Non-Fundamental Trunc-Simplex Tilings and Their Optimal Hyperball Packings and Coverings in Hyperbolic Space I

Non-fundamental trunc-simplex tilings and their optimal hyperball packings and coverings in hyperbolic space I. For families F1-F4

Emil Moln´ar1, Milica Stojanovi´c2, Jen˝oSzirmai1

1Budapest University of Tehnology and Economics Institute of Mathematics, Department of Geometry, H-1521 Budapest, Hungary [email protected], [email protected]

2Faculty of Organizational Sciences, University of Belgrade 11040 Belgrade, Serbia [email protected]

Abstract Supergroups of some hyperbolic space groups are classified as a contin- uation of our former works. Fundamental domains will be integer parts of truncated tetrahedra belonging to families F1 - F4, for a while, by the notation of E. Molnáret al. in 2006. As an application, optimal congruent hyperball packings and coverings to the truncation base planes with their very good densities are computed. This covering density is better than the conjecture of L. Fejes Tóthfor balls and horoballs in 1964.

1 Introduction arXiv:2003.13631v1 [math.MG] 30 Mar 2020 1.1 Short history Hyperbolic space groups are isometry groups, acting discontinuously on hyperbolic 3-space H3 with compact fundamental domains. It will be investigated some series of such groups by looking for their fundamental domains. Face pairing identiﬁcations

2000 Mathematics Subject Classiﬁcations. 51M20, 52C22, 20H15, 20F55. Key words and Phrases. hyperbolic space group, fundamental domain, isometries, truncated simplex, Poincar`ealgorithm.

1 2 Molnár,Stojanović,Szirmai on a given polyhedron give us generators and relations for a space group by the algorithmically generalized PoincaréTheorem [4], [6], [11] (as in subsection 1.2). The simplest fundamental domains are 3-simplices (tetrahedra) and their integer parts by inner symmetries. In the process of classifying the fundamental simplices, 64 combinatorially different face pairings of fundamental simplices were determined [28], [8], and also 35 solid transitive non-fundamental simplex identifications [10]. I. K. Zhuk [28] classified Euclidean and hyperbolic fundamental simplices of finite volume. An algorithmic procedure was given by E. Molnárand I. Prok [7]. In [8–10] the authors summarized all those results, arranging identified simplices into 32 families. Each of them is characterized by the so-called maximal series of simplex tilings, i.e. maximal symmetry groups with smallest fundamental domains. The other members are derived by symmetry breaking of this maximal one. These principles are also discussed by D-symbols in [10], see also 1.2. Some complete cases of supergroups with fundamental truncated simplices (shortly trunc-simplices) are discussed in [5,14–19]. As a classification, supergroups of the groups with fundamental trunc-simplices are given in [20]. Especially, supergroups of the groups with fundamental simplices belonging to families F1-F4, by the notations given in [10], are investigated in [5, 15, 18, 19]. For that purpose, the stabilizer group of the corresponding vertex figure is analyzed. The new method, for such analysis, established e.g. in the summary works [19, 20], is also confirmed by the results given in [3].

1.2 Preliminaries Generators and relations for a space group G with a given polyhedron P as a fundamental domain can be obtained by the Poincarètheorem, applying Poincarè algorithm. In this paper such polyhedra will be simplices and trunc-simplices, for overview see e.g. [19, 20], and here mainly their integer parts. It is necessary to consider all face pairing identifications of such domains. These are isometries which generate a group G and induce subdivision of faces and directed edge segments of P into equivalence classes, such that an edge segment does not contain two G-equivalent points in its interior. The Poincarèalgorithm (see [4], [6]) gives us for each edge segment class one cycle transformation of the form c = gr . . . g2g1, where gi, i = 1, 2, . . . r are face pairing identifications. Since each of these cycle transformations is a rotation of order ν, the cycle relations are of the ν form (gr . . . g2g1) = 1. The Poincarètheorem guarantees that these cycle relations, 2 −1 together with relations, gi = 1 to the occasional involutive generators gi = gi , form a complete set of defining relations for G. Remark 1. As an intuitive picture (e.g. from [6]) we shall use in the following, the space tiling P G by fundamental domains (polyhedra) represents the space group G itself after having chosen an arbitrary identity domain P . For any generator g ∈ G g domain P will be adjacent with the g-image domain P along face fg denoted simply either by g on the side (half space) of P g, or by g−1 on the side of P (we prefer the first). Involutive generators g = g−1 we write just on the corresponding face. Thus for any γ ∈ G, domain P γ will be adjacent with domain P gγ along an image Non-fundamental trunc-simplex tilings ... 3

γ gγ −1 γ face fg , signed simply either by g on side of P , or by g on the side of P . γ So domain P is obtained by γ = gr . . . g2g1 from P through the path in domains g1 g2g1 γ P , P , P , ... , P entered finely through gr. Imagine also the group graph by coloured directed edges with so many colours as many generators we have. From g and g−1 we use only g, as to direction of the corresponding graph edge. Classification all non-fundamental trunc-simplex tilings means the following finite process (7 steps as a natural sketchy generalization of our earlier works by D−symbol method or barycentric subsimplex orbits, see e.g. [10] and [11], you can skip this difficult part at the first read). 1. We consider a 3-simplex T (as tetrahedron) with its barycentric subdivision into 24 subsimplices. Any subsimplex has a 3-centre (solid centre, first for- mally, later also metrically), a 2-centre in a simplex face, a 1-centre in an edge of the provious face, a 0-centre as a vertex of the previous edge, as usual. The simplex has first the piece-wise linear (PL) topology. Any subsimplex has a 3−face. 2−, 1−, 0−faces, opposite to the corresponding cen- tres. Then 2−, 1−, 0− (involutive (involutory)) adjacency operations σi can be introduced. Then occasional symmetry operations come acting on points, segments ... , preserving the subsimplices and incidences with adjacency operations. E.g. we can write by convention σiCg := (σiC)g = σi(Cg), (1.1) that means for any subsimplex C: the adjacency operation σi (i ∈ {0, 1, 2} first) and mapping g commute. So, σi’s act on left, symmetries act on right. 2. The σi operation, as above (i ∈ {0, 1, 2} at the first considerations), can be considered as a local reflection in the i−side face of any subsimples C. Then any symmetry mapping g (if it exists) can be given by a starting subsimplex and its image subsimplex and by simplex-wise extension, so that (1.1) holds further step-by-step, as requirement. The axiomatically defined symmetric D−matrix function (mij = mji) and the consequences for D−symbol “theory” (e.g. in [10], [11]) can guarantee the extension of g first to the first simplex T , then – introducing σ3 operation for the tile adjacency in the next step – onto the whole tiling T .

3. The face pairing generator symmetry mappings of T , i.e. g1, g2,... (may be involutive one as well), introduce also σ3 adjacency operations and generate a simplex tiling T = T G under a symmetry group G, finitely generated by the 23 previous g1, g2,... . The D−matrix function has free parameters yet for m = m32 entries, so we shall have infinite series of tilings for a given D−diagram ij with certain free rotational orders. The D-matrix function kD 7→ m (kD) of the D-symbol says, by requirement, that the ij−edge at the meeting of σi, σj side faces of any subsimplex kC in the D−set of the D−symbol, schould be ij G surrounded by 2m (kD) subsimplices in the tiling T = T . (Note, e.g. for the 3−simplex tiling T that m01 = m12 = 3 constant, and m02 = m03 = m13 = 2 constant, because of the barycentric subdivision of T .) 4 Molnár,Stojanović,Szirmai

4. Around the half-edge G−equivalence classes we get subsimplices providing defining relations for G (besides the involution relations), just in the sense of Poincaré.However, it is our main consideration, we can take the D0 sub- symbols, obtained by leaving σ0 operations from the D−diagram (and the D−matrix entries), the components describing the fundamental domains of the vertex stabilizer subgroups for the vertex equivalence classes induced, as a cutting 2−surface. The curvature formula ( [10], [11]) says whether a vertex stabilizer (depending on the free parameters) is either a finite spherical group (of positive curvature), or Euclidean group (of zero curvature), or hyperbolic one (of negative curvature).

5. In this paper we concentrate on the last cases, where the parameters involve only outer vertices of hyperbolic space H3. Than we cut (truncate) this vertex(ices) orthogonally by polar plane(s) in H3. The vertex stabilizer group acts then, preserving the two half spaces of the polar plane(s). Introducing half-space changing isometries to this H2−stabilizer, preserving the original simplex tiling T , so we get a space group with compact fundamental domain. The procedure induces face pairing at truncating vertex polars as well.

6. This new face pairings can be formulated by a new adjacency operation σ0∗ commuting with other σj’s (j ∈ {1, 2, 3}) but not with σ0. Computer can also help, but we have made careful case-by-case discussions, ﬁrst for later experiences.

7.A D−symbol isomorphism leads to equivariant group extension. Morphism does it onto a smaller D−diagram, or opposite procedures, lead to symmetry breaking members in the same family, where the free parameters need extra attention (see [10] and [11]).

