprismatic hexagons. Cyclically, at each vertex, there is a hexagon, square, hexagon, and a square. Thus the vertex configuration is (6s.4.6s.4) and the vertex figure is a rectangle. The faces are all regular polygons so this is a uniform apeirohedron.

P 2 The Wythoffian is the dual apeirohedron to {4, 6|4}, the regular apeirohedron {6, 4|4}. Each {1,2} face is a regular, convex hexagon of type F2 . Four come together at each vertex yielding an antiprismatic quadrilateral as the vertex figure with a vertex configuration (6.6.6.6).

01 {0,1} P With this apeirohedral Wythoffian, the faces of type F2 are convex octagons and the faces {1,2} of type F2 are regular, antiprismatic hexagons. The vertex configuration is (6s.8.8) and an isosceles triangle is the vertex figure. For a specific choice of initial vertex the octagons are regular and the Wythoffian is uniform. Note that for an initial vertex chosen outside of the convex hull of the base face of {4, 6|4}, the octagon would become a truncated antiprismatic quadrilateral which could not be made regular and so in this case the Wythoffian is not uniform.

02 {0,1} P In this apeirohedral Wythoffian the faces of type F2 are convex squares; the faces of {1,2} {0,2} type F2 are regular, convex hexagons; and the faces of type F2 are convex rectangles. Cyclically, at each vertex, there is a square, a rectangle, a hexagon, and a second rectangle giving a vertex configuration (4.4.6.4). The vertex figure is an antiprismatic quadrilateral. For certain initial vertex choices the rectangles can be made into squares making the Wythoffian uniform.

12 {0,1} {1,2} P In this apeirohedron, the faces of type F2 are convex squares and the faces of type F2 are skew dodecagons (truncated antiprismatic hexagons). The vertex configuration is (4.t6s.t6s) which corresponds to an isosceles triangle as the vertex figure. The skew dodecagons cannot be made regular by any vertex choice and thus this Wythoffian is not a uniform apeirohedron for any initial vertex choice.

P 012 Finally, consider some initial vertex which is transient under all symmetries of G({4, 6|4}). {0,1} {1,2} In this apeirohedron, the faces of type F2 are convex octagons, the faces of type F2 are {0,2} 12 skew dodecagons, and the faces of type F2 are convex rectangles. As with P the skew dodecagons are never regular so the apeirohedron is not uniform. There is one face of each

type at each vertex, (4.8.t6s), yielding a triangular vertex ﬁgure.

6.2 {6, 4|4}

This apeirohedron is the dual of {4, 6|4}. As such, its automorphism group is

Γ({6, 4|4}) = hτ2, τ1τ3, τ0i

108 and the corresponding symmetry group is

G({4, 6|4}) = ht2, t1t3, t0i =: hr0, r1, r2i

6 4 2 such that (r0r1) = (r1r2) = (r0r2) = 1. As with {4, 6|4} we will only consider initial vertices which are contained within the convex hull of the base face and the fundamental region of {6, 4|4}. As before, choosing the vertices in this way makes the faces of the Wythoffians more geometrically similar to the the faces of {6, 4|4}. The {1,2} planarity of the faces of type F2 may be different for those initial vertices which are transient under r1, but combinatorially they will be isomorphic. Due to the duality between {6, 4|4} and {4, 6|4} we can interchange 0 and 2 in the superscripts of the Wythoffians of {6, 4|4} and get the Wythoffians of {4, 6|4}. Images of the Wythoffians can be found in the Appendix in Table 7.38.

(a) Fundamental region in honeycomb. (b) Fundamental region in {6, 4|4}.

Figure 6.3: Figure (a) shows the fundamental region of {6, 4|4} within {4, 3, 4}. Figure (b) shows the fundamental region and the generating reﬂections of the symmetry group of {6, 4|4} within {6, 4|4}. The red plane is r0, the blue line is r1, and the green plane is r2. The base face of {6, 4|4} is outlined with dashed edges.

P 0 The initial Wythoffian is the regular apeirohedron {6, 4|4} itself whose faces are convex, regular hexagons. Four such hexagons meet at each vertex yielding a regular, antiprismatic quadrilateral as the vertex figure with vertex configuration (6.6.6.6).

1 {0,1} {1,2} P In this apeirohedron the faces of type F2 are convex hexagons and the faces of type F2 are regular, antiprismatic quadrilaterals. The vertex conﬁguration is (4s.6.4s.6) so the vertex ﬁgure is a rectangle. This is a uniform apeirohedron.

P 2 The resulting Wythoﬃan is the dual apeirohedron to {6, 4|4}, the regular apeirohedron {1,2} {4, 6|4}. The faces are convex squares of type F2 and there are six circling each vertex

109 with vertex conﬁguration (4.4.4.4.4.4). The vertex ﬁgure is a regular, antiprismatic hexagon.

01 {0,1} P In this apeirohedron the faces of type F2 are convex dodecagons (truncated hexagons) and {1,2} the faces of type F2 are regular, antiprismatic quadrilaterals. The vertex configuration is (4s.12.12) yielding an isosceles triangle as a vertex figure. For a carefully chosen initial vertex the dodecagons are regular and this Wythoffian is uniform.

02 {0,1} P For this apeirohedron, the faces of type F2 are convex, regular hexagons; the faces of type {1,2} {0,2} F2 are convex squares; and the faces of type F2 are convex rectangles. At each vertex there is a rectangle, a square, a rectangle, and a hexagon, in cyclic order yielding a vertex conﬁguration of (4.4.6.4). The vertex ﬁgure is then a convex quadrilateral. For certain choices of initial vertex the faces are all regular and the apeirohedron is uniform.

12 {0,1} P In this apeirohedron the faces of type F2 are convex hexagons and the faces of type {1,2} F2 are skew octagons (truncated antiprismatic quadrilaterals). The vertex configuration is (t4s.t4s.6) resulting in an isosceles triangle as a vertex figure. For an initial vertex choice outside the convex hull of the base face of {6, 4|4} the skew octagons would become convex octagons and the convex hexagons become skew and they could both be made regular. In this case the Wythoffian would be uniform.

P 012 Finally, let the initial vertex be transient under all symmetries of G({6, 4|4}). In the resulting {0,1} apeirohedron the faces of type F2 are convex dodecagons (truncated hexagons), the faces {1,2} of type F2 are skew octagons (truncated antiprismatic quadrilaterals), and the faces of {0,2} type F2 are convex rectangles. The vertex conﬁguration is (4.t4s.12) corresponding to a triangular vertex ﬁgure.

6.3 {∞, 6}4,4

The apeirohedron {∞, 6}4,4 is the Petrial of {4, 6|4} and its automorphism group is

Γ({∞, 6}4,4) = hτ0τ2, τ1τ3, τ2i.

The corresponding symmetry group is

G({∞, 6}4,4) = ht0t2, t1t3, t2i =: hr0, r1, r2i

6 2 such that (r1r2) = (r0r2) = 1. Now r0 and r1 are half-turns and r2 is a plane reflection. Here we will examine initial vertices which lie in the convex hull of the faces of {4, 6|4}. Choosing other initial vertices in the same fundamental region but not in that face will produce no fundamental combinatorial change in the nature of the Wythoffians produced though it may change the planarity of the faces. Furthermore, any point held invariant under r0 is also invariant under r2 excluding P 2 and P 12. For images of the Wythoffians see Table 7.39 in the Appendix.

110 (a) Fundamental region in honeycomb. (b) Fundamental region in {∞, 6}4,4.

Figure 6.4: Figure (a) shows the fundamental region of {∞, 6}4,4 within {4, 3, 4}. Figure (b) shows the fundamental region and the generating reﬂections of the symmetry group of {∞, 6}4,4 within {∞, 6}4,4. The red line is r0, the blue line is r1, and the green plane is r2. The base face of {∞, 6}4,4 is outlined with dashed edges.

0 P This Wythoﬃan is the regular apeirohedron {∞, 6}4,4 itself. The faces are regular helices about triangular bases. Six helices meet at each vertex giving a regular, antiprismatic hexagon

as the vertex ﬁgure and vertex conﬁguration (∞3.∞3.∞3.∞3.∞3.∞3).

1 {0,1} P In this apeirohedron the faces of type F2 are regular helices about triangular bases. The {1,2} faces of type F2 are regular, antiprismatic hexagons. There are two hexagons and two helices alternating about each vertex giving a vertex conﬁguration of (6s.∞3.6s.∞3) and a crossed quadrilateral for a vertex ﬁgure. Both face types are regular polygons so this is a uniform apeirohedron. For an initial vertex chosen from outside the base face of {4, 6|4} the {1,2} polyhedron is still uniform but the faces of type F2 are regular, convex hexagons.

01 {0,1} P In this apeirohedron the faces of type F2 are helices about hexagonal bases (truncated {1,2} helices over triangles) and the faces of type F2 regular, antiprismatic hexagons. The vertex configuration is (6s.∞6.∞6) giving an isosceles triangle as the vertex figure. When the initial vertex is chosen so that the helices are regular, the Wythoffian is uniform.

02 {0,1} P For this Wythoﬃan the faces of type F2 are regular helices about triangular bases; the {1,2} {0,2} faces of type F2 are convex, regular hexagons; and the faces of type F2 are crossed quadrilaterals. In this generic case when the initial vertex is invariant under r1 but not t0 the Wythoﬃan is an apeirohedron. Cyclically, at each vertex, there is a quadrilateral, a hexagon,

a quadrilateral, and a helix yielding a vertex configuration of (4.6.4.∞3). The resulting vertex {0,2} figure is a quadrilateral. The faces of type F2 are not regular for any choice of initial vertex so this Wythoffian can never be uniform.

111 Now consider an initial vertex which is invariant under r1 and t0. In this case the Wythoﬃan {0,1} is a non-polyhedral complex. The faces of type F2 are regular helices over triangles. The {1,2} {0} {2} faces of type F2 are convex hexagons. Since the base edges F1 and F1 are equal {0,2} there is now no face of type F2 . Consequently, there are four faces incident to each edge. There are four helices and four hexagons meeting at each vertex with a double edged, regular, antiprismatic quadrilateral as the vertex ﬁgure.

012 P Finally, let the initial vertex be transient under all the symmetries of G({∞, 6}4,4). In the {0,1} resulting Wythoffian the faces of type F2 are hexagonal helices (truncated helices over {1,2} triangles), the faces of type F2 are skew dodecagons (truncated antiprismatic hexagons), {0,2} and the faces of type F2 are crossed quadrilaterals. In this generic case where the initial vertex is not invariant under any reflections the Wythoffian is an apeirohedron. The vertex {1,2} configuration is (4.t6s.∞6) and the vertex figure is then a triangle. The faces of type F2 {0,2} and F2 are not regular for any choice of initial vertex so this Wythoffian is not uniform. {0,1} Now look at a point which is stabilized by t0 alone. The helices of type F2 go around a triangular base but while moving through the helix it takes six vertices to come back to one {1,2} in the same relative position. The faces of type F2 are still skew dodecagons. There are {0} {2} four faces incident to each edge of type F1 = F1 . The figure is a semi-complex and not an apeirohedron in this case. The vertex figure consists of two digons with a common vertex.

6.4 {∞, 4}6,4

This apeirohedron is the Petrial of {6, 4|4} and thus its automorphism group is

Γ({∞, 4}6,4) = hτ2τ0, τ1τ3, τ0i.

The corresponding symmetry group is then

G({∞, 4, }6,4) = ht2t0, t1t3, t0i =: hr0, r1, r2i

4 2 such that (r1r2) = (r0r2) = 1. Again r0 and r1 are half-turns and r2 is a plane reflection. Here we will examine initial vertices which lie in the convex hull of the base face of {6, 4|4}. Choosing other initial vertices in the same fundamental region but not in that face will produce no fundamental change in the combinatorial nature of the Wythoffians produced but may change the planarity of the faces. Furthermore, any point held invariant under r0 is also invariant under r2. Pictures of the Wythoffians can be found in Table 7.40 in the Appendix.

0 P The ﬁrst Wythoﬃan is the regular apeirohedron {∞, 4}6,4 itself. The faces are regular helices about triangular bases. There are four such helices which meet at each vertex, with

vertex conﬁguration (∞3.∞3.∞3.∞3). The resulting vertex ﬁgure is a regular, antiprismatic quadrilateral.

112 (a) Fundamental region in honeycomb. (b) Fundamental region in {∞, 4}6,4.

Figure 6.5: Figure (a) shows the fundamental region of {∞, 4}6,4 within {4, 3, 4}. Figure (b) shows the fundamental region and the generating reﬂections of the symmetry group of {∞, 4}6,4 within {∞, 4}6,4. The red line is r0, the blue line is r1, and the green plane is r2. The base face of {∞, 4}6,4 is outlined with dashed edges.

1 {0,1} P In this apeirohedron the faces of type F2 are again regular helices spiraling about triangular {1,2} bases while the faces of type F2 are regular, antiprismatic quadrilaterals. The vertex figure is (4s.∞3.4s.∞3) yielding a crossed quadrilateral vertex figure. The faces are all regular polygons so the Wythoffian is a uniform apeirohedron. There is an initial vertex outside of {1,2} the base face of {6, 4|4} which still yields a uniform polyhedron except the faces of type F2 are convex squares.

01 {0,1} P For this apeirohedron the faces of type F2 are truncations of the faces of {∞, 4}6,4 and {1,2} as such, appear as helices about hexagonal bases. The faces of type F2 are regular, antiprismatic quadrilaterals. There are two helices and one quadrilateral meeting at each

vertex, with vertex configuration (4s.∞6.∞6), yielding an isosceles triangle as the vertex figure. When the initial vertex is chosen so that the helices are regular helices about regular hexagons the Wythoffian is uniform.

02 {0,1} P With this Wythoﬃan the faces of type F2 are regular helices over triangular bases; the {1,2} {0,2} faces of type F2 are regular, antiprismatic quadrilaterals; and the faces of type F2 are crossed quadrilaterals. Meeting at each vertex is a crossed quadrilateral, a skew quadrilateral,

a crossed quadrilateral, and a helix yielding a vertex configuration of (4.4s.4.∞3). The vertex figure is then a planar quadrilateral. The crossed quadrilaterals prevent this Wythoffian from

ever being uniform. In the generic case when the initial vertex is stabilzed by r1 but not by

t2 the Wythoﬃan is an apeirohedron.

Now let the initial vertex be invariant under r1 and t2. The resulting complex is not an

113 02 {0,2} apeirohedron but is similar to the generic P except that now the faces of type F2 are single edges, which are compressions of the crossed quadrilaterals of the previous case. This is {0} {2} a result of F1 = F1 . Then there are two skew quadrilaterals and two helices sharing each edge and so the ﬁgure is a complex but not an apeirohedron. Six helices and six quadrilaterals meet at each vertex yielding a double edged hexagon as the vertex ﬁgure.

P 012 Here we look at an initial vertex which is not stabilized by any of the distinguished gener- {0,1} ators of G({∞, 4}6,4). In the resulting Wythoﬃan the faces of type F2 are helices over {1,2} hexagons (truncated helices over triangles), the faces of type F2 are skew octagons (trun- {0,2} cated antiprismatic quadrilaterals), and the faces of type F2 are crossed quadrilaterals. In this generic case when the initial vertex is not invariant under any symmetries, including t2,

the Wythoﬃan is an apeirohedron. One face of each type meets at each vertex, (4.t4s.∞6), yielding a triangular vertex ﬁgure. As with P 02 the crossed quadrilaterals prevent this apeirohedron from being uniform.

Finally, consider an initial vertex which is only held invariant under t2. This ﬁgure is a semi- {0,1} {1,2} complex but not an apeirohedron. The faces of type F2 and F2 are similar to their 012 {0,2} corresponding faces in the more generic P , but here there is no face of type F2 . This {0} {2} {0} is a result of F1 = F1 . Consequently, each edge of type F1 is incident to two octagons and two helices. The vertex ﬁgure is then composed of two digons with a common vertex.

6.5 {∞, 4}·,∗3

By performing the skewing operation on the automorphism group of {6, 4|4} we get the automorphism group of {∞, 4}·,∗3. Then, letting τ0, τ1, τ2, and τ3 be as before,

2 2 Γ({∞, 4}·,∗3) = hτ1τ3, τ2τ0, (τ1τ3τ0) i = hτ1τ3, τ2τ0, (τ1τ0) i.

The apeirohedron then has symmetry group

2 G({∞, 4}·,∗3) = ht1t3, t2t0, (t1t0) i =: hr0, r1, r2i

4 2 which is subject to the relations (r1r2) = (r0r2) = 1. Now all three generators ri are half-turns.

Note that there is no point which is stabilized by both r0 and r1. For choice of initial vertex we will focus on points lying in the fundamental region and when possible in the plane spanned by {1,2} the line reflections r1 and r2. We will use this restriction since it will force all faces F2 to be planar similar to the vertex figure of {∞, 4}·,∗3. Other points in the fundamental region will yield combinatorially similar figures but the planarity of the faces may differ. In P 12 we cannot use this {1,2} restriction so the faces of type F2 are not planar. For images of the facets and vertex figures of the Wythoffians and of the Wythoffians themselves see Table 7.41 in the Appendix.

0 P The ﬁrst Wythoﬃan is the regular apeirohedron {∞, 4}·,∗3 itself. The faces are regular helices

114 (a) Fundamental region in honeycomb. (b) Fundamental region in {∞, 4}·,∗3.

Figure 6.6: Figure (a) shows the fundamental region of {∞, 4}·,∗3 within {4, 3, 4}. Figure (b) shows the fundamental region and the generating reﬂections of the symmetry group of {∞, 4}·,∗3 within {∞, 4}·,∗3. The red line is r0, the blue line is r1, and the green line is r2. The base face of {∞, 4}·,∗3 is outlined with dashed edges.

over triangular bases. There are four such helices meeting at each vertex, (∞3.∞3.∞3.∞3), yielding a convex square as the vertex ﬁgure.

1 {0,1} P In this apeirohedron the faces of type F2 are regular helices about triangles and the faces {1,2} of type F2 are convex squares. There are two squares and two helices meeting at each vertex giving a vertex configuration of (4.∞3.4.∞3). This results in a regular, antiprismatic quadrilateral as the vertex figure. All the faces are regular polygons so the Wythoffian is a uniform apeirohedron.

01 {0,1} P With this apeirohedron the faces of type F2 are truncations of the helical faces of {∞, 4}·,∗3 {1,2} which makes them helices about hexagons. The faces of type F2 are convex squares. The vertex configuration is (4.∞6.∞6) yielding an isosceles triangle as the vertex figure. There is no initial vertex which makes the truncated helix regular, so this Wythoffian is never uniform.

02 {0,1} P For this apeirohedron the faces of type F2 are regular helices about triangular bases, the {1,2} {0,2} faces of type F2 are convex squares, and the faces of type F2 are skew quadrilaterals. Cyclically at each vertex there is a skew quadrilateral, a square, a skew quadrilateral, and a

helix giving vertex configuration (4s.4.4s.∞3). The vertex figure is then a skew quadrilateral. {0,2} For a carefully chosen initial vertex the faces of type F2 are regular and the Wytoffian is uniform.

12 {0,1} P In this apeirohedron the faces of type F2 are regular helices about triangles and the faces {1,2} of type F2 are skew octagons. The vertex conﬁguration is (8s.∞3.8s.∞3) giving a skew

115 quadrilateral as the vertex figure. For a specific initial vertex choice the faces are all regular and the Wythoffian is uniform.

P 012 Finally look at an initial vertex which is not stabilized by any of the generating symmetries {0,1} of G({∞, 4}·,∗3). For this apeirohedron the faces of type F2 are helices about hexagons {1,2} (truncations of helices about triangles), the faces of type F2 are convex octagons (trun- {0,2} cated squares), and the faces of type F2 are skew quadrilaterals. There is one face of each type meeting at each vertex, (4s.8.∞6), yielding a triangular vertex ﬁgure. For a speciﬁcally chosen initial vertex within the fundamental region but not within the plane spanned by

r1 and r2, the faces are all regular and the apeirohedron is uniform, but the faces of type {1,2} F2 are now antiprismatic octagons. If we allow an initial vertex choice from outside the {1,2} fundamental region we get another uniform apeirohedron, but the faces of type F2 are antiprismatic octagrams.

The regular apeirohedron {∞, 4}·,∗3 is one of three regular pure apeirohedra with only half-turns as generators of its symmetry group (the others are {∞, 3}(a) and {∞, 3}(b) and will be discussed later). Accordingly, {∞, 4}·,∗3 occurs in two enantiomorphic forms, one with left-handed helices and the other with right-handed helices. One form can be obtained from the other through a plane reflection. Similarly, the Wythoffians of {∞, 4}·,∗3 also occur in enantiomorphic pairs. The orientation of the helices in the Wythoffians matches the orientation of the helices of the copy of

{∞, 4}·,∗3 on which it is based.

6.6 {6, 6}4

By performing the halving operation on {4, 6|4} we get the regular apeirohedron {6, 6}4. It has automorphism group

Γ({6, 6}4) = hτ0τ1τ3τ0, τ2, τ1τ3i and symmetry group

G({6, 6}4) = ht0t1t3t0, t2, t1t3i =: hr0, r1, r2i

6 6 2 where (r0r1) = (r1r2) = (r0r2) = 1. Now r0 and r2 are half-turns and r1 is a plane reﬂection.

As with {∞, 4}·,∗3 the initial vertices are simply chosen from the fundamental region without 0 2 12 01 further restriction. Note that {6, 6}4 is self-dual and so P ' P and P ' P . Pictures of the Wythoﬃans can be found in Table 7.42 in the Appendix.

0 P The first Wythoffian is the regular apeirohedron {6, 6}4 itself. The faces are regular, antiprismatic hexagons. There are six antiprismatic hexagons meeting at each vertex yielding a convex, regular hexagon as a vertex figure.

1 {0,1} {1,2} P In this apeirohedron the faces of types F2 and F2 are regular, convex hexagons. The resulting ﬁgure is the regular {6, 4|4} with a regular, antiprismatic quadrilateral for the vertex

116 (a) Fundamental region in honeycomb. (b) Fundamental region in {6, 6}4.

Figure 6.7: Figure (a) shows the fundamental region of {6, 6}4 within {4, 3, 4}. Figure (b) shows the fundamental region and the generating reﬂections of the symmetry group of {6, 6}4 within {6, 6}4. The red line is r0, the blue plane is r1, and the green line is r2. The base face of {6, 6}4 is outlined with dashed edges.

ﬁgure and vertex symbol (6.6.6.6). The reason P 1 is {6, 4|4} is that this is the dual of {4, 6|4}

from which {6, 6}4 was obtained through the halving operation.

2 P Since {6, 6}4 is self-dual this Wythoﬃan is the regular apeirohedron {6, 6}4. The faces of {1,2} type F2 are regular antiprismatic hexagons with six meeting at each vertex. The vertex ﬁgure is a convex hexagon.

01 {0,1} P In this apeirohedron the faces of type F2 are skew dodecagons which appear as truncated {1,2} antiprismatic hexagons. The faces of type F2 are regular, convex hexagons. The vertex configuration is (6.t6s.t6s) yielding an isosceles triangle for a vertex figure. The skew dodecagons are not regular so the Wythoffian is not uniform.

02 {0,1} {1,2} P In this apeirohedron the faces of types F2 and F2 are both regular, antiprismatic {0,2} hexagons. The faces of type F2 are convex rectangles. At each vertex there are two rectangles and one hexagon of each type in the vertex configuration (4.6s.4.6s). The vertex {0,2} figure is then a planar rectangle. For a specific choice of initial vertex the faces of type F2 are squares and the Wythoffian is a uniform apeirohedron.

12 {0,1} P In this apeirohedron the faces of type F2 are regular, convex hexagons while the faces {1,2} of type F2 are skew dodecagons which appear as truncations of regular, antiprismatic hexagons. There are two dodecagons and one hexagon, (6.t6s.t6s), at each vertex yielding an isosceles triangle as the vertex ﬁgure. As with P 01 the dodecagons are not regular so the Wythoﬃan is not uniform.

117 P 012 Finally, let the initial vertex be a point which is not stabilized under any generating sym- {0,1} {1,2} metries of G({6, 6}4). In the resulting apeirohedron the faces of type F2 and F2 are both skew dodecagons which appear as truncations of antiprismatic hexagons. The faces of {0,2} type F2 are convex rectangles. There is one face of each type meeting at each vertex, (t6s.t6s.4), yielding a triangular vertex ﬁgure. Again, the skew dodecagons are not regular so the Wythoﬃan is not uniform.

6.7 {4, 6}6

The Petrial of {6, 6}4 is the apeirohedron {4, 6}6. Accordingly, the automorphism group is

2 Γ({4, 6}6) = h(τ0τ1τ3) , τ2, τ1τ3i.

2 2 Note that (τ0τ1τ3) = (τ0τ1) so the expression simpliﬁes to

2 Γ({4, 6}6) = h(τ0τ1) , τ2, τ1τ3i.

Then the corresponding symmetry group is

2 G({4, 6}6) = h(t0t1) , t2, t1t3i =: hr0, r1, r2i

4 6 2 with the relations (r0r1) = (r1r2) = (r0r2) = 1. Here r0 and r2 are half-turns and r1 is a plane reﬂection. Note that {4, 6}6 has the same symmetry group as {6, 6}4 so they share a fundamental region. Again, the only restriction on the initial vertex choices is that they must come from the fundamental region. For images of the Wythoﬃans, see Table 7.43 in the Appendix.

0 P The ﬁrst Wythoﬃan is the regular apeirohedron {4, 6}6 itself. The faces are regular, antipris-

matic quadrilaterals, with six meeting at each vertex, (4s.4s.4s.4s.4s.4s). The vertex ﬁgure is a regular, convex hexagon.

1 {0,1} {1,2} P In this apeirohedron the faces of type F2 are convex squares, and the faces of type F2 are regular, convex hexagons. The vertex configuration is (4.6.4.6) and the vertex figure is a skew quadrilateral. Each face is a regular polygon so the Wythoffian is a uniform apeirohedron.

2 P This apeirohedron is the dual to {4, 6}6, the regular apeirohedron {6, 4}6. The faces of type {1,2} F2 are regular, antiprismatic hexagons. Four hexagons meet at each vertex giving a convex square for the vertex ﬁgure.

01 {0,1} P In this apeirohedron the faces of type F2 are skew octagons which appear as truncations {1,2} of the faces of {4, 6}6. The faces of type F2 are convex, regular hexagons. There is one hexagon and two octagons at each vertex, (6.t4s.t4s), yielding an isosceles triangle as the vertex ﬁgure. The skew octagons are not regular so the Wythoﬃan is not uniform.

118 (a) Fundamental region in honeycomb. (b) Fundamental region in {4, 6}6.

Figure 6.8: Figure (a) shows the fundamental region of {4, 6}6 within {4, 3, 4}. Figure (b) shows the fundamental region and the generating reﬂections of the symmetry group of {4, 6}6 within {4, 6}6. The red line is r0, the blue plane is r1, and the green line is r2. The base face of {4, 6}6 is outlined with dashed edges.