1.3 Results In Section 2 we illustrate our method with some cases from family F1, then the leading representative group series will be detailed from each family (Sections 2-5), sometimes with characteristic examples, also accompanied with ﬁgures. Finally we have proved the following summary

Theorem 1. For integer parts of the trunc-simplices of families F1 - F4 there are given the 73 group extensions all together. Namely, ∗233 43¯ m F1: Representing series Γ(u) = 24Γ(u) have 14 extensions, 2∗2 42¯ m F2: Representing series Γ3(u, 2v) = 8Γ(u, 2v) have 21 extensions, ∗33 3m F3: Representing series Γ(2u, v) = 6Γ(2u, v) have 17 extensions, ∗22 mm2 F4: Representing series Γ1(u, 2v, w) = 4Γ1(u, 2v, w) have 21 extensions. There are given all the space group series in ﬁgures and by group presentations, namely, the face pairing generators and deﬁning relations to the edge-segment equivalence classes in Sections 2-5. The other non-fundamental series for families F5 - F12 will be published later. Non-fundamental trunc-simplex tilings ... 5

The truncation plane of a trunc-simplex will be a natural base plane for a hypersphere (constant distance surface from the base plane in both sides), whose ball (the inner part of the hypersphere) with its images under the symmetry group of the corresponding trunc-simplex tiling, can fil (pack) the whole space H3 without common interior point, or can cover H3 without gap, respectively. As a by-product, we have determined the optimal, densest packing for each family from F1-F4, moreover, the optimal thinnest (loosest) covering for families F1- F2. Now this is straightforward, because the Coxeter-Schläfli(extended) reflection groups {u, v, w} occur for these families. See tables to Section 6. However, the coverings for families F3-F4 deserve a separate publication with worser densities than the optimal covering at F2. Theorem 2. Among the hyperball packings and coverings to the truncation base planes of hyperballs in the trunc-simplex tilings in our families F1-F4 there is a densest packing in F1 to group ∗233Γ (u = 7) (as complete Coxeter group {3, 3, 7}) with density ≈ 0.82251 (in Fig. 1); and there is a loosest covering in F2 to group 2∗2Γ (u = 7, 2v = 6) (extended {7, 3, 7}) with density ≈ 1.26829 (in Fig. 8 and Fig. 25). We conjecture (see also investigations [21–26] of the third author in References) that we have found among all arrangements of congruent hyperballs in H3 the densest hyperball packing with density ≈ 0.82251 above (related to Dirichlet-Voronoi cell subdivision) to the regular trunc-simplex group {3, 3, 7}; furthermore the loosest hyperball covering with density ≈ 1.26829 above, to the extended trunc-orthoscheme group {7, 3, 7}. This last result is published first time here (surprising a little bit). This covering density is less than that in the conjecture of L. Fejes Tóth[1] for balls and horoballs.

2 Family F1

The maximal series of family F1 is characterized by the simplex tiling whose group 43¯ m series is 24 Γ(u) with previous (crystallographic) notation from [10], and now with the new (orbifold) notation ∗233Γ(u) (see [3]). It has the trivial extension by the reﬂection a0 =m ¯ 0 (Fig. 1:F1-1)

∗233 2 2 2 2 3 2 Γ(u) = (m0, m1, m2, m3 − 1 = m0 = m1 = m2 = m3 = (m0m1) = (m0m2) =

2 3 2 u = (m0m3) = (m1m2) = (m1m3) = (m2m3) ; 6 < u ∈ N) 2 1 2 2 2 3 2 with extenion (a0 − 1 = a0 = (a0m1) = (a0m2) = (a0m3) ).

Other non-fundamental cases in F1:

233 3 2 2 −1 3 u • Γ1(2u) = (r, r2, m3−1 = r = r2 = m3 = m3rm3r = (rr2) = (m3r2m3r2) , 23 3 < u ∈ N) = 12Γ1(2u), with 2 extensions 2 2 1 −1 2 3 2 4 2 (1) (m ¯ 0, m¯ 1 − 1 =m ¯ 0 =m ¯ 1 =m ¯ 0r m¯ 1r =m ¯ 0r2m¯ 1r2 = (m ¯ 0m3) = (m ¯ 1m3) ) 6 Moln´ar,Stojanovi´c,Szirmai

1 2 2 2 3 −1 (2) (¯s − 1 = (¯sr) = (¯sr2) = m3sm¯ 3s¯ ). 0 2 The pictures are given in Fig. 1: F1-2 with Γ (A0,A1) = 3 ∗ u as H group with half-turn extension.

233 3 2 2 3 2 u • Γ2(u) = (r, r2, r3 − 1 = r = r2 = r3 = (rr2) = (rr3) = (r2r3) , 6 < u ∈ 23 N) = 12Γ2(u), with 2 extensions 2 2 1 −1 2 3 (1) (m ¯ 0, m¯ 1 − 1 =m ¯ 0 =m ¯ 1 =m ¯ 0r m¯ 1r =m ¯ 0r2m¯ 1r2 =m ¯ 0r3m¯ 1r3) 1 2 2 2 3 2 (2) (¯s − 1 = (¯sr) = (¯sr2) = (¯sr3) ). In Fig. 2: F1-3 the pictures are given with Γ0 = 23u as H2 group with half-turn extension.

2× 2 2 −1 u • Γ3(3u) = (z, r1, r3 − 1 = r1 = r3 = zzr1 = (r1r3zr3z r3) , 2 < u ∈ N) = 4¯ 4Γ3(3u), with 2 extensions 2 2 2 1 2 −1 3 4 (1) (m ¯ 0, m¯ 1, m¯ 2 − 1 =m ¯ 0 =m ¯ 1 =m ¯ 2 =m ¯ 0r1m¯ 2r1 =m ¯ 0z m¯ 1z =m ¯ 0r3m¯ 1r3 = −1 5 2 m¯ 1z m¯ 2z = (m ¯ 2r3) ) ¯ ¯2 1 −1¯ 2 2 3 2 5 ¯ 2 (2) (¯g, h2 − 1 = h2 =gz ¯ h2r1 = (¯gz) = (¯gr3) = (h2r3) ). The pictures are given in Fig. 2: F1-4 with Γ0 = 2u× as H2 group with half-turn extension.

33 3 2 2 2 −1 u 3 • Γ5(4u) = (r, r2, r3 − 1 = r = r2 = r3 = (rr2) = (r3rr3r r3r2) , 2 < u ∈ 3 N) = 3Γ5(4u), with 2 extensions 2 2 2 1 −1 2 3 (1) (m ¯ 0, m¯ 1, m¯ 2 − 1 =m ¯ 0 =m ¯ 1 =m ¯ 2 =m ¯ 0r m¯ 1r =m ¯ 0r2m¯ 1r2 = 4 2 5 −1 m¯ 0r3m¯ 2r3 = (m ¯ 1r3) =m ¯ 2rm¯ 2r ) ¯ ¯ ¯ ¯2 ¯2 ¯2 1 ¯ ¯ 3 ¯ ¯ 4 ¯ 2 5 ¯ 2 (2) (h0, h1, h2 − 1 = h0 = h1 = h2 = h0r2h1r = h0r3h2r3 = (h1r3) = (h2r) ). The pictures with Γ0 = 223 as H2 group with two half-turn extensions are given in Fig. 3: F1-5.

33 3 2 2 −1 −1 2u 3 • Γ6(8u) = (r, m2, r3 − 1 = r = m2 = r3 = m2rm2r = (m2r3rr3r r3) , 4 < 3 u ∈ N) = 3Γ6(8u), with trivial extension 2 2 2 1 −1 2 2 3 4 (1) (m ¯ 0, m¯ 1, m¯ 2 − 1 =m ¯ 0 =m ¯ 1 =m ¯ 2 =m ¯ 0r m¯ 1r = (m ¯ 0m2) =m ¯ 0r3m¯ 3r3 = 2 5 2 6 −1 (m ¯ 1m2) = (m ¯ 1r3) =m ¯ 2rm¯ 2r ). See Fig. 3: F1-6 for the pictures with Γ0 = 23 ∗ u.

22 2 −1 −1 −1 u 2 • Γ11(6u) = (z, h − 1 = h = (hzzhzhz hz z ) , 1 < u ∈ N) = 2Γ11(6u), with 4 extensions, each in two variants, given in Fig. 4 - 6 to F1-7. These so-called Gieseking orbifolds are derived from the famous non-orientable manifold with an inﬁnite vertex class, originally if u = 1 (cusp, but 1 < u now). The half-turn extension h allows various half simplices and fundamental domains, e.g. for the trivial extension as follows (Fig. 4: F1-71). 2 2 2 2 1 2 −1 3 (1) (m ¯ 0, m¯ 1, m¯ 2, m¯ 3 − 1 =m ¯ 0 =m ¯ 1 =m ¯ 2 =m ¯ 3 =m ¯ 0hm¯ 1h =m ¯ 0z m¯ 1z = −1 4 5 −1 m¯ 1z m¯ 2z =m ¯ 2hm¯ 3h =m ¯ 3zm¯ 3z ). Fig. 6: F1-73 shows the simpler version of extensions (2)-(4) by (additional) presentations: Non-fundamental trunc-simplex tilings ... 7

¯ ¯ ¯2 ¯2 2 ¯ −1 2 3 ¯ −1¯ 5 ¯ 2 (2) (h0, h2 − 1 = h0 = h2 = (h0z h) = h0hz h2zh = (h2hzh) ), (3) (¯g − 1 =2 ghzh ¯ g¯−1z−1h =3 (¯gzh)2), (4) (ˆs − 1 =2 shzh ˆ sˆ−1hz =3 (ˆszh)2).

2.1 The 4 extensions to Gieseking orbifolds as fundamental trunc-simplex tilings

Our Fig. 7: F1-7+ to Γ62(6u) shows this famous simplex face pairing, as repetition from [20], with orientation reversing transforms −1 −1 z1 : z1 → z1 fixing vertex A3 and z2 : z2 → z2 fixing A0 All vertices lie at the infinity (absolute) of H3, iff the “rotation order” parameter at the edge equivalence class is u = 1. At the polar plane of A0 (say, e.g.) we look the fundamental domain F 0 of the vertex class stabilizer subgroup Γ0 = uu×× that is the Euclidean Klein bottle group 4.pg (with two cross caps ×, or projective planes), iff u = 1. Moreover, we get interesting “surgery effects” that lead e.g. to the famous Fomenko-Matveev-Weeks hyperbolic space form of minimal volume [27]. 2 0 4π Otherwise, iff 1 < u ∈ N we get H plane group (area of F is u less then 4π [3]). Then the generators, reversing the half-spaces of the polar plane a0, but preserving the original simplex tiling T , lead to 4 group extensions with compact 1 4 fundamental domains as trunc-simplices (or octahedra O62,... O62). Presentation of the original simplex tiling group is −1 −1 u Γ62(6u) = (z1, z2 − 1 = (z1z2z2z1 z2 z1) ; 1 < u ∈ N). (1) Then comes the trivial extension with plane reflections:

2 2 2 2 1 −1 2 −1 3 (m0, m1, m2, m3 − 1 = m0 = m1 = m2 = m3 = m0z1 m2z1 = m0z2m0z2 = 3 −1 4 −1 5 −1 6 −1 = m1z1m2z1 = m1z2m2z2 = m1z2 m3z2 = m3z1m3z1 ). (2) The half turns extension, changing the half spaces at F 0 (inducing point reﬂections at edges 2, 6, moreover, a 2−knot u−segment combination):

2 2 2 2 1 −1 2 2 3 (h0, h1, h2, h3 − 1 = h0 = h1 = h2 = h3 = h0z1 h2z1 = (h0z1) = 3 −1 5 −1 6 2 = h1z1h2z2 = h1z2 h3z2 = (h3z1) . (3) The most interesting half-screw extension, changing the half-spaces at F 0 yields a “non-orientable u−knot” (by glide reﬂections g1, g2): 1 −1 2 −1 3 (g1, g2 − 1 = g1z2g2 z1 = g1z1g1 z2 = g2z1g2z2). (4) The 4th extension is a point reﬂection at edges 3, 4 of F 0 (inducing screw motions s1, s2 and u−knot):

1 −1 2 −1 −1 3 2 4 2 (s1, s2 − 1 = s1z2s2 z1 = s1z1s1 z2 = (s2z1) = (s2z2) ). We think that these phenomena are remarkable! 8 Moln´ar,Stojanovi´c,Szirmai

233 * G(u) trivial extension Coxeter diagram

A0123 A 3 m0 3 3 u

m1 m2 m3 A2 a0 =m0 m2 33 m1 * 3m= 33 *33 22 0 m0 * * F 2 1 P3 m a0 A012 A

2 1 2 A 0 m 22 22 * 3 * m3 * uu um= uu 2m=*22 A P A A * 0 A0 1 01 A1 1 m G0 3 =*23u A01 A1

m0 * 22

22 m m * 3 3 2 2 a0 1 A 0 A0

m * 33 * 22 1 * 33 A A0123 A 012 A2 3 233 G 1 (2u) 2 extensions

A0123 -1 r G0 r 0 = 3* u A F 2 - 3

33 r2 r2

33 1 r 33 2

r2 A0 r A1 2

- 3 - - u m 4 u 3 * m3

A012 * * * * uu uu A0 A01 A1 m3

1 1

2 m A s 2 A1 4 1 1 3

* uu

uu uu

* * 33 A 33

A0 r * 0 r

* m 2 m3 2 3 3

3 2 -1 2 m0 s 1 1

33 33 * r * r

33 33 -1 -1 A012 r A0123 A012 r A0123

∗233 233 Figure 1: F1-1 to Γ(u) and F1-2 to Γ1(2u) Non-fundamental trunc-simplex tilings ... 9

233 G 2 (u) 2 extensions

A0123 -1 r G0 = 23u r F 0(A , A ) A 0 1 2 - 3

33 r r

2 1 2

33 33 A 2 r2 A0 r 1

u- - 3 - r3 3 2 r3 u

22 A012 22 uu uu A0 A01 A1 r3

1 A 1 A

1 s 1 2 m 2 3 1 3

33 r A 33

A 22 3 0 r 22 r 0 r2 2 3 3

3 2 -1 2 m0 s 1 1

r 33 r 33 22 22

33 33 -1 -1 A012 r A0123 A012 r A0123 2×G 3 (3u) 2 extensions

0 A0123 F u-

G0 = 2u 4 × r1 -1 1 A r3 z 2 5 z

uu r 1 3 1 uu 2 r1 zz z

z A2= A2 A1

1 3

uu 5 z uu 2 r1 1 A uu uu 0 A A - 0 01 A1 u r3 u-

r3 3

2 2

m1 A g A 1 3 1

3 4 3 1

uu uu

uu r3 r3 A z 1 A z 1 0 3 0

3 2 -1 2 m0 3 g 1 1 z-1 z-1 uu uu uu uu 1 1

uu uu

5 5

5 r1 r1

1 1

m2 A0123 A0123 4 h 1 2 1 1

A2 A2

233 2× Figure 2: F1-3 to Γ2(u) and F1-4 to Γ3(3u) 10 Moln´ar,Stojanovi´c,Szirmai

33G 5 (4u) 2 extensions

A3 0 - F 3 r 0

5 G = 223u

r r u- A 3

2 2

A 5 r3 2 3 A A0 r2 013 3 ext. uu - - r u 1 2 r-1 uu r 4 r

22 r2

22 A 2 1 ext.

uu r2 uu r3 4 u- uu uu

A0 A01 A1 r3

1 1

m1 A h A 4

1 4 1 2 2 1

4 4

4 4 11

uuu u uu r3

uu r3 uu rr 22

A 22 r2 222

0 A0

1 3 3 3

2 m 3 0 h0 1 1

u u uu uu uu u 222 uu 222

uu uuu 3 r 3 3 r

5 5 m2 A013 A013 333 -1 h2 333 5 r 5 r-1

A2 A2 33G 6 (8u) trivial extension G0 A3 = 23* u

0 3 - F 6 . . 6 r r 3 A2 r

A2 - 3 - u r3 u A013 33 2 uu r * A

r uu 5 0 -1 * 1 r m2

r 3 r * * A1 - uu uu 5 4 * * m2 * m2 - * uu *uu u A0 A01 A1 r3 1

m1 A

5 1 5 4

* uu r3 * *

A0 3 m2 2 m0 1

uu * uu * *

3 * r 6

m2 A013 6 . 33 . r-1 A2

33 33 Figure 3: F1-5 to Γ5(4u) and F1-6 to Γ6(8u) Non-fundamental trunc-simplex tilings ... 11

22 G (6u)

11 4 extensions 2 0 0 A1 G

m1 = u×× . 3 F 1 6 8 uu 4 z u- 5 u- uu h uu z hzh 3 uu 5 A 3 zh h

A A2 2

5 0 4 4

1 z 3 h h 1 2 uu -1 z m z A1 0 11 z 3 7 z 2 uu uu u- u- 2 2 1 3 half turn half uu ext. 4 point reﬂ.

5 turn ext.

4 ext.

4 m

m2 3 5 3 uu uu A A3 2 9 10 1 trivial extension G (6u) 22G

62 Gieseking = 2 11(6u) simple variant 2 A1 A1 2

1 with various side

2 1 3

faces of h 3 uu

A01 A01 22u

uu 22u A z A z uu z 0 22u 0 -1

3 1 2 3 2 z -1 m0

2 z 2 uu h 22u uu -1 5 5

z 5 A

4 0123

5 5

4 5 3 5 m2 A2 3 3 uu 22u A3 A3 A2 A23

h axis through A01,A23 and simplex centre A0123 h z z h = z = hzh h h A1 =A0 , A3 =A2 , the side faces of h can be varied G (6u) This is a modiﬁed version of 62

22 Figure 4: F1-71 to Γ11(6u) as Gieseking orbifolds 12 Moln´ar,Stojanovi´c,Szirmai

22 G (6u)

11 cont.

2 A1 2 A1 -1

s1 g

1 2 1

3 3

A01 A01

22u 22u

22u 22u A A 22u 0 z 22u 0 z 2 3 3 -1

-1 -1 g -1 2

s 1 1 z 1 1 2 z

h 22u h 22u

2 5

5 5

A 2 A

-1 0123 4 0123 1

s 4 5 g

3 2 3 1 1

3 g

s2 2 2

5 5 3

3 3 22u A3 22u A3 A2 A2

g1 g2 s1 s2 3 ext. A A A 2 ext. A A , A A , 0 2 , 1 A3 , 0 1 2 3 , 1 1, 4 4 1 4, (3,3) (3,3) 2 5 3,3 3 3, 2 2 2,

5 5 5, 2 A1 -1 s2

1 3 3

A01 22u

22u A z 22u 3 0

-1 -1 s11 2 z

h 22u

2 A0123

s 1

2 1 2 3 5 3 s1

3 22u A3 A2 s 4 1 s2 ext. A0 A3 , A1 A2 , 1 4, 3,3 3 3, 2 5.5

22 Figure 5: F1-72 to Γ11(6u), ﬁrst version of ext. (2)-(4) Non-fundamental trunc-simplex tilings ... 13

22 G (6u)

11 cont. 2

2 2 2

3 3 3

A01 01

22u 22u

A 22u z 22u 0 z 3 2 -1

-1 3 2 2 -1 3 z g z h0 2

2 2 h 22u h 22u

2 5

5 5 A 2 A0123 0123 5

3 5 g

h2 2 2

5 5 3 3 22u 22u

2 ext. 3 3 3, 2 2 2 3 ext. 3 3 , 2 5, g

5 5 5, h0 , h2 2

3 3

A01 22u

22u z 22u 3 -1

-1 2 3 s z

h 22u

2 2

A0123

2 3 s 2

3 22u

4 ext. 3 3 3, 2 5, s

22 Figure 6: F1-73 to Γ11(6u), second simpler version of ext. (2)-(4) 14 Moln´ar,Stojanovi´c,Szirmai

G (6u) 62 fundamental 4 extensions

Gieseking orbifolds

3 2 3

m1 h1

2 3 3 4 3 5

5 5

uu uu

uu uu uu uu -1 -1

uu z2 uu z2

z2 A0 z2 A0 2 2 2

3 2 2 m 2 0 h0

12 1 1 uu uu uuuu uuuu

5 5 5 5 5

53 54 .

z1 z1 6 . 4 6 4

5 m3 6 h 5

3 . 1 6 3

m2 3 . h 3

6 2 .

1 . 6 3 1 uu u+ uu -1 -1 z zz z1 1 2 G0

= uu×× 2 half turns ext. to

1 trivial ext. 0 2 F 2 h0 , h1 , h2 , h3 A0 zz2 + + u 1 u z1 h z zz z1 1 zz 2 A 2 3

2 4

4 z 2 half 3 1 A1 screw ext. 3 half screw ext. to u- u- 5 z2 4 point reﬂection ext. to 2 2 . z2 z1 z1 z1 g : A g : 6 2 1 0 A3 , 2 A2 A z z z1 s : * * *

1 2 1 1 A0 A , s2 : A A

A 3 2 1 3 6

3 z1 3

g s 4 3

2 2 2

2 2 3 - 3 u 4 1 1 half turn ext.

uu uu uu uu -1 -1 uu z

uu z A 2

z A 2 zz2 0 2 0 2 2 -1 3 -1 3 2 2 s1 g1 12 12 uu uu uu uuuu uu

15 5 1

5 z

z 4 1

3 1 4

2 -1 3 s 5 g 3 1 -1 2 5

3 1 5 s 32

g 3 2 3 2 2 1 1 3 3 uu uu -1 z-1 z1 1

Figure 7: F1-7+ to Γ62(6u), the 4 extensions to Gieseking fundamental orbifolds Non-fundamental trunc-simplex tilings ... 15

3 Family 2

In family F2 the maximal series is characterized by the simplex tiling whose group 2∗2 42¯ m series is Γ(u, 2v) (or 8 Γ(u, 2v) by notations in [10]). Remember that in case u = 2v we order this tiling (and similarly the later ones) to family F1 with parameter u = u = 2v, since so many simplices (i.e. 2u barycentric subsimplices) surround every simplex edge.