02 {0,1} P In this apeirohedron the faces of type F2 are regular quadrilaterals, the faces of type {1,2} {0,2} F2 are regular hexagons, and the faces of type F2 are quadrilaterals. Different choices of initial vertex in the fundamental region (transient under r0 and r2) yield faces with different planarity. The number of edges and the regularity of the faces remains constant throughout, {0,2} as does the vertex configuration. Cyclically at each vertex there is one face of type F2 , {1,2} {0,2} {0,1} one face of type F2 , one face of type F2 , and one face of type F2 . The resulting vertex figure is always a convex quadrilateral. For certain choices of initial vertex the faces {0,2} of type F2 are regular and the Wythoffian is uniform. There are initial vertex choices 02 {0,1} {1,2} {0,2} that yield a uniform P with skew F2 and F2 and planar F2 , and there are initial 02 {0,1} {1,2} {0,2} vertex choices that yield a uniform P with planar F2 and F2 and skew F2 .

12 {0,1} {1,2} P With this apeirohedron the faces of type F2 are convex squares and the faces of type F2 are skew dodecagons which appear as truncated, antiprismatic hexagons. The vertex conﬁg-

uration is (4.t6s.t6s) giving an isosceles triangle as the vertex ﬁgure. The skew dodecagons are not regular so the Wythoﬃan is not uniform for any choice of initial vertex.

012 P Finally, let the initial vertex be transient under all generating symmetries of G({4, 6}6). Sim- ilarly to P 02, the particular choice of initial vertex within the fundamental region determines {0,1} {1,2} {0,2} the planarity and convexity of the faces of type F2 ,F2 , and F2 . For this apeirohe- {0,1} {1,2} dron the faces of type F2 are octagons (truncated squares), the faces of type F2 are {0,2} dodecagons (truncated hexagons), and the faces of type F2 are quadrilaterals. There is one face of each type surrounding each vertex yielding a triangular vertex ﬁgure. There is no {0,1} {1,2} initial vertex for which F2 and F2 are planar and have the same edge lengths. They will

119 only have the same edge lengths when at least one of them is skew. A skew dodecagon appearing as a truncated antiprismatic hexagon is not regular, similarly a skew octagon appearing as a truncated antiprismatic square is not regular. Thus P 012 is not uniform.

6.8 {6, 4}6

This apeirohedron is the dual to {4, 6}6 so its automorphism group is

2 Γ({6, 4}6) = hτ1τ3, τ2, (τ0τ1) i and its symmetry group is

2 G({6, 4}6) = ht1t3, t2, (t0t1) i =: hr0, r1, r2i,

6 4 2 with the relations (r0r1) = (r1r2) = (r0r2) = 1. All initial vertices are chosen from the fundamental region. Since {6, 4}6 is dual to {4, 6}6 they share a fundamental region. Furthermore, their Wythoffians are related in that interchanging the superscripts 0 and 2 in a Wythoffian of one gives a Wythoffian of the other. For images of the Wythoffians see Table 7.44 in the Appendix.

(a) Fundamental region in honeycomb. (b) Fundamental region in {6, 4}6.

Figure 6.9: Figure (a) shows the fundamental region of {6, 4}6 within {4, 3, 4}. Figure (b) shows the fundamental region and the generating reﬂections of the symmetry group of {6, 4}6 within {6, 4}6. The red line is r0, the blue plane is r1, and the green line is r2. The base face of {6, 4}6 is outlined with dashed edges.

0 P The ﬁrst Wythoﬃan is the regular apeirohedron {6, 4}6 itself. The faces are regular, antipris-

matic hexagons. There are four such hexagons meeting at each vertex, (6s.6s.6s.6s), yielding a convex square as the vertex ﬁgure.

120 1 {0,1} P In this apeirohedron the faces of type F2 are regular, antiprismatic hexagons while the {1,2} faces of type F2 are convex squares. Alternating around each vertex are two squares and two hexagons giving a vertex conﬁguration of (4.6s.4.6s). The vertex ﬁgure is then a skew quadrilateral. All faces are regular polygons so this is a uniform apeirohedron.

2 P This Wythoﬃan is {4, 6}6, the dual apeirohedron to {6, 4}6. The faces, which are all of type {1,2} F2 , are regular, antiprismatic quadrilaterals such that six meet at each vertex. The vertex ﬁgure is a regular, convex hexagon.

01 {0,1} P In this apeirohedron the faces of type F2 are skew dodecagons which appear as trun- {1,2} cated, antiprismatic hexagons. The faces of type F2 are convex squares. There are two dodecagons and one square meeting at each vertex, (4.t6s.t6s), yielding an isosceles triangle as a vertex ﬁgure. The skew dodecagons are not regular so the Wythoﬃan is not uniform.

02 {0,1} {1,2} P In this apeirohedron the faces of type F2 are regular hexagons; the faces of type F2 are {0,2} squares; and the faces of type F2 are quadrilaterals. At each vertex there is a hexagon, a quadrilateral, a square, and a quadrilateral yielding a convex quadrilateral as the vertex figure. The planarity of the faces is different for different choices of initial vertex within the {0,2} fundamental region. For certain initial vertices the faces of type F2 are squares and the Wythoffian is uniform. There are initial vertex choices that yield a uniform P 02 with skew {0,1} {1,2} {0,2} F2 and F2 and planar F2 , and there are initial vertex choices that yield a uniform 02 {0,1} {1,2} {0,2} P with planar F2 and F2 and skew F2 .

12 {0,1} P For this apeirohedron the faces of type F2 are convex hexagons and the faces of type {1,2} F2 are skew octagons (truncated, antiprismatic quadrilaterals). There are two octagons and one hexagon meeting at each vertex, (6.t4s.t4s), yielding an isosceles triangle as the vertex ﬁgure. The truncated antiprismatic squares are not regular polygons so this Wythoﬃan is not uniform.

012 02 P Finally, let the initial vertex be transient under all symmetries of G({6, 4}6). As with P the exact choice of initial vertex determines the planarity of the faces. In this apeirohedron the {0,1} {1,2} faces of type F2 are truncated hexagons, the faces of type F2 are truncated squares, {0,2} and the faces of type F2 are quadrilaterals. One face of each type meets at each vertex {0,1} {1,2} yielding a triangular vertex ﬁgure. There is no initial vertex for which F2 and F2 are planar and have the same edge lengths. They will only have the same edge lengths when at least one of them is skew. A skew dodecagon appearing as a truncated antiprismatic hexagon is not regular, similarly a skew octagon appearing as a truncated antiprismatic square is not regular. Thus P 012 is not uniform.

121 6.9 {6, 6|3}

By performing the halving operation on {4, 6}6 we arrive at the self-dual {6, 6|3}, one of the Petrie- Coxeter polyhedra [32]. Its automorphism group is then

2 2 Γ({6, 6|3}) = h(τ0τ1) τ2(τ0τ1) , τ1τ3, τ2i.

The symmetry group is

2 2 G({6, 6|3}) = h(t0t1) t2(t0t1) , t1t3, t2i =: hr0, r1, r2i.

6 6 2 The generators of the symmetry group satisfy the relations (r0r1) = (r1r2) = (r0r2) = 1. In particular, r0 and r2 are plane reflections and r1 is a half-turn. The initial vertices are chosen from where the fundamental region coincides with the base face of {6, 6|3}. We place this restriction on the initial vertex choices to make the geometry of the Wythoffian similar to the geometry of {6, 6|3}. Note that the self-duality of {6, 6|3} implies that P 0 ' P 2 and P 12 ' P 01. Images of the Wythoffians can be found in Table 7.45 in the Appendix.

(a) Fundamental region in honeycomb. (b) Fundamental region in {6, 6|3}.

Figure 6.10: Figure (a) shows the fundamental region of {6, 6|3} within {4, 3, 4}. Figure (b) shows the fundamental region and the generating reﬂections of the symmetry group of {6, 6|3} within {6, 6|3}. The red plane is r0, the blue line is r1, and the green plane is r2. The base face of {6, 6|3} is outlined with dashed edges.

P 0 This Wythoﬃan is the regular apeirohedron {6, 6|3} itself. It has regular, convex hexagons for faces. Six such hexagons meet at each vertex yielding a regular, antiprismatic hexagon for the vertex ﬁgure.

1 {0,1} P In this apeirohedron the faces of type F2 are regular, convex hexagons while the faces

122 {1,2} of type F2 are regular, antiprismatic hexagons. Alternating about each vertex are two skew hexagons and two convex hexagons so the vertex configuration is (6.6s.6.6s). The vertex figure is then a convex rectangle. All faces are regular so this Wythoffian is uniform.

P 2 In this case the resulting ﬁgure is again the regular apeirohedron {6, 6|3}, thanks to the self-duality of {6, 6|3}.

01 {0,1} P In this apeirohedron the faces of type F2 are convex dodecagons (truncated hexagons) and {1,2} the faces of type F2 are regular, antiprismatic hexagons. There are two dodecagons and one hexagon meeting at each vertex, (6.t6s.t6s), yielding an isosceles triangle as the vertex figure. For a specific choice of initial vertex the dodecagons are regular and the Wythoffian is uniform.

02 {0,1} P In this apeirohedron the faces of type F2 are regular, convex hexagons; the faces of type {1,2} F2 are regular hexagons (convex or antiprismatic depending on the initial vertex); and the {0,2} faces of type F2 are convex rectangles. At each vertex there is a hexagon of the ﬁrst type, a rectangle, a hexagon of the second type, and another rectangle giving a vertex conﬁguration

of (6.4.6s.4). The resulting vertex figure is a skew quadrilateral. If a certain initial vertex is {0,2} chosen the faces of type F2 are squares and the Wythoffian is uniform. With this specific {1,2} initial vertex the faces of type F2 are convex hexagons.

12 {0,1} P In this apeirohedron the faces of type F2 are regular, convex hexagons. The faces of {1,2} type F2 are skew dodecagons which appear as the truncations of regular, antiprismatic hexagons. There are two dodecagons and one hexagon at each vertex yielding an isosce-

les triangle as the vertex figure corresponding to the vertex configuration (6.t6s.t6s). The dodecagons are not regular so the Wythoffian is not uniform.

P 012 Now consider an initial vertex which is not stabilized by any generating symmetry of G({6, 6|3}). {0,1} In the apeirohedron the faces of type F2 are convex dodecagons (truncated hexagons), the {1,2} faces of type F2 are skew dodecagons (truncated, anstiprismatic hexagons), and the faces {0,2} of type F2 are convex rectangles. The vertex configuration is (4.6.t6s) yielding a triangular vertex figure. As before, the skew dodecagons are not regular so the Wythoffian is not uniform.

6.10 {∞, 6}6,3

The Petrial of {6, 6|3} is the apeirohedron {∞, 6}6,3. Its automorphism group is then

2 2 Γ({∞, 6}6,3) = h[(τ0τ1) τ2] , τ1τ3, τ2i.

The symmetry group is

2 2 G({∞, 6}6,3) = h[(t0t1) t2] , t1t3, t2i =: hr0, r1, r2i,

123 and has the same fundamental region as {6, 6|3} by Petrie duality. The generators r0 and r1 are half-turns and r2 is a plane reﬂection. Due to the nature of the symmetries, there is no point which is stabilized by r0 and r1. The initial vertices are chosen from the fundamental region in such a way that they are also in the base face of {6, 6|3}. Choosing other initial vertices from the fundamental region will result in combinatorially similar objects but the planarity of the faces may change. Additionally, any point which is stabilized by r0 is also stabilized by r2. For pictures of the Wythoﬃans see Table 7.46 in the Appendix.

(a) Fundamental region in honeycomb. (b) Fundamental region in {∞, 6}6,3.

Figure 6.11: Figure (a) shows the fundamental region of {∞, 6}6,3 within {4, 3, 4}. Figure (b) shows the fundamental region and the generating reﬂections of the symmetry group of {∞, 6}6,3 within {∞, 6}6,3. The red line is r0, the blue line is r1, and the green plane is r2. The base face of {∞, 6}6,3 is outlined with dashed edges.

0 P The ﬁrst Wythoﬃan is the regular apeirohedron {∞, 6}6,3 itself. The faces are regular helices

about square bases. There are six helices meeting at each vertex, (∞4.∞4.∞4.∞4.∞4.∞4), yielding a regular, antiprismatic hexagon as the vertex ﬁgure.

1 {0,1} P In this apeirohedron the faces of type F2 are regular helices over square bases. The faces {1,2} of type F2 are regular, antiprismatic hexagons. Alternating about each vertex are two helices and two hexagons with vertex configuration (6s.∞4.6s.∞4). The vertex figure is a planar crossed quadrilateral. Since all faces are regular polygons the Wythoffian is a uniform apeirohedron. For an initial vertex chosen outside of the base face of {6, 6|3} this polyhedron {1,2} is still uniform but the faces of type F2 are regular convex hexagons.

01 {0,1} P In this apeirohedron the faces of type F2 are helices over octagons (truncations of the {1,2} helices over squares). The faces of type F1 are regular, antiprismatic hexagons. The vertex conﬁguration is (6s.∞8.∞8) yielding an isosceles triangle as the vertex ﬁgure. For a

124 speciﬁc initial vertex the helices are regular helices over regular octagons and the Wythoﬃan is a uniform apeirohedron.