2∗2 2 2 2 2 2 2 Γ(u, 2v) = (m0, m2, m3, r1 − 1 = m0 = m2 = m3 = r1 = (m0m2) = (m0m3) = 1 1 1 = m r m r = (m m )u = (m r m r )v, u 6= 2v ∈ , + < ) 0 1 2 1 2 3 3 1 3 1 N u v 2 with the trivial extension 2 2 1 2 2 3 2 4 2 5 (m ¯ 0, m¯ 2 − 1 =m ¯ 0 =m ¯ 2 =m ¯ 0r1m¯ 2r1 = (m ¯ 0m2) = (m ¯ 0m3) = (m ¯ 2m0) = 2 (m ¯ 2m3) ).

0 2 Pictures are given in Fig. 8: F2-1 with Γ (A0,A2) = ∗2uv as H group. Other non-fundamental cases in F2

∗22 2 2 2 −1 • Γ2(2u, 2v) = (m0, m2, z − 1 = m0 = m2 = (m0m2) = m2zm0z = 1 2 (m zm z−1)u = (zz)v, u 6= v ∈ , + < 1) = mm2 Γ (2u, 2v) 0 2 N u v 4 2 with 2 extensions in Fig. 8: F2-2 2 2 1 −1 2 2 3 −1 4 2 (1) (m ¯ 0, m¯ 2 − 1 =m ¯ 0 =m ¯ 2 =m ¯ 0zm¯ 2z = (m ¯ 0m2) =m ¯ 0z m¯ 2z = (m ¯ 2m0) ) 1 −1 2 2 −1 3 2 (2) (¯g − 1 = (¯gz ) = m2gm¯ 0g¯ = (¯gz) ). Γ0 = v ∗ u is a H2 group with half-turn extension yielding glide reﬂectiong ¯.

2× 2 2 u • Γ1(2u, 2v) = (r2, m3, z − 1 = r2 = m3 = zzr2 = (m3r2m3r2) = 1 2 (m zm z−1)v, u 6= v ∈ , + < 1) = 4¯Γ (2u, 2v) with 2 extensions 3 3 N u v 4 1 2 2 2 1 −1 2 3 2 4 (1) (m ¯ 0, m¯ 1, m¯ 2 − 1 =m ¯ 0 =m ¯ 1 =m ¯ 2 =m ¯ 0z m¯ 2z =m ¯ 0r2m¯ 1r2 = (m ¯ 0m3) = −1 5 2 6 2 m¯ 1zm¯ 2z = (m ¯ 1m3) = (m ¯ 2m3) ) ¯ ¯2 1 ¯ 2 2 3 −1 6 ¯ 2 (2) (h2, s¯ − 1 = h2 =sz ¯ h2z = (¯sr2) = m3sm¯ 3s¯ = (m3h2) ). Pictures are given in Fig. 9: F2-3 with Γ0 = ∗uvv as H2 group with half-turn extension.

2× 2 2 u v • Γ2(u, 2v) = (r2, r3, z − 1 = r2 = r3 = zzr2 = (r2r3) = (zr3zr3) , u 6= 2v ∈ 1 1 1 , + < ) = 4¯Γ (u, 2v) with 2 extensions N u v 2 4 2 2 2 2 1 −1 2 3 4 (1) (m ¯ 0, m¯ 1, m¯ 2 − 1 =m ¯ 0 =m ¯ 1 =m ¯ 2 =m ¯ 0z m¯ 2z =m ¯ 0r2m¯ 1r2 =m ¯ 0r3m¯ 1r3 = −1 5 2 m¯ 1zm¯ 2z = (m ¯ 2r3) ) ¯ ¯2 1 ¯ 2 2 3 2 5 ¯ 2 (2) (h2, s¯ − 1 = h2 =sz ¯ h2z = (¯sr2) = (¯sr3) = (h2r3) ). Pictures are given in Fig. 9: F2-4 with Γ0 = 2uv as H2 group with half-turn extension. 16 Moln´ar,Stojanovi´c,Szirmai

222 2 2 2 2 u • Γ2(u, 2v) = (r0, r1r2, r3 − 1 = r0 = r1 = r2 = r3 = r0r1r2 = (r2r3) = 1 1 1 (r r r r )v, u 6= 2v ∈ , + < ) = 222 Γ (u, 2v) with 2 extensions 0 3 1 3 N u v 2 4 2 2 2 2 1 2 3 (1) (m ¯ 0, m¯ 1, m¯ 2 − 1 =m ¯ 0 =m ¯ 1 =m ¯ 2 =m ¯ 0r1m¯ 2r1 =m ¯ 0r2m¯ 1r2 = 4 5 2 m¯ 0r3m¯ 1r3 =m ¯ 1r0m¯ 2r1 = (m ¯ 2r3) ) ¯ ¯2 1 ¯ 2 2 3 2 5 ¯ 2 (2) (h2, s¯ − 1 = h2 =sr ¯ 0h2r1 = (¯sr2) = (¯sr3) = (h2r3) ). See Fig. 10: F2-5 with Γ0 = 2uv as H2 group with half-turn extension.

3.1 Truncated half-simplices from F2

22 2 2 2 u v • Γ7(2u, 4v) = (r2, r3, h − 1 = r2 = r3 = h = (r2r3) = (r2hr3hr3hr2h) , 1 1 u 6= 2v ∈ , + < 1) =2 Γ (2u, 4v) with 4 extensions N u v 2 7 2 2 2 2 1 2 3 (1) (m ¯ 0, m¯ 1, m¯ 2, m¯ 3 − 1 =m ¯ 0 =m ¯ 1 =m ¯ 2 =m ¯ 3 =m ¯ 0hm¯ 3h =m ¯ 0r2m¯ 1r2 = 4 5 2 6 2 m¯ 0r3m¯ 1r3 =m ¯ 1hm¯ 2h = (m ¯ 2r3) = (m ¯ 3r2) ) 1 2 5 −1 (2) (¯g1, g¯2 = 1 =g ¯1hg¯2h =g ¯1r3g¯1r2 = r3g¯2r2g¯2 ) 1 2 2 3 2 5 −1 (3) (¯s1, s¯2 = 1 =s ¯1hs¯2h = (¯s1r2) = (¯s1r3) = r3s¯2r2s¯2 ) ¯ ¯ ¯ ¯ ¯2 ¯2 ¯2 ¯2 1 ¯ ¯ 2 ¯ ¯ 4 ¯ ¯ 5 (4) (h0, h1, h2, h3 − 1 = h0 = h1 = h2 = h3 = h0hh3h = h0r2h1r3 = h1hh2h = ¯ 2 6 ¯ 2 (h2r3) = (h3r2) ). 0 2 Look at Fig. 10: F2-61 for these presentations with Γ = 22uv as H group. At Fig. 11 : F2-62 you look pictures for the simpler versions. E.g. for the third extension (by a half-turn about edge 3, so 2 as well, yielding screw motions ¯) we get in addition: 2 2 3 2 5 −1 (3) (¯s − 1 = (¯sr2) = (¯sr3) =shr ¯ 3hs¯ hr2h). 2 1 • 22Γ (u, 4v) = (r, h − 1 = h2 = ru = (hrhrhr−1hr−1)v, 2 < u 6= 4v ∈ , + < 8 N u v 2 1) =2 Γ8(u, 4v) with 4 extensions 2 2 2 2 1 2 −1 3 (1) (m ¯ 0, m¯ 1, m¯ 2, m¯ 3 − 1 =m ¯ 0 =m ¯ 1 =m ¯ 2 =m ¯ 3 =m ¯ 0hm¯ 3h =m ¯ 0rm¯ 0r = 4 −1 5 −1 m¯ 1hm¯ 2h =m ¯ 1rm¯ 1r =m ¯ 2r m¯ 3r) 1 2 −1 −1 5 2 (2) (¯s1, s¯2 = 1 =s ¯1hs¯2h =s ¯1rs¯1 r = (rs¯2) ) ¯ ¯ ¯ ¯ ¯2 ¯2 ¯2 ¯2 1 ¯ ¯ 2 ¯ 2 3 ¯ ¯ 4 (3) (h0, h1, h2, h3 − 1 = h0 = h1 = h2 = h3 = h0hh3h = (h0r) = h1hh2h = ¯ 2 5 ¯ −1¯ (h1r) = h2r h3r) 1 2 −1 5 2 (4) (¯g1, g¯2 = 1 =g ¯1hg¯2h =g ¯1rg¯1 r = (rg¯2) ). Look at Fig. 12: F2-7 with Γ0 = uuv as H2 group. Simpler presentation is not pictured.

22 2 −1 u v • Γ9(2u, 2v) = (s, h − 1 = h = (shs h) = (ssh) = 1, 1 < u ∈ N, 1 < v ∈ 1 2 , + < 1) =2 Γ (2u, 2v) with 2 extensions N u v 2 9 2 2 2 2 1 2 −1 3 (1) (m ¯ 0, m¯ 1, m¯ 2, m¯ 3 − 1 =m ¯ 0 =m ¯ 1 =m ¯ 2 =m ¯ 3 =m ¯ 0hm¯ 1h =m ¯ 0s m¯ 3s = −1 4 −1 5 m¯ 1sm¯ 2s =m ¯ 1s m¯ 2s =m ¯ 2hm¯ 3h) Non-fundamental trunc-simplex tilings ... 17

1 −1 2 3 2 (2) (¯g1, g¯2 = 1 =g ¯1hg¯2 h =g ¯1sg¯2s = (¯g2s) ). 0 2 Look at Fig. 13: F2-81 for these presentations with Γ = uvv as H group. For the simpler second version look at Fig. 14 F2-82. For instance the second extension occurs with point reﬂection at edge 3, yielding glide reﬂectiong ¯ as follows in addition: (2) (¯g − 1 =2 gsh ¯ ghs¯ =3 (¯ghs−1h)2).

3.2 The 8 extensions for fundamental trunc-simplex tiling and group Γ58(2u, 4v) These top number extensions are pictured in Fig. 15 and 16 as to F 2 − 8+. Maybe, the numbered edges at truncation are more convenient than in the original publication [19]. The stabilizer subgroup Γ0 = uuvv of the vertex class are pictured in Fig. 15, where every vertex triangle has one u-point (u+ or u−) and v+-,v−-point as vertices to the spatial rotation axes, respectively. Thus the 8 possible correspon- dences are upper bound, indeed, that can be realized.

1 1 Γ (2u, 4v) = (s , s − 1 = (s s−1)u = (s s s s )v, u 6= 2v ∈ , + < 1). 58 1 2 1 2 1 1 2 2 N u v

1 (1) The trivial extension is for O58 as octahedron:

2 2 2 2 1 −1 2 −1 3 (m0, m1, m2, m3 − 1 = m0 = m1 = m2 = m3 = m0s2m2s2 = m0s1 m3s1 = 3 −1 4 −1 5 −1 6 −1 = m0s2 m3s2 = m1s1m3s1 = m1s1 m2s1 = m1s2 m2s2).

(2) The second extension will be derived from the two half-turns at u+ and u− 2 that ﬁx the vertex triangles by h0,..., h3 as follows for O58 in addition:

2 2 2 2 1 −1 2 −1 4 (h0, h1, h2, h3 − 1 = h0 = h1 = h2 = h3 = h0s2h2s2 = h0s1 h3s2 = 4 −1 5 −1 = h1s1h3s1 = h1s1 h2s2).

(3) The 3rd extension will be by rotatory reﬂection at v− (and v+ as well) rotating −1 −2 s1 s2 s2 0 A3 → A2 , A0 → A1 and changing the half-spaces at F . This leads to glide reﬂection g1, g2 and additional presentation:

1 −1 −1 2 −1 3 −1 (g1, g2 − 1 = g1s1g2 s2 = g1s2 g2s1 = g1s1 g2s2).

(4) The 4th extension is derived by a half screw motions through 5 and 5’ −1 2 s2 s1 s1 0 A2 → A3 , A0 → A1 changing the half-spaces at F . This leads to screw motions s1, s2 and additional presentation:

1 −1 −1 2 −1 3 −1 2 (s1, s2 − 1 = s1s1s2 s2 = s1s1 s2s1 = s1s2 s2s2) ). 18 Moln´ar,Stojanovi´c,Szirmai

Here we meet with ﬁxed-point-free actions beyond the u− and v− rotations whose axes close. Thus we found a double (u, v) link as an interesting topological phenomenon. (The previous and next ones are also remarkable). E.g. the case u = 3, v = 2 leads to hyperbolic metric. The case u = 2 = v leads to ideal vertices at the absolute (with E2 metric to F 0 domain, Fig. 15), i.e. we get H3 realization with so-called cusp phenomenon. Imagine a fundamental tetrahedron in H3 with two rectangles at opposite u = 2 rotational edges and π/4 angles at the remaining four v = 2 rotational edges. We also cite from [10] the (u, 1) link cases with spherical S3 realization. The “pure topological (1, 1) link, as manifold with “lens realization, moreover, the 4−truncated lens as S3 manifold are extremely remarkable, also to our context and Fig. 16. (5) The 5th extension is obtained by point-reﬂection in 5 at edge 1 (or in 5’ at −2 −1 s2 s2 s2 4): A1 → A3 , A0 → A2 . These lead gb1, gb2 pairings:

1 −1 2 2 3 4 −1 2 (gb1, gb2 − 1 = (gb1s2 ) = gb1s1gb2s1 = gb1s2gb2s2) = (gb2s1 ) ).

−2 s1 s2 (6) The 6th extension is by half turn about edge 1 (also about 4) A3 → A1 , −1 s2 A0 → A2 , leading to screw motions sb1, sb2:

1 −1 2 2 3 4 −1 2 (sb1, sb2 − 1 = (sb1s2 ) = sb1s2sb2s1 = sb1s1sb2s2) = (sb2s1 ) ).

∗ ∗ ∗ − (7) The 7th extension will be A0 → A3, A1 → A2 by rotatory reﬂection at u + and u (1 ↔ 4, 2 ↔ 3, 5 ↔ 6) yielding glide reﬂections ge1, ge2:

1 −1 −1 2 5 (ge1, ge2 − 1 = ge1s1 ge2s2 = ge1s2ge1s1 = ge2s1ge2s2).

∗ ∗ ∗ (8) The 8th extension is again A0 → A3, A1 → A2, but by half turns about edges 2, 3, 5, 6 with 1 ↔ 4 inducing screw motions se1, se2: 1 −1 −1 2 2 3 2 5 2 6 2 (se1, se2 − 1 = se1s1 se2s2 = (se1s1) = (se1s2) = (se2s1) = (se2s2) ).

4 Family F3

This family is also characterized by a Coxeter reflection group {u,¯ v,¯ 3} with Cox- eter Schläflidiagram in Fig. 17: F3-1 with doubly truncated simplex (trunc- orthoscheme). ∗33 2 2 2 2 3 • Γ(2u, v) = (m0, m1, m2, m3 − 1 = m0 = m1 = m2 = m3 = (m0m1) = 1 1 = (m m )2 = (m m )2 = (m m )2 = (m m )v = (m m )u; 2u 6= v ∈ , + < 0 2 0 3 1 2 1 3 2 3 N u v 1 1 1 1 , + + < 1) = 3mΓ(2u, v), with trivial extensions both for vertices A and 2 v 3 2 6 0 A2 Non-fundamental trunc-simplex tilings ... 19

2 2 * G(u,2v) trivial extension Extended Coxeter diagram A 0123 r1 r1 r1 m2 m3 m3 m0= m2 a2 u v u r 1 * A2 m0

22 * * m 22 2 v a2 =m2 a0 = m0 a0 2* *

A012

* uu A0 A01 A1 m3

A01 F 0(A, A ) G0 = 2uv 0 2 u- *

uu m3

* 2 3

22 m A m 2 *22 0 2

* m3 3 - m 2 * 0 A 1 1 0

m m0 0 r 4 r 1 1 A2 v * 2* - 5 - v m3 2

A02 r 5 1 1

m2 A0123

4 * A 2 h 0 1 2 3 b u b v b w=u b From the Coxeter-Schläﬂi 01 12 23 01 orthoschem b b b = b

a0 a3

22 * G 2 (2u,2v) 2 extensions 0 G0=v u F (A0 ,A2 ) *

A3 m 4 - A * 0 u z-1 23 z z

uu 1 A * 2 3 2 A2 A0 m2 m z 1 0 z v- 22 * *

m g z-1

* 2 3 m0 v× * A2 4 z

u- * uu A A01 A 0 z 1

A01 A01

m m

* 2 *22 * 2 *22

3 z 3 z m 2 g -1 2 0 A A 1 0 1 0 m0 m0 * v× * v×

z-1 z-1

1 1

m2 3 A23 g 3 A23 2

4 * uu * uu

A2 A2

2∗2 ∗22 Figure 8: F2-1 to Γ(u, 2v) and F2-2 to Γ2(2u, 2v) 20 Moln´ar,Stojanovi´c,Szirmai

2 ×G 1 (2u,2v) 2 extensions

4 A1 1 A1

m s 1 2

5 3 2

2 2

vv vv* 1 1

3 m r2 m 3 r 3 m 2 3 -1 2 2 0 A s A 1 0 1 0 z-1 z-1

v 1 v

* 1 v v*

z 6 . z

. 6

. 6 4 1

m2 A0123 A0123

1 1 1 h2 1

A2 A2

u- 0 0 G uvv F =*

m m

3 2 3

5 3 r 2 r z A 2 z 0 A1=A1 1-

4 1 z A z z 2

+ v . 6 - m3 v

h2 s 2 ×G 2 (u,2v) 2 extensions

4 A1 1 A1

m s 1 2

3 3 2

22u 22

1 v× 1

v× 3

2 r 3 2 r3 m 2 s-1 r 0 A r3 A 2 1 0 1 0

1 1 v× v×

5 z 5 z

5 4 1

m2 A0123 -1 A0123

1 1 h z 1 z-1 2 1

A2 A2

0 - F u G0 = 2uv

2 r3 r3

r2 r2 zz

A0 A 3 =A 1 1 3 1-

4 1 z

A2 z z 5 - - v 5 v

h2 , s ext.

2× 2× Figure 9: F2-3 to Γ1(2u, 2v) and F2-4 to Γ2(u, 2v) Non-fundamental trunc-simplex tilings ... 21

222 G 2 (u,2v) 2 extensions

4 A1 1 A1

m s 1 2

3 3 2

r0 u 22u r0 22

1 22v 1

22v 3

2 r 3 r r m 2 r3 -1 2 2 3 0 A s A 1 0 1 0

1 1 22v 22v

5 r1 5 r1 5

r0 5 1 1

m2 A0123 A0123 4

1 1 h2 1

A2 A2

0 - F u G0 = 2uv 2 r3 r3

r2 r2

3 A0 A1 3 1- 1 r1 4 r0 r1 A2 5

v- v- 5 r3

h2 , s ext. 22 G 7 (2u,4v) 4 extensions

4 A A1

1 1

2 g m 1 2 1 2 3

22v 22v

22u 22u

22v 22v 3 r 2 r r m 2 2 r g -1 2 2 3 0 A 3 1 A 1 1 0 0

22v 22v

22v 6 . 22v 5 5

. 6 5

5 5 g

5 m3 2

1 1 A A A2 A3 -1 2 3

m g2 4 2

1 h h

- - 2 v r2 u 0 r 0 F 4 3 r G = 22uv A1 3 2 5 hr3

A2 3 A

r3 0 5 h r2

h 1

Ah 6 3 - . - v 6 . v r2

1 4 A

A1 1 4

s1 2 2 h1 3

22v

22v 22u 22u

22v 22v 3 r r2 2 2 -1 2 2 s1

r A r h0 A0

3 1 0 3 1 1

22v 22v

22v 5 22v 6 .