02 {0,1} P For this Wythoﬃan the faces of type F2 are regular helices over square bases; the faces of {1,2} {0,2} type F2 are regular, antiprismatic hexagons; and the faces of type F2 are planar, crossed quadrilaterals. Cycling about each vertex are a quadrilateral, a hexagon, a quadrilateral,

and a helix giving a vertex configuration of (4.6s.4.∞4). The vertex figure is then a skew quadrilateral. The crossed quadrilaterals are not regular for any choice of initial vertex so the resulting figure is not uniform. This Wythoffian is an apeirohedron, except when the initial 2 2 vertex is also stabilized by (t0t1) t2(t0t1) . 2 2 Now consider an initial vertex which is stabilized by r1 and (t0t1) t2(t0t1) but not by r0 {0,1} or r2. In this non-polyhedral complex the faces of type F2 are still regular helices over {1,2} squares and the faces of type F2 are still regular, skew hexagons. Here the base edges {0} {2} {0,2} F1 and F1 are the same, and so the base face F2 gets compressed into a single edge. There are two helices and two hexagons incident to each edge making the figure a complex but not an apeiorhedron. The figure has a double edged antiprismatic hexagon as the vertex figure.

012 P Let the initial vertex be transient under all generating symmetries of G({∞, 6}6,3). For this {0,1} Wythoffian the faces of type F2 are helices about octagons (truncations of helices over {1,2} squares), the faces of type F2 are skew dodecagons (truncated antiprismatic hexagons), {0,2} and the faces of type F2 are crossed quadrilaterals. There is one face of each type meeting at each vertex, (4.t6s.∞8), resulting in a triangular vertex figure. The crossed quadrilaterals and the skew dodecagons are not regular so the Wythoffian is not uniform. This Wythoffian 2 2 is an apeirohedron, except when the initial vertex is stabilized by (t0t1) t2(t0t1) . 2 2 Finally, consider an initial vertex which is stabilized by (t0t1) t2(t0t1) but is transient under

all distinguished generators of G({∞, 6}6,3). The resulting figure is a semi-complex rather {0,1} than an apeirohedron. The face F2 is a little different in this case; it is a helix whose base is a square. As with all helices it behaves cyclically, repeating itself every eight vertices similarly 012 {1,2} to the helices for the more generic P . The faces of type F2 are skew dodecagons. Here, {0} {2} {0,2} F1 = F1 so the faces of type F2 get compressed into edges which are then incident to {0,1} {1,2} two faces of type F2 and two faces of type F2 . This semi-complex has a vertex figure consisting of two double edges with a common vertex.

6.11 {∞, 3}(a)

There are two regular apeirohedra of this equivelar type that diﬀer in the nature of their helical faces. We distinguish between them by labeling them {∞, 3}(a) and {∞, 3}(b). First, we will look at {∞, 3}(a). For this apeirohedron the helical faces are over triangles.

125 The automorphism group of this apeirohedron is found by performing the facetting operation on {6, 6}4. The group is then

(a) Γ({∞, 3} ) = hτ0τ1τ3τ0, τ2τ1τ3τ2, τ1τ3i.

Its symmetry group is

(a) G({∞, 3} ) = ht0t1t3t0, t2t1t3t2, t1t3i =: hr0, r1, r2i, where the three generators are half-turns. Note that there is no point which is stabilized by both r0 and r1. The initial vertices are chosen from within the fundamental region. Pictures of the Wythoﬃans can be found in Table 7.47 in the Appendix.

(a) Fundamental region in honeycomb. (b) Fundamental region in {∞, 3}(a).

Figure 6.12: Figure (a) shows the fundamental region of {∞, 3}(a) within {4, 3, 4}. Figure (b) shows the fundamental region and the generating reﬂections of the symmetry group of {∞, 3}(a) within (a) (a) {∞, 3} . The red line is r0, the blue line is r1, and the green line is r2. The base face of {∞, 3} is outlined with dashed edges.

P 0 The ﬁrst Wythoﬃan is the regular apeirohedron {∞, 3}(a) itself. The faces are regular helices

about triangular bases with three such helices meeting at each vertex, (∞3.∞3.∞3). The vertex ﬁgure is then a regular triangle.

1 {0,1} P In this apeirohedron the faces of type F2 are regular helices about triangles and the faces {1,2} of type F2 are regular triangles. Alternating about each vertex are two triangles and two

126 helices, (3.∞3.3.∞3). The vertex ﬁgure is a skew quadrilateral. All faces are regular polygons so this is a uniform apeirohedron.

01 {0,1} P In this apeirohedron the faces of type F2 are helices about hexagonal bases which appear (a) {1,2} as truncations of the faces of {∞, 3} . The faces of type F2 are regular triangles. The vertex configuration is (3.∞6.∞6) yielding an isosceles triangles as a vertex figure. There is no initial vertex choice which makes the truncated helices regular so this Wythoffian is not uniform.

02 {0,1} P In this apeirohedron the faces of type F2 are helices about triangular bases, the faces of {1,2} {0,2} type F2 are regular triangles, and the faces of type F2 are skew quadrilaterals. Cycling about each vertex are a triangle, a quadrilateral, a helix, and a quadrilateral giving a vertex

configuration of (3.4s.∞3.4s). The vertex figure is a skew quadrilateral. If a certain initial {0,2} vertex is chosen then the faces of type F2 are regular and the Wythoffian is uniform.

12 {0,1} P In this apeirohedron the faces of type F2 are regular helices about triangular bases and {1,2} the faces of type F2 are antiprismatic hexagons. There are two hexagons and one helix at each vertex, (6s.6s.∞3) yielding an isosceles triangle as the vertex ﬁgure. For a certain choice of initial vertex the skew hexagons are regular and then the Wythoﬃan is uniform.

P 012 Finally let the initial vertex be such that it is transient under all distinguished generators of (a) {0,1} G({∞, 3} ). For this apeirohedron the faces of type F2 are helices about hexagonal bases {1,2} (truncations of helices over triangles), the faces of type F2 are convex hexagons (truncated {0,2} triangles), and the faces of type F2 are skew quadrilaterals. The vertex configuration is (4s.6.∞6) yielding a triangular vertex figure. As with the previous Wythoffians, for a carefully chosen initial vertex the faces can all be made regular, and in this case the Wythoffian is a uniform aperiohedron. One note to make about the uniform version of this Wythoffian, {1,2} the planarity of the face of type F2 changes. In the uniform apeirohedron it is a regular prismatic hexagon.

(a) As with {∞, 4}·,∗3, since the generators of {∞, 3} are all half-turns this apeirohedron occurs in two enantiomorphic forms. The Wythoﬃans of {∞, 3}(a) also occur in enantiomorphic pairs. In one of the forms the helical facets are left-handed and in the other form the helical facets are right-handed.

6.12 {∞, 3}(b)

The last apeirohedron we will examine is {∞, 3}(b). This apeirohedron has the same equivelar type as the previous apeirohedron we examined. In this case the helical faces are over squares. The automorphism group is found by performing the facetting operation on the group of {4, 6}6. The

127 group is then (b) 2 Γ({∞, 3} ) = h(τ0τ1) , τ2τ1τ3τ2, τ1τ3i.

Its symmetry group is

(b) 2 G({∞, 3} ) = h(t0t1) , t2t1t3t2, t1t3i =: hr0, r1, r2i

(a) where, again, all three generators are half-turns. As with {∞, 4}·,∗3 and {∞, 3} , having all half- turns as the generators makes {∞, 3}(b) occur in two enantiomorphic forms where the primary difference is the orientation of the helical faces. The Wythoffians also occur in enantiomorphic forms where in one form the helices are left-handed and in the other form the helices are right- handed. Due to the nature of the symmetries, there is no point which is stabilized by both r0 and 2 r1, excluding P . All initial vertices are chosen from the fundamental region. The fundamental region for {∞, 3}(b) is the same as for its Petrie dual {∞, 3}(a). For pictures of the Wythoffians see Table 7.48 in the Appendix.

(a) Fundamental region in honeycomb. (b) Fundamental region in {∞, 3}(b).

Figure 6.13: Figure (a) shows the fundamental region of {∞, 3}(b) within {4, 3, 4}. Figure (b) shows the fundamental region and the generating reﬂections of the symmetry group of {∞, 3}(b) within (b) (b) {∞, 3} . The red line is r0, the blue line is r1, and the green line is r2. The base face of {∞, 3} is outlined with dashed edges.

P 0 The ﬁrst Wythoﬃan is the regular apeirohedron {∞, 3}(b) itself. The faces are regular helices

about square bases with three helices meeting at each vertex, (∞4.∞4.∞4). The vertex ﬁgure is a regular triangle.

128 1 {0,1} P For this apeirohedron the faces of type F2 are regular helices about square bases and the {1,2} faces of type F2 are regular triangles. There are two triangles and two helices alternating about each vertex, (3.∞4.3.∞4), yielding a skew quadrilateral as the vertex ﬁgure. All faces are regular polygons so this is a uniform apeirohedron.

01 {0,1} (b) P In this apeirohedron the faces of type F2 are truncations of the faces of {∞, 3} which {1,2} makes them helices about octagonal bases. The faces of type F2 are regular triangles. There are two helices and one triangle coming together at each vertex giving a vertex con-

figuration (3.∞8.∞8) and yielding an isosceles triangle as the vertex figure. The truncated helices cannot be made regular in this case so there is no uniform version of this Wythoffian.

02 {0,1} P In this apeirohedron the faces of type F2 are regular helices over square bases, the faces of {1,2} {0,2} type F2 are regular triangles, and the faces of type F2 are skew quadrilaterals. At each vertex the vertex configuration is (3.4s.∞4.4s). The vertex figure is a skew quadrilateral. For {0,2} a certain initial vertex the faces of type F2 are regular and the Wythoffian is uniform.

12 {0,1} P For this apeirohedron the faces of type F2 are regular helices about square bases and the {1,2} faces of type F2 are antiprismatic hexagons. There are two hexagons and one helix at each vertex, (6s.6s.∞4), yielding an isosceles triangle as a vertex figure. For a specific initial {1,2} vertex the faces of type F2 are regular and the Wythoffian is uniform.

P 012 Finally let the initial vertex be transient under all generating symmetries of G({∞, 3}(b)). In {0,1} this apeirohedron the faces of type F2 are helices about octagonal bases (truncations of {1,2} helices over squares). The faces of type F2 are hexagons which may be skew or convex depending on the specific choice of initial vertex within the fundamental region. The faces {0,2} of type F2 are skew quadrilaterals. There is one face of each type meeting at each vertex yielding at triangular vertex figure. As with the previous Wythoffians an initial vertex can be chosen so that the faces are all regular and the Wythoffian is a uniform apeirohedron. In {1,2} the uniform case the faces of type F2 are skew.

129 Chapter 7

Uniform polyhedra

In this chapter we will examine several new uniform polyhedra. See Section 1.6 for a discussion of 3 uniform polyhedra. For our purposes here, a uniform polyhedron is a geometric object in E where the following properties are satisﬁed, as detailed in Chapter 1.

(U1) The edge graph is connected.

(U2) The vertex ﬁgure at each vertex is connected.

(U3) The object is discrete.

(U4) Each edge is incident to exactly two polygonal faces.

(U5) Every 2-face is a regular polygon.

(U6) Every vertex is surrounded in the same manner.

Properties (U1)-(U4) ensure that the object is a polyhedron ([35]) while properties (U5) and (U6) ensure it is uniform ([9]). The polygonal faces can be any of the types discussed in [19] including planar, non-planar, and infinite. All regular polyhedra are also uniform. Historically, examinations of uniform polyhedra have focused primarily on finite polyhedra with planar faces. The uniform polyhedra we will examine in this chapter all have a non-planar face, infinite face, or non-planar vertex figure. Coxeter, Longuet-Higgins, and Miller [9] gave an enumeration of all uniform polyhedra with planar faces. This enumeration was later proved to be complete by Skilling [43] and Har’El [24]. The list includes 75 uniform polyhedra and two infinite families, the uniform prisms and uniform antiprisms. In the polyhedra in their list the faces are either regular convex polygons or regular star polygons. Coxeter, Longuet-Higgins, and Miller used Wythoff’s construction applied to Schwarz triangles to find uniform polyhedra ([9]), see Section 1.6 for a description of their methodology. This method has two limitations. First, it can only be used to find finite polyhedra. Second, it only generates planar faces.

130 To prove the completeness of the list, Skilling used an algorithm to generate a complete list of possible vertices on a sphere and all possible edge lengths. These sets of vertices all must be centrally symmetric under one of the finite symmetry groups, that is, the tetrahedral group, the octahedral group, or the icosahedral group. Then for a given vertex set he constructs all possible edges and uses a program to determine if any regular planar polygons can be formed using only those edges. Once all the possible polygons have been found his program determines which can be faces of a uniform polyhedron [43]. As with Coxeter, Longuet-Higgins, and Miller, the work of Skilling only looks at finite polyhedra with planar faces. When we allow the faces to be any type of regular polygon we can find many new uniform polyhedra. Wythoff’s construction as described in this thesis can be used to generate uniform polyhedra and apeirohedra with planar, non-planar, and infinite faces. In Chapter 3 the uniform polyhedra we generated all belong to Coxeter, Longuet-Higgins, and Miller’s list. In Chapters 4-6 we saw many uniform apeirohedra which do not appear on any previous lists of uniform polyhedra. In Section 7.2 we will see that an extension of Wythoff’s construction motivated by Skilling’s methods will generate even more new uniform polyhedra.

7.1 New uniform polyhedra resulting from Wythoﬀ’s construction

In this section we will list all of the new uniform polyhedra found in this thesis through Wythoff’s construction. All of the finite uniform polyhedra with planar faces from Chapter 3 can be found in [9]. The only finite uniform polyhedra with non-planar faces generated through Wythoff’s construction are the nine regular polyhedra

5 10 10 5 {4, 3}3, {6, 4}3, {6, 3}4, {10, 5}3, {10, 3}5, {6, 2 }, {6, 5}, { 3 , 3}, and { 3 , 2 }.