6 5 .

5 1 5

5 s2 1 h 3 1 A -1 A2 A3 2

s2 h 4 A 1 4 2 3 h h

222 22 Figure 10: F2-5 to Γ2(u, 2v) and F2-61 to Γ7(2u, 4v) 22 Moln´ar,Stojanovi´c,Szirmai

22 G 7 (2u,4v) 4 extensions cont. simpler

versions with less generators and relations

5 5 5

m 5

1 2 g 3 2

22v

22u 22v 22u

22v 3 22v 2 r2 r r m 2 -1 2 2 3 0 r3 g

. 6 . 5 5

22v 22v

22v 6 .

22v 5

6 . 5

5 5 h h - - 2 v r2 u 0 G0 F h r = 22uv 3 2 5 A1 = A2

3 A = A3

r3 0 5

- . 6 - v 6 . v

s 5

2 3 h1 2

22v 22u

22v 3 r 22v -1 2 2 r r3 s 2 2 2 5 r3

5 h0

. 6 . 22v

22v 5 22v 5 22v 6 .

5 . 6

5 5 5 h h

22 Figure 11: F2-62 to Γ7(2u, 4v) simpler version Non-fundamental trunc-simplex tilings ... 23

22 G 8 (u,4v) 4 extensions

3 A A1 1 1

4 g m 1 2 1 4

22v 22v

u u

-1 22v 22v r r r -1 2 2 -1 m 2 r g 2 0 A 1 A 1 1 0 0

22v 22v 22v 22v 5

5 5

5 5 g

m3 2

1 1 A A A2 A3 -1 2 3 m g2

3 2 h h 1 u+

2 2 0 0 F A0 G = uuv r r h h 1 A3

5 r v- v- -1 Ar h h 2 r 3 r

-1

A hr h

4 1

1 4 3 A

A1 1 3

s1 4 2 h u- 1

22v 22v

u u

22v -1 22v -1 r r 2 r r 2 s -1 2 2

1 A h0 A

1 0 1 0 1

22v 22v

22v 5

5 22v 5

5 1 5

5 s2 1 h3 1 A -1 A2 A3 2

s2 h2 A 1 3 3 3 h h

22 Figure 12: F2-7 to Γ8(u, 4v), simpler version is not pictured 24 Moln´ar,Stojanovi´c,Szirmai

22 G 9 (2u,2v) 4 extensions

A A 3 3 1 1 3

m g -1 2 1 4 2

1 1

22u

22u 22u

vv vv -1 s -1 s s vv s vv -1 2 m 2 g

0 A0 1 A0 1 1

vv vv 3

h 3

1 g 2

5 m3 2 2

2 1 4 m2 5 g1

22u 22u A3 A3 A2 A2

s 4 v+ u- 2.ext. by point reﬂection

h * 1 0 3 A1 F h 0 * 4 G A0 2 = uvv -1 s Ah s h

-1 2 s s h 5 s A3 half turn - 2. + v 3 v s ext. 4. ext.

s 1 A s

A0 1

2 h 4

u- 3.ext. half turn

3 A A 3 1 1 3

1 4 2

22u

22u 22u

vv vv

s vv s

-1 s-1 s vv 1 2 4 -1 2 -1 s A

s1 A0 0

3 3

3 vv 3 vv

h h 3 3

-1 4

s s 2 5 2

s2 5 2

4 2

22u 22u A3 A3 A2 A2

22 Figure 13: F2-81 to Γ9(2u, 2v) ﬁrst version Non-fundamental trunc-simplex tilings ... 25

22 G 9 (2u,2v) cont. simpler

A 3 3 1 1 3

1 4

22u

22u 22u

vv vv -1 s -1 s

s vv s vv 4 1 2 -1 2 g 2 m A0 A0

0 3

3 3

vv vv 3

h 3 3

3 3 h 3

m 5 5 2 g

2 2 4 2

22u 22u A3 A3 A2 A2 g v+ u- 2.ext. by point reﬂection

* A 1 F 0 1 0 * 4 G A0 2 = uvv -1 Ah s h s -1 h 2 s

s A3 5 half turn v- 2. v+ s ext. 3 4. ext.

s s h 3 A A1 s s 0 1 5

A 3 2 *

4 A2

3 * 5 A3 s hs

A 4 s 2 - 2 s v+ u v- 3.ext. half turn

3 1 A 3 3 1

22u

22u 22u vv s vv -1 -1 s vv s s vv 2 2 4 -1 2 h0 s

3 3

vv 3 vv

3 3

h 3 h 3

2 2

s 2

2 h2 4

22u A 22u A 3 A3 2 A2

22 Figure 14: F2-82 to Γ9(2u, 2v) second simpler version 26 Moln´ar,Stojanovi´c,Szirmai

G (2u, 4v) 58 fundamental 8 extensions

2 half turns 4 4 half A1 ext. screw ext. 0 s 1 - + + 2 -

m 2

1 F v u 3 v 5 v 2

5 . 3 s1 s2 6 s1 3 . s2 s2 s s1 s1 2 1 4 6

A3 A A2

3 5 2 5

4 s1 3 s uu A0 A 2 s1 s1 1 1 6 ext. by s2

uu v+ 5 v- u- .

vvuu half turn 6 s s2 7 ext. by 2 -1 . 3 ext. by s s1 rot. reﬂ.

uuvv s1 4 2 6 rot. reﬂ.

s2 A0

3 A 5 2 4 3 s -1s -1 s s -2 m 2 1 2 2 2 -1 0 A1 A1 s1 12 s1

v- u- v+

s 5 s1 vv uu 2 uuvv

0 5 4 G

-1 = uuvv

51 s

2 4

5 m Every vertex triangle has one u 3 2 point m . 3 5 2 6 +

5 and v v- poits.

3 uu A uu 3 A2

1 trivial ext. 1

4 A A1 4 1

g1 2

5 h 3 3

1 5

uu uu vvuu vvuu s

s uuvv A 1

uuvv s A 1 s2 0 2 0 3 3 -1 23 2 2 2 g1 0 -1 h 2 -1 s 1

s 1 1

1 1

vv uu vv uu vv uuvv uu 4 51

5 4 -1

-1 51

1 s

s 2 4

2 4

-1 2 g 2

2 5 h 2 3 3 5 g 3

h2 5 2

3 uu A3 uu A 3 uu uu A2 A2

2 ext. by half turns at u to h0 ,h1 ,h2 ,h3 3 ext. by rot. reﬂ. at v to g1 ,g2

Figure 15: F2-8+ for Γ58(2u, 4v) the 8 fundamental extensions (repetition) Non-fundamental trunc-simplex tilings ... 27

G (2u, 4v) 58 fundamental 8 extensions cont.

4 A1 A1 4 -1

s1 g 2

2 2 3 2 3

uu uu vvuu vvuu

s uuvv s1

uuvv s A 1 s A0 2 0 2 3

3 3 s-1 2 -1 2 -1 1 -1 g1 1

12 s

s1 1 1

uu vv uu vv vv uuvv uu

1 5 5 4 -1 -1 51 1

s 1 s2 2 4

5 s

-1 2 2 2 5 3 5 g 3 g2

s2 3 1 3 2 2

3 3 uu A uu A 3 uu 3 uu A2 A2

4 ext. by half scew at 5 5’ to s 1 ,s 2 5 ext. by point reﬂ. at 1,4 to g1 , g2

4 1 A1 A1 -1 -1

2 s2 2 5

3 5

3 3

uu uu vvuu vvuu s

uuvv s A 1 uuvv s1

2 0 s A0 3 2 3 -1 2 -1 s 2 3 g~ 2 -1 1 -1 1 12

s 12

1 s1

vv uu vv vv uu

uu uuvv

5 4 -1 5 1 51 -1

s2 51 s

2 4

2 ~

2 5 5

2 g 5

s2 2 s 3 ~ 3 1 5

1 g2 5

33 3 uu A 3 uu A uu uu 3 A2 A2

6 ext. by half turn at 1,4 to s , ~ ~ 1 s2 7 ext. by rot. reﬂ. at u to g1 ,g2 G (2u, 4v) 58 fundamental 8 extensions cont.2

A1 1

~-1

3 . s2 5

uu uu vvuu

uuvv s1

s A0 2 3 ~-1 2 -1 s1

s1 1 vv uu uuvv 51 -1 1

5 ~ 2 ~ . 3 s 5

s2 6 1 5

3 uu A uu 3 A2 ~ ~ 8 ext. by half turns at 2,3,5,6 to s1 , s2

Figure 16: F2-8+ for Γ58(2u, 2v), important topological phenomena 28 Moln´ar,Stojanovi´c,Szirmai

*33 Coxeter-Schläﬂi diagram G (2u,v) trivial extension m m 2 u 3 v m1 3 m0 A3

a2 A2 a2 =m2 a0 = m0 T2 A013 A3 33 * * vv m1 0 * uu F (A ) 0 22 0 F (A2 ) * m2 vv m3 22 * 5

22 * *

m0 2

m2 3

6 m . 1 uu m2 * 1 T0 A m1 m0 4 A0 A0 m A01 A 1 a 3 1 * vv *22 *22 *33 0 0 G0 G =*2uv =*23v

A01

22 * * 22

A0 m 3 m2 3 2 m0 1

m0 vv * 22

. m1 6

m2 5 A013 4 * 33

A2 33 G 1 (2u,2v) 4 (=2×2) extensions

1 A1 1 A1

m s 1 4 2

5 3 u

* * *

uu uu2 uu uu

* vv * * * vv

m * * 3 2 3 m m m 3 m 2 3 -1 2 2 0 A s A 1 0 1 0

v v

* v * v* *

r 7 r

7 7

6 6

m2 . A013 . A013 6

6 h

331 -1 2 33 -1 . r . r

A2 A2

u+ 0 G0 F (A0 , A1 ) (A0 , A1 ) =*uuv half turn m3 ext. 2

0 0 3 G A F (A2 ) (A2 ) =3 v 0 m * half turn 2 6 + 6 v 1 ext r * r A2 r 5 r 4

A1 - 7 v m3 v-

m3 m2

∗33 33 Figure 17: F3-1 to Γ(2u, v) and F3-2 to Γ1(2u, 2v) Non-fundamental trunc-simplex tilings ... 29

33 G 2 (4u,2v) 4 (=2×2) extensions A1 1 1

m s 2 1 2

4 3 uu * * uu

* * uu

vv *

22 vv 22

3 m m 3 3 m 2 r 3 -1 2 0 A 2 s A 1 0 1 0 r2

v 22 v 22 *

v v*

r 6 . r

6 .

. 6

5 5

m2 A013 A013 5 5 h 331 r -1 2 33 r -1

A2 A2

u- 0 G0 half turn F (A0 , A1 ) (A0 , A1 )=2* uv m3 2 ext. 3-

0 G0

F (A2 ) (A2 ) =3 v 3 A * 0 r 2 half turn 5 5 + - v r 1 2 ext. r A2 r 4 r r2

- 6

A1 v m3 . v- 2

u- 33 G 3 (4u,v) 4 (=2×2) extensions A1 1 1

m s 2 1 4

3 3 uu * * uu

uu * * uu

vv m

m2 * 2 * 3

r3 2 3 m r3 -1 2 0 A s A 1 0 1 0

vv * vv *

6 r 6 . r

. .

6 6

5 5

m2 A013 A013 5 5 h 331 r -1 2 33 r -1

A2 A2

u- 3 0 0 0 G0 G F (A2 ) (A2 ) =23v F (A0 , A1 ) (A0 , A1 ) =v* u r

3 2 5 5 3 A A0 m r 2 r half turn 2

6 .

+ - . - v 1 v v

. r * 6 ext r3

3 r m2 half turn

A1 ext. r3 2

33 33 Figure 18: F3-3 to Γ2(4u, 2v) and F3-4 to Γ3(4u, v) 30 Moln´ar,Stojanovi´c,Szirmai

2 1 2 2 2 3 2 (1) for A0: (m ¯ 0 − 1 =m ¯ 0 = (m ¯ 0m1) = (m ¯ 0m2) = (m ¯ 0m3) ). 2 4 2 5 2 6 2 (1) for A2: (m ¯ 2 − 1 =m ¯ 2 = (m ¯ 2m0) = (m ¯ 2m1) = (m ¯ 2m3) ). 0 0 The pictures are given in Fig. 17: F3-1 with Γ (A0) = ∗2uv and Γ (A2) = ∗23v as H2 groups.

33 2 2 3 u −1 v • Γ1(2u, 2v) = (r, m2, m3 − 1 = m2 = m3 = r = (m2m3) = (m3rm3r ) = 2 1 2 1 m rm r−1, u 6= v ∈ , + < 1, + < 1) = 3Γ (2u, 2v), with 2 extensions 2 2 N u v 3 v 3 1 both for vertices A0, A1 and A2 2 2 1 −1 2 2 3 2 4 (1) for A0, A1: (m ¯ 0, m¯ 1 − 1 =m ¯ 0 =m ¯ 1 =m ¯ 1rm¯ 0r = (m ¯ 0m2) = (m ¯ 0m3) = 2 5 2 (m ¯ 1m2) = (m ¯ 1m3) ) 2 6 −1 7 2 (1) for A2: (m ¯ 2 − 1 =m ¯ 2 =m ¯ 2rm¯ 2r = (m ¯ 2m3) ) 1 2 2 −1 3 −1 (2) for A0, A1: (¯s − 1 = (sr) =sm ¯ 2s¯ m2 =sm ¯ 3s¯ m3) ¯ ¯2 6 ¯ 2 7 ¯ ¯ (2) for A2:(h2 − 1 = h2 = (h2r) = h2m3h2m3). 0 0 The pictures are given in Fig. 17: F3-2 with Γ (A0,A1) = ∗uuv and Γ (A2) = 3 ∗ v as H2 groups.

33 2 2 3 u −1 v • Γ2(4u, 2v) = (r, r2, m3 − 1 = r2 = m3 = r = (m3r2m3r2) = (m3rm3r ) = 1 1 2 1 (r r)2, 2u 6= v ∈ , + < 1, + < 1) = 3Γ (4u, 2v), with 2 extensions both 2 N u v 3 v 3 2 for vertices A0, A1 and A2 2 2 1 −1 2 3 2 4 (1) for A0, A1: (m ¯ 0, m¯ 1 − 1 =m ¯ 0 =m ¯ 1 =m ¯ 1rm¯ 0r =m ¯ 0r2m¯ 1r2 = (m ¯ 0m3) = 2 (m ¯ 1m3) ) 2 5 −1 6 2 (1) for A2: (m ¯ 2 − 1 =m ¯ 2 =m ¯ 2rm¯ 2r = (m ¯ 2m3) ) 1 2 2 2 3 −1 (2) for A0, A1: (¯s − 1 = (sr) = (sr2) =sm ¯ 3s¯ m3) ¯ ¯2 5 ¯ 2 7 ¯ ¯ (2) for A2:(h2 − 1 = h2 = (h2r) = h2m3h2m3). 0 0 The pictures are given in Fig. 18: F3-3 with Γ (A0,A1) = 2 ∗ uv and Γ (A2) = 3 ∗ v as H2 groups. 33 2 2 3 u v • Γ3(4u, v) = (r, m2, r3 − 1 = m2 = r3 = r = (m2r3m2r3) = (r3r) = 1 2 2 2 m rm r−1, 4u 6= v ∈ , + < 1, + < 1) = 3Γ (4u, v), with 2 exten- 2 2 N u v 3 v 3 3 sions both for vertices A0, A1 and A2 2 2 1 −1 2 2 3 4 (1) for A0, A1: (m ¯ 0, m¯ 1 − 1 =m ¯ 0 =m ¯ 1 =m ¯ 1rm¯ 0r = (m ¯ 0m2) =m ¯ 0r3m¯ 1r3 = 2 (m ¯ 1m2) ) 2 5 −1 6 (1) for A2: (m ¯ 2 − 1 =m ¯ 2 =m ¯ 2rm¯ 2r =m ¯ 2r3m¯ 2r3) 1 2 2 −1 3 2 (2) for A0, A1: (¯s − 1 = (sr) =sm ¯ 2s¯ m2 = (¯sr3) ) ¯ ¯2 5 ¯ 2 6 ¯ 2 (2) for A2:(h2 − 1 = h2 = (h2r) = (h2r3) ). 0 0 The pictures are given in Fig. 18: F3-4 with Γ (A0,A1) = v∗u and Γ (A2) = 23v as H2 groups.

33 2 2 3 u v 2 • Γ4(2u, v) = (r, r2, r3 − 1 = r2 = r3 = r = (r2r3) = (r3r) = (rr2) , 2u 6= v ∈ 2 2 2 2 , + < 1, + < 1) = 3Γ (2u, v), with 2 extensions both for vertices A , N u v 3 v 3 4 0 A1 and A2 Non-fundamental trunc-simplex tilings ... 31

33 G 4 (2u,v) 4 (=2×2) extensions

1 1

m s 2 1 2

3 3

uu uu

vv 22 vv 22

3 r 3 2 r2 r3 2 2 m r3 -1 0 A s A 1 0 1 0

22 22

vv vv

5 . r 5 r

5 5

4 4

m2 A013 A013 4 4 h 331 r -1 2 33 r -1

A2 A2

u- 3 0 G0 0 G0 F (A0 , A1 ) (A0 , A1 ) =2uv F (A2 ) (A2 ) =23v r

3 2 4 4 3 A A0 r r 2 r half turn 2 5 + - v- - v 1 2 v

. r 5 ext r3 3 r r2 half turn

A1 ext. r3 2

33 Figure 19: F3-5 to Γ4(2u, v)

2 2 1 −1 2 3 (1) for A0, A1: (m ¯ 0, m¯ 1 − 1 =m ¯ 0 =m ¯ 1 =m ¯ 1rm¯ 0r =m ¯ 0r2m¯ 1r2 =m ¯ 0r3m¯ 1r3) 2 4 −1 5 (1) for A2: (m ¯ 2 − 1 =m ¯ 2 =m ¯ 2rm¯ 2r =m ¯ 2r3m¯ 2r3) 1 2 2 2 3 2 (2) for A0, A1: (¯s − 1 = (sr) = (¯sr2) = (¯sr3) ) ¯ ¯2 4 ¯ 2 5 ¯ 2 (2) for A2:(h2 − 1 = h2 = (h2r) = (h2r3) ). 0 0 The pictures are given in Fig. 19: F3-5 with Γ (A0,A1) = 2uv and Γ (A2) = 23v as H2 groups.

5 Family F4

The family F4 is characterized by ∗22 2 2 2 2 w Γ1(u, 2v, w) = (m0, m1, m2, m3 − 1 = m0 = m1 = m2 = m3 = (m0m1) = 1 = (m m )2 = (m m )2 = (m m )2 = (m m )v = (m m )u; u 6= w, v ∈ , + 0 2 0 3 1 2 1 3 2 3 N u 1 1 1 1 1 < , + < ) = mm2Γ (u, 2v, w), with trivial extensions both for vertices v 2 v w 2 4 1 A0 and A2 2 1 2 2 2 3 2 (1) for A0: (m ¯ 0 − 1 =m ¯ 0 = (m ¯ 0m1) = (m ¯ 0m2) = (m ¯ 0m3) ). 2 4 2 5 2 6 2 (1) for A2: (m ¯ 2 − 1 =m ¯ 2 = (m ¯ 2m0) = (m ¯ 2m1) = (m ¯ 2m3) ). 0 0 The pictures are given in Fig. 20: F4-1 with Γ (A0) = ∗2uv and Γ (A2) = ∗2vw as H2 groups. Other non-fundamental cases from F4 32 Moln´ar,Stojanovi´c,Szirmai

∗ 2 2 u −1 v w • Γ2(u, 4v, w) = (r, m0, m1 − 1 = m0 = m1 = r = (m1rm1r ) = (m0m1) = 2 1 2 1 m rm r−1, u 6= w, v ∈ , + < 1, + < 1) = mΓ (u, 4v, w), with 2 0 0 N u v w v 2 2 extensions both for vertices A0 and A2, A3 2 1 2 2 −1 (1) for A0: (m ¯ 0 − 1 =m ¯ 0 = (m ¯ 0m1) =m ¯ 0rm¯ 0r ) 2 2 3 2 4 2 5 −1 6 (1) for A2, A3: (m ¯ 2, m¯ 3 − 1 =m ¯ 2 =m ¯ 3 = (m ¯ 2m0) = (m ¯ 2m1) =m ¯ 3rm¯ 2r = 2 7 2 (m ¯ 3m0) = (m ¯ 3m1) ) ¯ ¯2 1 ¯ 2 2 ¯ 2 (2) for A0:(h0 − 1 = h0 = (m1h0) = (h0r) ) 3 −1 4 −1 5 2 (2) for A2, A3: (¯s − 1 = m0sm¯ 0s¯ = m1sm¯ 1s¯ = (rs¯) ). 0 0 The pictures are given in Fig. 20: F4-2 with Γ (A0) = u ∗ v and Γ (A2,A3) = ∗vww as H2 groups.