In Chapter 4 we examined planar apeirohedra. All of the uniform planar apeirohedra with ﬁnite faces examined in Chapter 4 can be found in [23]. The new uniform planar apeirohedra are all derived from {∞, 4}4, {∞, 3}6, and {∞, 6}3. The other new uniform apeirohedra are all derived from the blended and pure regular apeirohedra and were discussed in Chapters 5 and 6, respectively. Here we will only list the uniform polyhedra not discussed in either [9] or [19], including regular polyhedra. Note that some of these uniform polyhedra are Wythoﬃans of multiple regular polyhedra. In these cases we only list one of the regular polyhedra as a reference.

131 7.1.1 Uniform polyhedra derived from ﬁnite regular polyhedra

Figure 7.1: The regular polyhedron {4, 3}3. Figure 7.2: The regular polyhedron {6, 4}3. The base face is an antiprismatic square and The base face is an antiprismatic hexagon is outlined in red. and is outlined in red.

Figure 7.3: The regular polyhedron {6, 3}4. Figure 7.4: The regular polyhedron {10, 5}3. The base face is an antiprismatic hexagon The base face is an antiprismatic decagon and is outlined in red. and is outlined in red.

5 Figure 7.5: The regular polyhedron {10, 3}5. Figure 7.6: The regular polyhedron {6, 2 }. The base face is an antiprismatic decagon The base face is an antiprismatic hexagon and is outlined in red. and is outlined in red.

132 10 5 Figure 7.7: The regular polyhedron {6, 5}. Figure 7.8: The regular polyhedron { 3 , 2 }. The base face is an antiprismatic hexagon The base face is an antiprismatic decagram and is outlined in red. and is outlined in red.

10 Figure 7.9: The regular polyhedron { 3 , 3}. The base face is an antiprismatic decagram and is outlined in red.

7.1.2 Uniform apeirohedra derived from planar regular apeirohedra

1 Figure 7.10: The regular {∞, 4}4. The base Figure 7.11: P from {∞, 4}4. The base face is a zigzag apeirogon and is outlined in faces are a linear apeirogon outlined in red red. and a convex square ﬁlled in with blue. Some of the square faces are ﬁlled with a lighter blue to show the nature of the shape.

133 1 Figure 7.12: The regular {∞, 6}3. The base Figure 7.13: P from {∞, 6}3. The base face is a zigzag apeirogon and is outlined in faces are a linear apeirogon outlined in red red. and a convex hexagon ﬁlled in with blue. Some of the hexagons are ﬁlled with a lighter blue to show the nature of the shape.

1 Figure 7.14: The regular {∞, 3}6. The base Figure 7.15: P from {∞, 3}6. The base face is a zigzag apeirogon and is outlined in faces are a linear apeirogon outlined in red red. and a triangle ﬁlled in with blue. Some of the triangles have been ﬁlled with a lighter blue to show the nature of the shape.

7.1.3 Uniform apeirohedra derived from blended regular apeirohedra

Figure 7.16: The regular {4, 4}#{}. The Figure 7.17: P 02 from {4, 4}#{}. The base base face is an antiprismatic square and is faces are an antiprismatic square (outlined outlined in red. in red), a convex square (ﬁlled in with blue), and another convex square (ﬁlled in with green).

134 Figure 7.18: The regular {∞, 4}4#{}. The Figure 7.19: The regular {4, 4}#{∞}. The base face is a zigzag and is outlined in red. base face is a helix over a square which has been outlined in red.

Figure 7.20: P 1 from {4, 4}#{∞}. The base Figure 7.21: P 01 from {4, 4}#{∞}. The faces are a helix over a square (outlined in base faces are a helix over an octagon (out- red) and an antiprismatic square (outlined lined in red) and an antiprismatic square in blue). (outlined in blue).

02 Figure 7.22: P from {4, 4}#{∞}. The Figure 7.23: The regular {∞, 4}4#{∞}. base faces are a linear apeirogon (outlined The base face is a zigzag which is outlined in red), a convex square (ﬁlled in with in red. blue), and another convex square (outlined in green).

135 Figure 7.24: The regular {6, 3}#{}. The Figure 7.25: P 02 from {6, 3}#{}. The base base face is an antiprismatic hexagon and is faces are an antiprismatic square (outlined outlined in red. in red), a triangle (ﬁlled in with blue), and a convex square (ﬁlled in with green).

Figure 7.26: The regular {∞, 3}6#{}. The Figure 7.27: The regular {6, 3}#{∞}. The base face is a zigzag and is outlined in red. base face is a helix over a hexagon which is outlined in red.

Figure 7.28: P 1 from {6, 3}#{∞}. The base Figure 7.29: P 01 from {6, 3}#{∞}. The faces are a helix over a hexagon (outlined base faces are a helix over a dodecagon (out- in red) and a prismatic hexagon (outlined in lined in red) and a prismatic hexagon (out- blue). lined in blue).

136 1 Figure 7.30: The regular {∞, 3}6#{∞}. Figure 7.31: P from {∞, 3}6#{∞}. The The base face is a zigzag which is outlined base faces are a linear apeirogon (outlined in red. in red) and a prismatic hexagon (outlined in blue).

Figure 7.32: The regular {3, 6}#{}. The Figure 7.33: P 02 from {3, 6}#{}. The base base face is a prismatic hexagon and is out- faces are a prismatic hexagon (outlined in lined in red. red), a convex hexagon (ﬁlled in with blue), and a convex square (ﬁlled in with green).

Figure 7.34: The regular {∞, 6}3#{}. The Figure 7.35: The regular {3, 6}#{∞}. The base face is a zigzag and is outlined in red. base face is a helix over a triangle and is outlined in red.

137 Figure 7.36: P 1 from {3, 6}#{∞}. The base Figure 7.37: P 01 from {3, 6}#{∞}. The faces are a helix over a triangle (outlined in base faces are a helix over a hexagon (out- red) and an antiprismatic hexagon (outlined lined in red) and an antiprismatic hexagon in blue). (outlined in blue).

02 Figure 7.38: P from {3, 6}#{∞}. The Figure 7.39: The regular {∞, 6}3#{∞}. base faces are a helix over a triangle (out- The base face is a zigzag which is outlined lined in red), a convex hexagon (ﬁlled in with in red. blue), and a convex square (ﬁlled in with green).

1 Figure 7.40: P from {∞, 6}3#{∞}. The base faces are a linear apeirogon (in red) and an antiprismatic hexagon (in blue).

138 7.1.4 Uniform apeirohedra derived from pure regular apeirohedra

Figure 7.41: The regular {4, 6|4}. The base Figure 7.42: P 1 from {4, 6|4}. The base face is a convex square which is ﬁlled in with faces are an antiprismatic hexagon (outlined blue. in red) and a convex square (ﬁlled in with blue).

Figure 7.43: P 01 from {4, 6|4}. The base Figure 7.44: P 02 from {4, 6|4}. The base faces are an antiprismatic hexagon (outlined faces are a convex square (filled in with red), in red) and a convex octagon (filled in with a convex hexagon (filled in with blue), and blue). another convex square (filled in with green).

Figure 7.45: The regular {6, 4|4}. The base Figure 7.46: P 1 from {6, 4|4}. The base faces face is a convex hexagon which is ﬁlled in are an antiprismatic square (outlined in red) with blue. and a convex hexagon (ﬁlled in with blue).

139 01 Figure 7.47: P from {6, 4|4}. The base Figure 7.48: The regular {∞, 6}4,4. The base faces are a antiprismatic square (outlined in face is a helix about a triangle which is out- red) and a convex dodecagon (ﬁlled in with lined in red. blue).

1 1 Figure 7.49: P from {∞, 6}4,4. The base Figure 7.50: Alternate P from {∞, 6}4,4. faces are an antiprismatic hexagon (outlined The base faces are a convex hexagon (ﬁlled in blue) and a helix over a triangle (outlined in blue) and a helix over a triangle (outlined in red). in red).

01 Figure 7.51: P from {∞, 6}4,4. The base Figure 7.52: The regular {∞, 4}6,4. The base faces are an antiprismatic hexagon (outlined face is a helix over a triangle and is outlined in blue) and a helix over a hexagon (outlined in red. in red).

140 1 01 Figure 7.53: P from {∞, 4}6,4. The base Figure 7.54: P from {∞, 4}6,4. The base faces are an antiprismatic square (outlined faces are an antiprismatic square (outlined in in blue) and a helix over a triangle (outlined blue) and a helix over a hexagon (outlined in in red). red).

1 Figure 7.55: The regular {∞, 4}·,∗3. The Figure 7.56: P from {∞, 4}·,∗3. The base base face is a helix over a triangle which is faces are a convex square (ﬁlled in with blue) outlined in red. and a helix over a triangle (outlined in red).

02 12 Figure 7.57: P from {∞, 4}·,∗3. The base Figure 7.58: P from {∞, 4}·,∗3. The base faces are a convex square (ﬁlled in with blue), faces are an antiprismatic octagon (outlined an antiprismatic square (outlined in green), in blue) and a helix over a triangle (outlined and a helix over a triangle (oulined in red). in red).

141 012 Figure 7.59: P from {∞, 4}·,∗3. The base Figure 7.60: Initial vertex outside the funda- faces are an antiprismatic octagon (outlined mental region of {∞, 4}·,∗3. The base faces by blue), an antiprismatic square (outlined are an antiprismatic octagram (outlined by in green), and a helix over a hexagon (oulined blue), an antiprismatic square (outlined in in red). green), and a helix over a hexagon (oulined in red).

Figure 7.61: The regular {6, 6}4. The base Figure 7.62: The regular {4, 6}6. The base face is an antiprismatic hexagon which is out- face is an antiprismatic square outlined in lined in red. red.

02 02 Figure 7.63: P from {4, 6}6. The base faces Figure 7.64: Alternate P from {4, 6}6. The are an antiprismatic square (outlined in red), base facs are an antiprismatic square (out- an antiprismatic hexagon (outlined in blue), lined in green), an convex hexagon (filled in and a convex square (filled in with green). with blue), and a convex square (filled in with red).

142 1 Figure 7.65: The regular {6, 4}6. The base Figure 7.66: P from {6, 4}6. The base faces face is an antiprismatic hexagon which is out- are a convex hexagon (ﬁlled in with red) and lined in red. a convex square (ﬁlled in with blue).

Figure 7.67: The regular {6, 6|3}. The base Figure 7.68: P 1 from {6, 6|3}. The base faces face is a convex hexagon which is ﬁlled in are a convex hexagon (ﬁlled in with blue) and with blue. an antiprismatic hexagon (outlined in red).

01 Figure 7.69: P from {6, 6|3}. The base Figure 7.70: The regular {∞, 6}6,3. The base faces are a convex dodecagon (ﬁlled in with face is a helix over a square which is outlined blue) and an antiprismatic hexagon (outlined in red. in red).

143 1 1 Figure 7.71: P from {∞, 6}6,3. The base Figure 7.72: Alternate P from {∞, 6}6,3. faces are an antiprismatic hexagon (outlined The base faces are a convex hexagon (ﬁlled in blue) and a helix over a square (outlined in blue) and a helix over a square (outlined in red). in red).

01 (a) Figure 7.73: P from {∞, 6}6,3. The base Figure 7.74: The regular {∞, 3} . The base faces are an antiprismatic hexagon (outlined face is a helix over a triangle which is out- in blue) and a helix over an octagon (outlined lined in red. in red).

Figure 7.75: P 1 from {∞, 3}(a). The base Figure 7.76: P 02 from {∞, 3}(a). The base faces are a triangle (ﬁlled in with blue) and faces are a triangle (ﬁlled in with blue), an a helix over a triangle (outlined in red). antiprismatic square (outlined in green), and a helix over a triangle (outlined in red).

144 Figure 7.77: P 12 from {∞, 3}(a). The base Figure 7.78: P 012 from {∞, 3}(a). The base faces are an antiprismatic hexagon (outlined faces are a prismatic hexagon (outlined in in blue) and a helix over a triangle (outlined blue), an antiprismatic square (outlined in in red). green), and a helix over a hexagon (outlined in red).

Figure 7.79: The regular {∞, 3}(b). The base Figure 7.80: P 1 from {∞, 3}(b). The base face is a helix over a square which is outlined faces are a triangle (ﬁlled in with blue) and in red. a helix over a square (outlined in red).

145 Figure 7.81: P 02 from {∞, 3}(b). The base Figure 7.82: P 12 from {∞, 3}(b). The base faces are triangle (ﬁlled in with blue), an an- faces are a prismatic hexagon (outlined in tiprismatic square (outlined in green), and a blue) and a helix over a square (outlined in helix over a triangle (outlined in red). red).

Figure 7.83: P 012 from {∞, 3}(b). The base faces are a convex hexagon (ﬁlled in with blue), an antiprismatic square (outlined in blue), and a helix over an octagon (outlined in red).

7.2 New construction for uniform polyhedra

Here we will detail a construction which will generate finite uniform polyhedra with non-planar faces, including examples which are not Wythoffians. It is motivated by Wythoff’s construction and the method used by Skilling in [43] to enumerate the finite uniform polyhedra with planar faces. First we generate a vertex set, next we define the edge length, then we find the types of polygonal faces, then we construct the object, and finally we check whether it is a polyhedron. 3 We start by describing the vertex set, V , in E . To generate V we first pick a full symmetry group, G, corresponding to a finite regular polyhedron. Then we look at all of the Wythoffians generated from a finite regular polyhedron with that symmetry group which can possibly be uniform polyhedra (see Chapter 3). We choose one such Wythoffian, and find the initial vertex from the fundamental region which makes all its edge lengths equal. Set V equal to the orbit of this vertex

146 under the symmetry group. We choose the vertex set this way to maximize the symmetry of the object and, in particular, achieve vertex-transitivity. Now that we have V we need to set an edge length for the polyhedron to be constructed. Pick a base vertex, u, in V . By hand, we search within V for a subset that forms the vertex set of either a prism or an antiprism with a regular base which contains u as a vertex. This search will always yield a result. Each prismatic (antiprismatic) vertex set contains the vertex set (possibly as a proper subset of the vertex set) of a regular prismatic (antiprismatic) polygon incident to u. Here we allow any regular skew polygon including skew star polygons. For example if we find the vertex set of an antiprismatic octagon ({8}#{}) then it is also the vertex set of an antiprismatic 8 octagram ({ 3 #{}), so we would have to choose one of the polygons to focus on. Then use the edge length of one such polygon, l, as the edge length for our polyhedron. Find all vertices, v ∈ V , such that d(u, v) = l, where d(u, v) is the Euclidean metric. For each of these vertices we check if {u, v} is an edge of a regular polygon, either planar or non-planar, with vertices in V . We now have the set of all regular polygons incident to u with vertices in V and edge length l. Then we perform some checks on these polygons to ensure that they can be faces of a uniform polyhedron. First we need to check if these polygons create a connected circuit about u, that is, we need to check if the vertex figure at u is connected. If the vertex figure is not connected we throw out this edge length and find a new edge length by the same procedure as described above. If the vertex figure is connected, then we check that each edge incident to u is incident to exactly two polygons. If any edge is only incident to one polygon then we throw out the edge length and find a new one, again as described above. If any edge is incident to more than two polygons then we need to decide which combinations of polygons, if any, we can use as faces. When this happens we will repeat the subsequent step for all combinations of polygons which form a connected circuit about u and have two polygons at each edge incident to u. Once we have such a combination of regular polygons incident to u we can build the polyhedron using these polygons as our base faces. Now we borrow from Wythoff’s construction and apply the symmetry group G to each of the base faces. Each of these polygons and all of their transforms under G are the faces of the newly constructed uniform object. All that remains is to check that properties (U1)-(U6) have been fulfilled for the resulting object. If the properties (U1)-(U6) have been satisfied then we have a finite uniform polyhedron with at least one type of non-planar face. In the next section we will list all of the polyhedra we generated in this fashion. Most of the uniform polyhedra in this section are not Wythoffians. Wythoff’s construction is very limiting in how it generates figures with skew faces. The distinguished generators of the finite regular polyhedra with planar faces are plane reflections. In Wythoff’s construction any face generated by plane reflections is a planar face. The finite regular polyhedra with non-planar faces have distinguished generators r0, r1, r2 which are, respectively, a half-turn, a plane reflection, and a plane reflection. In each of these cases the Wythoffian P 0 has non-planar faces and is also generated

147 through this construction; the Wythoffian P 1 has planar faces and will thus not be generated by this construction; the Wythoffians P 2 and P 12 do not exist; and the Wythoffians P 01, P 02, and P 012 have non-regular faces and are not uniform. Accordingly, any finite non-regular uniform polyhedra with non-planar faces are not Wythoffians of a regular polyhedron. This new construction generates many examples of uniform polyhedra with skew faces which are not Wythoffians.

7.3 Uniform polyhedra generated through alternate construction

In this section we list all uniform polyhedra with non-planar faces that we generated through the new construction. The polyhedra are classiﬁed by symmetry family. Some of them are regular polyhedra while the rest are not Wythoﬃans.

7.3.1 Tetrahedral family

The only uniform polyhedra generated by this construction in the tetrahedral family are the regular

{4, 3}3 and {6, 4}3, the Petrie duals of the tetrahedron and octahedron, respectively. For more on these two polyhedra see Chapter 3.

Figure 7.84: The regular polyhedron {4, 3}3. Figure 7.85: The regular polyhedron {6, 4}3. The base face is an antiprismatic square and The base face is an antiprismatic hexagon is outlined in red. and is outlined in red.

7.3.2 Octahedral family

In the octahedral family there are two regular polyhedra with non-planar faces generated by this construction. They are {6, 4}3 and {6, 3}4, the Petrie duals of the octahedron and cube, respectively. See Chapter 3 for more on these two regular polyhedra. The rest of the uniform polyhedra in this family generated by the construction are all new and are not Wythoﬃans described in previous sections.

148 Figure 7.86: The regular polyhedron {6, 4}3. Figure 7.87: The regular polyhedron {6, 3}4. The base face is an antiprismatic hexagon The base face is an antiprismatic hexagon and is outlined in red. and is outlined in red.

The ﬁrst polyhedron we will look at has six antiprismatic squares and four antiprismatic hexagons for faces. In total there are ten faces, twenty-four edges, and twelve vertices. The vertex symbol is (4s.6s.4s.6s). In Figure 7.88 the base antiprismatic square face is outlined in red and the base antiprismatic hexagonal face is outlined in blue. The polyhedron in Figure 7.89 has six antiprismatic squares, four antiprismatic hexagons, and six antiprismatic octagons for faces. The vertex symbol is (4s.6s.8s.8s). In total there are sixteen faces, forty-eight edges, and twenty-four vertices. In Figure 7.89 the base square is highlighted in red, the base hexagon is highlighted in green, and the base octagon is highlighted in blue.

Figure 7.88 Figure 7.89

The polyhedron in Figure 7.90 has four antiprismatic hexagons and six antiprismatic octagrams, 8 8 8 8 { 3 #{}, for faces with vertex conﬁguration (6s. 3 s. 3 s) where 3 s refers to the skew octagrams. It has ten faces, thirty-six edges, and twenty-four vertices. In Figure 7.90 the base antiprismatic hexagon is highlighted in blue and the base antiprismatic octagram is highlighted in red. The polyhedron in Figure 7.91 has six antiprismatic squares, four antiprismatic hexagons, and 8 8 six antiprismatic octagrams with vertex conﬁguration (4s. 3 s.6s. 3 s.). In total there are sixteen faces, forty-eight edges, and twenty-four vertices. In Figure 7.91 the base antiprismatic square is outlined in red, the base antiprismatic hexagon is outlined in green, and the antiprismatic octagram

149 is outlined in blue.

Figure 7.90 Figure 7.91

The polyhedron in Figure 7.92 has twelve antiprismatic squares and six antiprismatic octagrams 8 8 with vertex symbol (4s. 3 s.4s. 3 s). In total there are eighteen faces, forty-eight edges, and twenty-four vertices. In Figure 7.92 the base antiprismatic square is outlined in blue and the base antiprismatic octagram is outlined in red. The next polyhedron has four antiprismatic hexagons and six antiprismatic octagons with vertex symbol (6s.8s.8s). It has ten faces, thirty-six edges, and twenty-four vertices. In Figure 7.93 the base antiprismatic hexagon is blue and the base antiprismatic octagon is red.

Figure 7.92 Figure 7.93

The polyhedron in Figure 7.94 has twelve antiprismatic squares in one orbit, six antiprismatic squares in a diﬀerent orbit, and four antiprismatic hexagons with vertex symbol (4s.4s.4s.6s). There are a total of twenty-two faces, forty-eight edges, and twenty-four vertices. In Figure 7.94 one base antiprismatic square is outlined in blue, the other base antiprismatic square is outlined in red, and the base antiprismatic hexagon is outlined in green. The polyhedron in Figure 7.95 has planar triangle faces and non-planar antiprismatic hexagon faces. It has sixteen triangles and eight antiprismatic hexagons with vertex symbol (3.6s.6s). There

150 are twenty-four faces, forty-eight edges, and twenty-four vertices in total. In Figure 7.95 the base triangle is outlined in blue and the base antiprismatic hexagon is outlined in red.

Figure 7.94 Figure 7.95

The polyhedron in Figure 7.96 has six antiprismatic squares and eight prismatic hexagons with vertex symbol (4s.6s.6s). There are fourteen faces, thirty-six edges, and twenty-four vertices. In Figure 7.96 the base antiprismatic square is outlined in blue and the base prismatic hexagon is outlined in red. The ﬁnal polyhedron we examine in the octahedral family has twelve antiprismatic squares, 8 eight prismatic hexagons, and six antiprismatic octagrams with vertex symbol (4s.6s. 3 s). In total there are twenty-six faces, seventy-two edges, and forty-eight vertices. In Figure 7.97 the base antiprismatic square is outlined in green, the base prismatic hexagon is outlined in blue, and the base antiprismatic octagram is outlined in red.

Figure 7.96 Figure 7.97

7.3.3 Icosahedral family

In the icosahedral family the construction generates six regular polyhedra with non-planar faces. 5 10 5 10 They are {10, 5}3, {10, 3}5, {6, 2 }, {6, 5}, { 3 , 2 }, and { 3 , 3}. These polyhedra are the Petrie

151 5 5 5 5 duals of {3, 5}, {5, 3}, {5, 2 }, { 2 , 5}, {3, 2 }, and { 2 , 3}, respectively. For more on these regular polyhedra see Chapter 3.

Figure 7.98: The regular polyhedron Figure 7.99: The regular polyhedron {10, 5}3. The base face is an antiprismatic {10, 3}5. The base face is an antiprismatic decagon and is outlined in red. decagon and is outlined in red.

5 Figure 7.100: The regular polyhedron {6, 2 }. Figure 7.101: The regular polyhedron {6, 5}. The base face is an antiprismatic hexagon The base face is an antiprismatic hexagon and is outlined in red. and is outlined in red.

Figure 7.102: The regular polyhedron Figure 7.103: The regular polyhedron 10 5 10 { 3 , 2 }. The base face is an antiprismatic { 3 , 3}. The base face is an antiprismatic decagram and is outlined in red. decagram and is outlined in red.

152 The ﬁrst new polyhedron we will look at has both planar and non-planar faces. The planar faces are convex pentagons while the non-planar faces are antiprismatic hexagons. There are twelve pentagons and ten antiprismatic hexagons yielding twenty-two faces, sixty edges, and thirty vertices.

The polyhedron has vertex symbol (5.6s.5.6s). In Figure 7.104 the base pentagon is outlined in red and the base antiprismatic hexagon is outlined in blue. The next polyhedron also has planar and non-planar faces. The planar faces are convex pentagons while the non-planar faces are antiprismatic decagons. In Figure 7.105 the base pentagon is outlined in red and the base antiprismatic decagon is outlined in blue. In total there are twelve pentagons and six antiprismatic decagons yielding eighteen total faces, sixty edges, and thirty vertices.

The vertex symbol is (5.10s.5.10s).

Figure 7.104 Figure 7.105

The polyhedron in Figure 7.106 has twelve planar pentagrams and ten antiprismatic hexagons 5 5 for faces with vertex symbol ( 2 .6s. 2 .6s). All together there are twenty-two faces, sixty edges, and thirty vertices. In Figure 7.106 the base pentagram is outlined in red and the base antiprismatic hexagon is outlined in blue. The polyhedron in Figure 7.107 has twelve planar pentagrams and six antiprismatic decagrams, 10 5 10 5 10 (where a decagram has Schl¨aﬂisymbol { 3 }), with vertex symbol ( 2 . 3 s. 2 . 3 s). All together there are eighteen faces, sixty edges, and thirty vertices. In Figure 7.107 the base pentagram is outlined in red and the base antiprismatic decagram is outlined in blue.

153 Figure 7.106 Figure 7.107

The next polyhedron only has non-planar faces. It has ten antiprismatic hexagons and six antiprismatic decagons yielding sixteen faces, sixty edges, and thirty vertices. The vertex symbol is then (6s.10s.6s.10s). In Figure 7.108 the base antiprismatic hexagon is outlined in blue and the base antiprismatic decagon is outlined in red. The polyhedron in Figure 7.109 has ten antiprismatic hexagons and six antiprismatic decagrams 10 10 with vertex symbol (6s. 3 s.6s. 3 s). In total there are sixteen faces, sixty edges, and thirty vertices. In Figure 7.109 the base antiprismatic hexagon is outlined in red while the base antiprismatic decagram is outlined in blue.

Figure 7.108 Figure 7.109

The polyhedron in Figure 7.110 has six antiprismatic decagons and six antiprismatic decagrams 10 10 with vertex symbol (10s. 3 s.10s. 3 s). There are twelve faces, sixty edges, and thirty vertices in total. In Figure 7.110 the base antiprismatic decagon is oulined in blue and the base antiprismatic decagram is outlined in red. The polyhedron in Figure 7.111 has planar and non-planar faces. The planar faces are pentagrams and the non-planar faces are antiprismatic hexagons. There are twenty-four pentagrams and 5 5 twenty antiprismatic hexagons with vertex symbol ( 2 .6s. 2 .6s). In total there are forty-four faces, one hundred twenty edges, and sixty vertices. In Figure 7.111 the base pentagram is outlined in

154 blue and the base antiprismatic hexagon is outlined in red.

Figure 7.110 Figure 7.111

The polyhedron in Figure 7.112 has three types of face. There are twelve prismatic decagons isomorphic to {5}#{}, ten antiprismatic hexagons, and six antiprismatic decagrams with vertex 10 symbol (6s.10s. 3 s.10s). All together there are twenty-eight faces, one hundred twenty edges, and sixty vertices. In Figure 7.112 the base antiprismatic hexagon is outlined in green, the base prismatic decagon isomorphic to {5}#{} is outlined in red, and the base antiprismatic decagram is outlined in blue. 5 The polyhedron in Figure 7.113 has twenty-four prismatic decagons isomorphic to { 2 }#{}, see section 1.5.1, and ten antiprismatic hexagons. The polyhedron has twenty-two faces, ninety edges, and sixty vertices with vertex symbol (6s.10s.10s). In Figure 7.113 the base antiprismatic hexagon 5 is outlined in blue and the base prismatic decagon isomorphic to { 2 }#{} is outlined in red.

Figure 7.112 Figure 7.113

The polyhedron in Figure 7.114 has twenty prismatic hexagons and six antiprismatic decagons with vertex symbol (6s.6s.10s). All together there are twenty-six faces, ninety edges, and sixty vertices. In Figure 7.114 the base prismatic hexagon is outlined in blue and the base antiprismatic decagon is outlined in red.

155 The polyhedron in Figure 7.115 has thirty antiprismatic squares, ten antiprismatic hexagons, 10 and six antiprismatic decagrams. The resulting vertex symbol is (4s.6s.4s. 3 s). All together there are forty-six faces, one hundred twenty edges, and sixty vertices. In Figure 7.115 the base antiprismatic square is outlined in red, the base antiprismatic hexagon is outlined in blue, and the antiprismatic decagram is outlined in green.

Figure 7.114 Figure 7.115

The polyhedron in Figure 7.116 has thirty antiprismatic squares, six antiprismatic decagons, 10 and six antiprismatic decagrams with vertex symbol (4s.10s.4s. 3 s). In total there are forty-two faces, one hundred twenty edges, and sixty vertices. In Figure 7.116 the base antiprismatic square is outlined in red, the base antiprismatic decagon is oulined in green, and the base antiprismatic decagram is outlined in blue. The polyhedron in Figure 7.117 has twenty prismatic hexagons and six antiprismatic decagrams 10 with vertex symbol (6s.6s. 3 s). It has twenty-six faces, ninety edges, and sixty vertices. In Figure 7.117 the base prismatic hexagon is outlined in red and the base antiprismatic decagram is outlined in blue.

Figure 7.116 Figure 7.117

The polyhedron in Figure 7.118 has six antiprismatic decagons and twelve prismatic decagons 5 isomorphic to { 2 }#{}. The vertex symbol is then (10s.10s.10s). In total there are eighteen faces,

156 5 ninety edges, and sixty vertices. In Figure 7.118 the base prismatic decagon isomorphic to { 2 }#{} is outlined in blue and the antiprismatic decagon is outlined in red. We arrive at the final polyhedron in a slightly different manner. For a particular vertex set when we apply the construction the resulting figure does not have a connected edge graph. The figure is not a polyhedron, but upon closer examination we found that the figure can be separated into two disjoint figures which are mirror images of each other. Each of these figures is a uniform polyhedron incident to exactly half of vertices in the vertex set. This is the only time we found that the construction yielded disjoint polyhedra from one vertex set. Figure 7.119 shows one of the uniform polyhedra. It has fifteen antiprismatic squares, ten prismatic hexagons, and six prismatic 5 decagons isomorphic to { 2 }#{} with vertex symbol (4s.6s.10s). All together there are thirty-one faces, ninety edges, and sixty vertices. In Figure 7.119 the base antiprismatic square is outlined in blue, the base prismatic hexagon is outlined in green, and the base prismatic decagon isomorphic 5 to { 2 }#{} is outlined in red.

Figure 7.118 Figure 7.119

7.4 Summary of results

All Wythoffians which are finite uniform polyhedra with non-planar faces are actually regular polyhedra. Each one is the Petrie dual of either a Platonic solid or a Kepler-Poinsot polyhedron. As uniform Wythoffians we have found twenty-seven regular apeirohedra and forty-five non- regular apeirohedra with either skew faces, skew vertex figures, or infinite faces. Three of the non-regular uniform apeirohedra are planar apeirohedra with apeirogonal faces. From the blended apeirohedra we found as Wythoffians thirteen uniform apeirohedra which were not regular. We found twenty-nine non-regular uniform Wythoffians derived from the pure regular apeirohedra. In these uniform apeirohedra the faces are planar, skew, or infinite. Note that for these Wythoffians we placed restrictions on the choice of initial vertex such as placing it within the fundamental region. If we were to remove these restrictions, other uniform Wythoffians might be found. Combinatorially they would be isomorphic to the Wythoffians discussed here but the planarity and convexity of their faces might change.

157 Our new construction discussed in Section 7.2 generated twenty-six new finite uniform polyhedra with non-planar faces which are not Wythoffians. In the octahedral symmetry family we have found ten new finite uniform polyhedra and in the icosahedral symmetry family we have found sixteen new finite uniform polyhedra. Our construction is based on the vertex sets of the planar faced uniform Wythoffians examined in Chapter 3, in that their vertex sets are taken as input for the construction. If we expand the construction to the vertex sets of all the planar faced uniform polyhedra examined in [9] or to apeirohedra, then we should find more new uniform polyhedra with non-planar faces. In this thesis we have loosened the definition of polygon to include non-planar polygons. If we expand our definitions of adjacency, polyhedron, and polygon even further as discussed by Grünbaum in [21] we would expect to find even more new uniform polyhedra with non-planar faces. Some of the extensions suggested by Grünbaum include vertex doubling and face doubling where each vertex or face is replaced by two vertices or faces. These two possibilities could certainly apply to non-planar faced polyhedra and would allow for the construction of new uniform polyhedra. While Wythoff’s construction and the new construction detailed in this chapter yield many new uniform polyhedra, extensions of these methods and expanded definitions could allow for the creation of many more.

158 Appendix

We conclude this thesis with tables of images of the Wythoffians. On each page there are two tables. In the first table we show the different face types and the vertex figure. In the second table we show the Wythoffian overlaying a copy of the regular polyhedron from which it is derived. The base faces have been outlined and in cases where the planarity of a face may not be obvious we have filled in a membrane to emphasize its planarity and convexity. We have primarily done this for the apeirohedra with either skew or infinite faces in addition to planar faces. The base face {0,1} {1,2} F2 is outlined (and where appropriate, filled in) with red, the base face F2 is outlined (and {0,2} where appropriate, filled in) with blue, and the base face F2 is outlined (and where appropriate, filled in) with green. For the finite Wythoffians we have images of the full figure. For the infinite Wythoffians we obviously can only show a small portion of the figure. For the regular apeirohedra we show a localized picture of the base vertex, and for the Wythoffians we show a similarly localized image. We hope to show enough so that the reader can see what the basic face types are and how they will interact at each vertex.

159 Table 7.1: Faces and Vertex Figures for the Wythoﬃans of {3, 3}

{0,1} {1,2} {0,2} Wythoﬃan F2 F2 F2 Vertex Figure

P 0

P 1 P 0 P 1 P 2

P 2 160

P 01

P 01 P 02 P 12 P 02

P 12

P 012 P 012 Table 7.2: Faces and Vertex Figures for Wythoﬃans of {4, 3}3

{0,1} {1,2} {0,2} Wythoﬃan F2 F2 F2 Vertex Figure

P 0

P 0 P 1

P 1 161

P 01

P 01 P 02 P 02

P 012

P 012 Table 7.3: Faces and Vertex Figures for Wythoﬃans of {3, 4}

{0,1} {1,2} {0,2} Wythoﬃan F2 F2 F2 Vertex Figure

P 0

P 1 P 0 P 1 P 2

P 2 162

P 01

P 01 P 02 P 12 P 02

P 12

P 012 P 012 Table 7.4: Faces and Vertex Figures for Wythoﬃans of {4, 3}

{0,1} {1,2} {0,2} Wythoﬃan F2 F2 F2 Vertex Figure

P 0

P 1 P 0 P 1 P 2

P 2 163

P 01

P 01 P 02 P 12 P 02

P 12

P 012 P 012 Table 7.5: Faces and Vertex Figures for Wythoﬃans of {6, 4}3

{0,1} {1,2} {0,2} Wythoﬃan F2 F2 F2 Vertex Figure

P 0

P 0 P 1 P 1 164

P 01

P 01 P 02 P 02

P 012

P 012 Table 7.6: Faces and Vertex Figures for Wythoﬃans of {6, 3}4

{0,1} {1,2} {0,2} Wythoﬃan F2 F2 F2 Vertex Figure

P 0

P 0 P 1

P 1 165

P 01

P 01 P 02 P 02

P 012

P 012 Table 7.7: Faces and Vertex Figures for Wythoﬃans of {3, 5}

{0,1} {1,2} {0,2} Wythoﬃan F2 F2 F2 Vertex Figure

P 0

P 1

P 0 P 1 P 2

P 2 166

P 01

P 01 P 02 P 12

P 02

P 12

P 012 P 012 Table 7.8: Faces and Vertex Figures for Wythoﬃans of {5, 3}

{0,1} {1,2} {0,2} Wythoﬃan F2 F2 F2 Vertex Figure

P 0

P 1

P 0 P 1 P 2

P 2 167

P 01

P 01 P 02 P 12 P 02

P 12

P 012 P 012 Table 7.9: Faces and Vertex Figures for Wythoﬃans of {10, 5}3

{0,1} {1,2} {0,2} Wythoﬃan F2 F2 F2 Vertex Figure

P 0

P 0 P 1 P 1 168

P 01

P 01 P 02 P 02

P 012

P 012 Table 7.10: Faces and Vertex Figures for Wythoﬃans of {10, 3}5

{0,1} {1,2} {0,2} Wythoﬃan F2 F2 F2 Vertex Figure

P 0

P 0 P 1

P 1 169

P 01

P 01 P 02

P 02

P 012

P 012 5 Table 7.11: Faces and Vertex Figures for Wythoﬃans of {5, 2 }

{0,1} {1,2} {0,2} Wythoﬃan F2 F2 F2 Vertex Figure

P 0

P 1

P 0 P 1 P 2

P 2 170

P 01

P 01 P 02 P 12 P 02

P 12

P 012 P 012 5 Table 7.12: Faces and Vertex Figures for Wythoﬃans of { 2 , 5}

{0,1} {1,2} {0,2} Wythoﬃan F2 F2 F2 Vertex Figure

P 0

P 1

P 0 P 1 P 2

P 2 171

P 01

P 01 P 02 P 12 P 02

P 12

012 P P 012 5 Table 7.13: Faces and Vertex Figures for Wythoﬃans of {6, 2 }

{0,1} {1,2} {0,2} Wythoﬃan F2 F2 F2 Vertex Figure

P 0

P 0 P 1 P 1 172

P 01

P 01 P 02

P 02

012 P P 012 Table 7.14: Faces and Vertex Figures for Wythoﬃans of {6, 5}

{0,1} {1,2} {0,2} Wythoﬃan F2 F2 F2 Vertex Figure

P 0

P 0 P 1

P 1 173

P 01

P 01 P 02

P 02

P 012

P 012 5 Table 7.15: Faces and Vertex Figures for Wythoﬃans of {3, 2 }

{0,1} {1,2} {0,2} Wythoﬃan F2 F2 F2 Vertex Figure

P 0

P 1

P 0 P 1 P 2

P 2 174

P 01

01 02 12 P 02 P P P

P 12

P 012 P 012 5 Table 7.16: Faces and Vertex Figures for Wythoﬃans of { 2 , 3}

{0,1} {1,2} {0,2} Wythoﬃan F2 F2 F2 Vertex Figure

P 0

P 1

P 0 P 1 P 2 P 2 175

P 01

P 01 P 02 P 12 P 02

P 12

012 P P 012 10 5 Table 7.17: Faces and Vertex Figures for Wythoﬃans of { 3 , 2 }

{0,1} {1,2} {0,2} Wythoﬃan F2 F2 F2 Vertex Figure

P 0

P 0 P 1 P 1 176

P 01

P 01 P 02

P 02

P 012

P 012 10 Table 7.18: Faces and Vertex Figures for Wythoﬃans of { 3 , 3}

{0,1} {1,2} {0,2} Wythoﬃan F2 F2 F2 Vertex Figure

P 0

P 0 P 1 P 1 177

P 01

P 01 P 02

P 02

P 012

P 012 Table 7.19: Faces and Vertex Figures for the Wythoﬃans of {4, 4}

{0,1} {1,2} {0,2} Wythoﬃan F2 F2 F2 Vertex Figure

P 0

P 1 P 0 P 1 P 2

P 2 178

P 01

P 01 P 02 P 12 P 02

P 12

P 012 P 012 Table 7.20: Faces and Vertex Figures for Wythoﬃans of {∞, 4}4

{0,1} {1,2} {0,2} Wythoﬃan F2 F2 F2 Vertex Figure

P 0

P 0 P 1 P 1 179

P 01

P 01 P 02 P 02

P 012

P 012 Table 7.21: Faces and Vertex Figures for the Wythoﬃans of {3, 6}

{0,1} {1,2} {0,2} Wythoﬃan F2 F2 F2 Vertex Figure

P 0

P 1

P 0 P 1 P 2 P 2 180

P 01

P 01 P 02 P 12 P 02

P 12

P 012 P 012 Table 7.22: Faces and Vertex Figures for Wythoﬃans of {∞, 6}3

{0,1} {1,2} {0,2} Wythoﬃan F2 F2 F2 Vertex Figure

P 0

P 0 P 1

P 1 181

P 01

P 01 P 02

P 02

P 012

P 012 Table 7.23: Faces and Vertex Figures for the Wythoﬃans of {6, 3}

{0,1} {1,2} {0,2} Wythoﬃan F2 F2 F2 Vertex Figure

P 0

P 1 P 0 P 1 P 2

P 2 182

P 01

P 01 P 02 P 12 P 02

P 12

P 012 P 012 Table 7.24: Faces and Vertex Figures for Wythoﬃans of {∞, 3}6

{0,1} {1,2} {0,2} Wythoﬃan F2 F2 F2 Vertex Figure

P 0

P 0 P 1

P 1 183

P 01

P 01 P 02 P 02

P 012

P 012 Table 7.25: Faces and Vertex Figures for the Wythoﬃans of {4, 4}#{}

{0,1} {1,2} {0,2} Wythoﬃan F2 F2 F2 Vertex Figure

P 0

P 1 P 0 P 1 P 2

P 2 184

P 01

P 02 P 01 P 02 P 12

P 12

012 P P 012 Table 7.26: Faces and Vertex Figures for Wythoﬃans of {∞, 4}4#{}

{0,1} {1,2} {0,2} Wythoﬃan F2 F2 F2 Vertex Figure

P 0 P 0 P 1

P 1 185

P 01

P 01 P 02

P 02

P 012

P 012 Table 7.27: Faces and Vertex Figures for the Wythoﬃans of {4, 4}#{∞}

{0,1} {1,2} {0,2} Wythoﬃan F2 F2 F2 Vertex Figure

P 0

P 1

186 P P 0 P 1 P 01

P 02

P 12

02 12 012 P 012 P P P Table 7.28: Faces and Vertex Figures for Wythoﬃans of {∞, 4}4#{∞}

{0,1} {1,2} {0,2} Wythoﬃan F2 F2 F2 Vertex Figure

P 0

P 1

0 1 01 187 P P P

P 01

P 02

P 012

P 02 P 012 Table 7.29: Faces and Vertex Figures for the Wythoﬃans of {6, 3}#{}

{0,1} {1,2} {0,2} Wythoﬃan F2 F2 F2 Vertex Figure P 0

P 1

P 0 P 1 P 2 P 2 188 P 01

P 02 P 01 P 02 P 12

P 12

P 012 P 012 Table 7.30: Faces and Vertex Figures for Wythoﬃans of {∞, 3}6#{}

{0,1} {1,2} {0,2} Wythoﬃan F2 F2 F2 Vertex Figure P 0

P 1 P 0 P 1 189

P 01

02 P P 01 P 02

P 012

P 012 Table 7.31: Faces and Vertex Figures for the Wythoﬃans of {6, 3}#{∞}

{0,1} {1,2} {0,2} Wythoﬃan F2 F2 F2 Vertex Figure

P 0

P 1

190 P 01 P 0 P 1 P 01

P 02

P 12

P 02 P 12 P 012 P 012 Table 7.32: Faces and Vertex Figures for Wythoﬃans of {∞, 3}6#{∞}

{0,1} {1,2} {0,2} Wythoﬃan F2 F2 F2 Vertex Figure P 0

P 1 191 P 0 P 1 P 01 P 01

P 02

P 012 P 02 P 012 Table 7.33: Faces and Vertex Figures for the Wythoﬃans of {3, 6}#{}

{0,1} {1,2} {0,2} Wythoﬃan F2 F2 F2 Vertex Figure

P 0

P 1

P 0 P 1 P 2

P 2 192

P 01

P 01 P 02 P 12

P 02

P 12

P 012 P 012 Table 7.34: Faces and Vertex Figures for Wythoﬃans of {∞, 6}3#{}

{0,1} {1,2} {0,2} Wythoﬃan F2 F2 F2 Vertex Figure

P 0

P 0 P 1

P 1 193

P 01

P 01 P 02 P 02

P 012

P 012 Table 7.35: Faces and Vertex Figures for the Wythoﬃans of {3, 6}#{∞}

{0,1} {1,2} {0,2} Wythoﬃan F2 F2 F2 Vertex Figure

P 0

P 1

194 P 01 P 0 P 1 P 01

P 02

P 12

012 P P 02 P 12 P 012 Table 7.36: Faces and Vertex Figures for Wythoﬃans of {∞, 6}3#{∞}

{0,1} {1,2} {0,2} Wythoﬃan F2 F2 F2 Vertex Figure

P 0

P 1 195 P 0 P 1 P 01

P 01

P 02

P 012

P 02 P 012 Table 7.37: Faces and Vertex Figures for the Wythoﬃans of {4, 6|4}

{0,1} {1,2} {0,2} Wythoﬃan F2 F2 F2 Vertex Figure

P 0

P 1

P 0 P 1 P 2

P 2 196

P 01

P 01 P 02 P 12 P 02

P 12

P 012 P 012 Table 7.38: Faces and Vertex Figures for the Wythoﬃans of {6, 4|4}

{0,1} {1,2} {0,2} Wythoﬃan F2 F2 F2 Vertex Figure

P 0

P 1

P 0 P 1 P 2

P 2 197

P 01

P 01 P 02 P 12 P 02

P 12

P 012 P 012 Table 7.39: Faces and Vertex Figures for Wythoﬃans of {∞, 6}4,4

{0,1} {1,2} {0,2} Wythoﬃan F2 F2 F2 Vertex Figure

P 0

P 0 P 1

P 1 198

P 01

P 01 P 02 P 02

P 012

P 012 Table 7.40: Faces and Vertex Figures for Wythoﬃans of {∞, 4}6,4

{0,1} {1,2} {0,2} Wythoﬃan F2 F2 F2 Vertex Figure

P 0

P 0 P 1

P 1 199

P 01

P 01 P 02

P 02

P 012

P 012 Table 7.41: Faces and Vertex Figures for the Wythoﬃans of {∞, 4}·,∗3

{0,1} {1,2} {0,2} Wythoﬃan F2 F2 F2 Vertex Figure

P 0

P 1 200 P 01 P 0 P 1 P 01

P 02

P 12

P 02 P 12 P 012 P 012 Table 7.42: Faces and Vertex Figures for the Wythoﬃans of {6, 6}4

{0,1} {1,2} {0,2} Wythoﬃan F2 F2 F2 Vertex Figure

P 0

P 1

P 0 P 1 P 2

P 2 201

P 01

P 01 P 02 P 12 P 02

P 12

P 012 P 012 Table 7.43: Faces and Vertex Figures for the Wythoﬃans of {4, 6}6

{0,1} {1,2} {0,2} Wythoﬃan F2 F2 F2 Vertex Figure

P 0

P 1

P 0 P 1 P 2

P 2 202

P 01

P 01 P 02 P 12 P 02

P 12

P 012 P 012 Table 7.44: Faces and Vertex Figures for the Wythoﬃans of {6, 4}6

{0,1} {1,2} {0,2} Wythoﬃan F2 F2 F2 Vertex Figure

P 0

P 1

P 0 P 1 P 2

P 2 203

P 01

P 01 P 02 P 12 P 02

P 12

012 P 012 P Table 7.45: Faces and Vertex Figures for the Wythoﬃans of {6, 6|3}

{0,1} {1,2} {0,2} Wythoﬃan F2 F2 F2 Vertex Figure

P 0

P 1

P 0 P 1 P 2

P 2 204

P 01

P 01 P 02 P 12 P 02

P 12

P 012 P 012 Table 7.46: Faces and Vertex Figures for Wythoﬃans of {∞, 6}6,3

{0,1} {1,2} {0,2} Wythoﬃan F2 F2 F2 Vertex Figure

P 0

P 1

205 P 0 P 1 P 01 P 01

P 02

P 02 P 012 P 012 Table 7.47: Faces and Vertex Figures for Wythoﬃans of {∞, 3}(a)

{0,1} {1,2} {0,2} Wythoﬃan F2 F2 F2 Vertex Figure

P 0

P 1

206 P 01 P 0 P 1 P 01

P 02

P 12

P 02 P 12 P 012

P 012 Table 7.48: Faces and Vertex Figures for Wythoﬃans of {∞, 3}(b)

{0,1} {1,2} {0,2} Wythoﬃan F2 F2 F2 Vertex Figure

P 0

P 1

P 01 207 P 0 P 1 P 01

P 02

P 12

P 02 P 12 P 012

P 012 Bibliography

[1] C.T. Benson and L.C. Grove. Finite Reﬂection Groups. Springer, 2nd edition, 1985.

[2] A. Brønsted. An Introduction to Convex Polytopes. Spring-Verlag, 1983.

[3] H.S.M. Coxeter. Discrete groups generated by reﬂections. Annals of Mathematics, 35:588–621, 1934.

[4] H.S.M. Coxeter. Wythoﬀ’s construction for uniform polytopes. Proc. London Math. Soc. (2), 38:327–339, 1935.

[5] H.S.M. Coxeter. Regular and semi-regular polytopes, I. Mathematische Zeitschrift, 46:380–407, 1940.

[6] H.S.M. Coxeter. Regular Polytopes. Dover, 1973.

[7] H.S.M. Coxeter. Regular Complex Polytopes. Cambridge University Press, second edition edition, 1991.

[8] H.S.M. Coxeter and S.L. Greitzer. Geometry Revisited. Mathematical Association of America, 1967.

[9] H.S.M. Coxeter, M.S. Longuet-Higgins, and J.C.P. Miller. Uniform polyhedra. Phil. Trans. Roy. Soc. London Series A, 246:401–450, 1954.

[10] P. Cromwell. Polyhedra. Cambridge University Press, 2002.

[11] G. Cunningham. Constructing self-dual chiral polytopes. European Journal of Combinatorics, 33:1313–1323, 2012.

[12] A.M. Cutler, E. Schulte, and J.M. Wills. Icosahedral skeletal polyhedra realizing Petrie rela- tives of Gordan’s regular map. Contributions to Algebra and Geometry, 54:651–657, 2013.

[13] M. del Rio Francos, I. Hubard, A. Orbanic, and T. Pisanski. Medial symmetry type graphs. Electronic Journal of Combinatorics, 20(3):P29, 2013.

[14] M. Deza and M. Dutour. Goldberg-Coxeter construction for 3- and 4- valent plane graphs. Electronic Journal of Combinatorics, 11:R20, 2004.

208 [15] M. Deza, M.D. Sikiric, and S. Shpectorov. Hypercube embeddings of Wythoﬃans. Ars Math- ematica Contemporanea, 1:99–111, 2008.

[16] A.W.M. Dress. A combinatorial theory of Gr¨unbaum’s new regular polyhedra, Part I: Gr¨unbaum’s new regular polyhedra and their automorphism group. Aequationes Mathemati- cae, 23:252–265, 1981.

[17] A.W.M. Dress. A combinatorial theory of Gr¨unbaum’s new regular polyhedra, Part II: Com- plete enumeration. Aequationes Mathematicae, 29:222–243, 1985.

[18] S. L. Farris. Completely classifying all vertex-transitive and edge-transitive polyhedra, Part I: Necessary class conditions. Geometriae Dedicata, 26:111–124, 1988.

[19] B. Gr¨unbaum. Regular polyhedra - old and new. Aequationes Mathematicae, 16:1–20, 1977.

[20] B. Gr¨unbaum. Convex Polytopes. Springer, second edition edition, 2003.

[21] B. Gr¨unbaum. “New” uniform polyhedra. Discrete Geometry: In Honor of W. Kuperberg’s 60th Birthday, pages 331–350, 2003.

[22] B. Gr¨unbaum and N.W. Johnson. The faces of a regular-faced polyhedron. Journal of London Math. Soc., 40:577–586, 1965.

[23] B. Gr¨unbaum and G.C. Shephard. Tilings and Patterns. WH Freeman and Company, 1987.

[24] Z. Har’El. Uniform solution for uniform polyhedra. Geometriae Dedicata, 47:57–110, 1993.

[25] I. Hubard and A.I. Weiss. Self-duality of chiral polytopes. Journal of Combinatorial Theory, Series A, 111:128–136, 2005.

[26] Wolfram Reseach Inc. Mathematica 10.0. Wolfram Reseach, Inc., 2014.

[27] N.W. Johnson. Convex solids with regular faces. Canadian Journal of Mathematics, 18:69–200, 1966.

[28] P. McMullen. Realizations of regular apeirotopes. Aequationes Mathematicae, 47:223–239, 1994.

[29] P. McMullen and B. Monson. Realizations of regular polytopes, II. Aequationes Mathematicae, 65(1):102–112, 2003.

[30] P. McMullen and E. Schulte. Constructions for regular polytopes. Journal of Combinatorial Theory, Series A, 53:1–28, 1990.

[31] P. McMullen and E. Schulte. Regular polytopes in ordinary space. Discrete and Computational Geometry, 17:449–478, 1997.

[32] P. McMullen and E. Schulte. Abstract Regular Polytopes. Cambridge University Press, 2002.

209 [33] D. Pellicer. A construction of higher rank chiral polytopes. Discrete Mathematics, 310:1222– 1237, 2010.

[34] D. Pellicer and Weiss A.I. Combinatorial structure of Schulte’s chiral polyhedra. Discrete and Comuptational Geometry, 44:167–194, 2010.

[35] D. Pellicer and E. Schulte. Regular polygonal complexes in space, I. Transactions of the American Mathematical Society, 362:6679–6714, 2010.

[36] E. Schulte. Tilings. Handbook of Convex Geometry, pages 899–932, 1993.

[37] E. Schulte. Chiral polyhedra in ordinary space, I. Discrete and Computational Geometry, 32:55–99, 2004.

[38] E. Schulte. Symmetry of polytopes and polyhedra. Handbook of Discrete and Computational Geometry, 2004.

[39] E. Schulte. Chiral polyhedra in ordinary space, II. Discrete and Computational Geometry, 34:181–229, 2005.

[40] E. Schulte. Polyhedra, complexes, nets and symmetry. Acta Crystallographica Section A, 70(3):203–216, 2014.

[41] M. Senechal. Shaping Space: Exploring Polyhedra in Nature, Art and the Geometrical Imagi- nation. Springer, 2013.

[42] J. Skilling. The complete set of uniform polyhedra. Phil. Trans. Roy. Soc. London Series A, 278:111–135, 1975.

[43] J. Skilling. Uniform compounds of uniform polyhedra. Mathematical Proceedings of the Cam- bridge Philosophical Society, 79:447–457, 1976.

[44] M. Wenninger. Polyhedron Models. Cambridge University Press, 1971.

[45] M. Wenninger. Spherical Models. Cambridge University Press, 1979.

[46] W.A. Wythoﬀ. A relation between the polytopes of the C600-family. Koninklijke Akademie van Wetenschappen te Amsterdam, 20, 1918.

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