∗ 2 2 2 2 2 • Γ3(2u, 4v, 2w) = (r1, m0, m2, m3 − 1 = r1 = m0 = m2 = m3 = (m0m2) = 1 1 (m m )2 = (m m )u = (m r m r )v = (m r m r )w, u 6= w, v ∈ , + < 0 3 2 3 2 1 3 1 0 1 0 1 N u v 1 1 1, + < 1) = mΓ (2u, 4v, 2w), with 2 extensions both for vertices A and A , v w 2 3 0 2 A3 2 1 2 2 2 3 2 (1) for A0: (m ¯ 0 − 1 =m ¯ 0 = (m ¯ 0r1) = (m ¯ 0m2) = (m ¯ 0m3) ) 2 2 4 2 5 6 2 7 (1) for A2, A3: (m ¯ 2, m¯ 3 − 1 =m ¯ 2 =m ¯ 3 = (m ¯ 2m0) =m ¯ 2r1m¯ 3r1 = (m ¯ 2m3) = 2 8 2 (m ¯ 3m2) = (m ¯ 3m0) ) ¯ ¯2 1 ¯ 2 2 ¯ ¯ (2) for A0:(h0 − 1 = h0 = (r1h0) = m2h0m3h0) 4 −1 5 2 6 −1 (2) for A2, A3: (¯s − 1 = m0sm¯ 0s¯ = (r1s¯) = m3sm¯ 2s¯ ). 0 0 The pictures are given in Fig. 21: F4-3 with Γ (A0) = 2 ∗ uv and Γ (A2,A3) = ∗2v2w as H2 groups.

∗ 2 2 u v w • Γ5(u, 2v, 2w) = (m0, r1, r − 1 = m0 = r1 = r = (r1r) = (m0r1m0r1) = 1 1 1 2 1 m rm r−1, u 6= 2w, v ∈ , + < , + < 1) = mΓ (u, 2v, 2w), with 2 0 0 N u v 2 v w 2 5 extensions both for vertices A0 and A2, A3 2 1 2 2 −1 (1) for A0: (m ¯ 0 − 1 =m ¯ 0 = (m ¯ 0r1) =m ¯ 0rm¯ 0r ) 2 2 3 4 −1 5 2 6 (1) for A2, A3: (m ¯ 2, m¯ 3 − 1 =m ¯ 2 =m ¯ 3 =m ¯ 2r1m¯ 3r1 =m ¯ 3rm¯ 2r = (m ¯ 2m0) = 2 (m ¯ 3m0) ) ¯ ¯2 1 ¯ 2 2 ¯ 2 (2) for A0:(h0 − 1 = h0 = (h0r1) = (h0r) ) 3 2 4 2 5 −1 (2) for A2, A3: (¯s − 1 = (r1s¯) = (rs¯) = m0sm¯ 0s¯ ). 0 0 The pictures are given in Fig. 21: F4-4 with Γ (A0) = 2uv and Γ (A2,A3) = v∗w as H2 groups.

5.1 Truncated half-simplices from F4

22 2 2 2 u v • Γ3(u, 4v, 2w) = (m1, r3, h − 1 = m1 = r3 = h = (r3h) = (m1r3hm1hr3) = 2 1 1 1 (m hm h)w, u 6= 2w, v ∈ , + < 1, + < 1) =2 Γ (u, 4v, 2w), with 2 1 1 N u v v w 2 3 extensions both for vertices A0, A1 and A2, A3 Non-fundamental trunc-simplex tilings ... 33

2 2 1 2 2 3 (1) for A0, A1: (m ¯ 0, m¯ 1 − 1 =m ¯ 0 =m ¯ 1 = (m1m¯ 0) =m ¯ 0hm¯ 1h =m ¯ 0r3m¯ 1r3) 2 2 4 2 5 6 2 7 (1) for A2, A3: (m ¯ 2, m¯ 3 − 1 =m ¯ 2 =m ¯ 3 = (m ¯ 2r3) =m ¯ 2hm¯ 3h = (m ¯ 2m1) = 2 (m ¯ 3m1) ) ¯ ¯ ¯2 1 ¯ 2 3 ¯ 2 (2) for A0, A1 by half-turn h0:(h0 − 1 = h0 = (m1h0) = (h0hr3) ) ¯ ¯ 4 ¯ 2 6 ¯ ¯ (2) for A2, A3 by half-turn h2:(h2 − 1 = (h2r3) = m1h2hm1hh2). 0 0 The pictures are given in Fig. 22: F4-5 with Γ (A0,A1) = u∗v and Γ (A2,A3) = 2 ∗ vw as H2 groups.

22 2 2 2 u v w • Γ4(u, 2v, w) = (r1, r3, h − 1 = r1 = r3 = h = (r3h) = (r1r3h) = (r1h) , u 6= 1 1 1 1 1 1 w, v ∈ , + < , + < ) =2 Γ (u, 2v, w), with 2 extensions both for N u v 2 v w 2 2 4 vertices A0, A1 and A2, A3 2 2 1 2 2 3 (1) for A0, A1: (m ¯ 0, m¯ 1 − 1 =m ¯ 0 =m ¯ 1 = (m ¯ 0r1) =m ¯ 0hm¯ 1h =m ¯ 0r3m¯ 1r3) 2 2 4 2 5 6 (1) for A2, A3: (m ¯ 2, m¯ 3 − 1 =m ¯ 2 =m ¯ 3 = (m ¯ 2r3) =m ¯ 2hm¯ 3h =m ¯ 2r1m¯ 3r1) ¯ ¯ ¯2 1 ¯ 2 3 ¯ 2 (2) for A0, A1 by half-turn h0:(h0 − 1 = h0 = (h0r1) = (h0hr3) ) ¯ ¯ ¯2 4 ¯ 2 6 ¯ 2 (2) for A2, A3 by half-turn h2:(h2 − 1 = h2 = (h2r3) = (h2hr1) ). 0 0 The pictures are given in Fig. 23: F4-6 with Γ (A0,A1) = 2uv and Γ (A2,A3) = 2vw as H2 groups.

6 On packings and coverings with congruent hyperballs in Family 1-4

In this section we consider congruent hyperball packings and coverings for families F1-F2 and congruent hyperball packings for families F3-F4. The base planes of the hyperballs are always the truncation planes of the trunc-simplices.

1. The first we consider Family 1, with a simple truncated orthoscheme (e.g. CHLA1A2A3 in Fig. 24.a and Fig. 1:F1-1) with Schläflisymbols {u, v, w} = {3, 3, u} 6 < u ∈ N). This simple truncated orthoscheme can be derived also 0 1 2 3 by truncation from orthoscheme A0A1A2A3 = b b b b with outer essential vertex A0. The truncating plane a0(a0) = CLH is the polar plane of A0, 0 h that is the (ultraparallel to b ) base plane of hyperball H0 with height hp for packing or hc for covering. 2. For families F2-F4 (see Fig. 24.b and Fig. 8: F2-1, Fig. 17: F3-1, Fig. 20: F4-1, respectively) we consider a tiling T (O(u, v, w)) with Schläflisymbol {u, v, w}, 1 1 1 1 1 1 ( u + v < 2 , v + w < 2 , u, v, w ∈ N) whose fundamental domain is doubly truncated orthoscheme O(u, v, w) = CHLA1A2EJQ in Fig. 24.b. Let a truncated orthoscheme O(u, v, w) ⊂ H3 be a tile from the above tiling. 0 1 2 3 This can be derived also by truncation from A0A1A2A3 = b b b b with essential outer vertices A0 and A3. The truncating planes a0(a0) = CLH and a3(a3) = JEQ are the polar planes of A0 and A3, respectively, that will be (ul- 0 3 h traparallel to b and b ) base planes of hyperballs Hi with height h (i = 0, 3). 34 Molnár,Stojanović,Szirmai

22 * G (u,2v,w) trivial extension 1 Coxeter-Schläﬂi diagram u v w

m2 m3 m1 m0

A3 m A23 A3 m2 0 a *ww 2 T2 A A 0 G0 m0 2 2 F (A0 ) = 2uv u- * m1 m2 A0123 3 22 22 m 2

* * 2 3 A

0 m * vv 22 v+ 1 - * m1 2

uu * 0 0 A A01 A A G 0 A0 T0 1 1 F (A2 ) =*2vw a0 w- m3 m 1 4

A01 0 5 A 2 m - 6 - v m3 2

A012 22 m 22 A

* 2 * 0123 m 3 3 m 2 0 A 1 0 m0

vv * * 22

m1 . 6

m2 5 A23

4 *ww

A2 *G 2 (u,4v,w) 4 (=2×2) extensions

A01 A01

uu uu

* * * * A0 A0 -1 m -1 r 2 r 0 r 2 r m 2 2 0 h0

1 1 1

m vv vv 0 vv * * * * vv

m1 5 m1 5

5 5

-1 7 m3 4 s

m2 4 s 4 3 3 6

w . 3 ww A ww * 3 * A3 A2 A2 A23 A23

0 0 F (A ) u- G (A ) =u v 0 G0 0 0 * F (A2 ,A3 ) (A2 ,A3 ) =* vww 2 2 v- 4

r 7

r A 5 0 A2 A m r r m21 3 1

+ 6 w 3 . - m0 m0 w 1 + * v+ m v 1 half turn half turn ext. ext.

∗22 ∗ Figure 20: F4-1 to Γ1(u, 2v, w) and F4-2 to Γ2(u, 4v, w) Non-fundamental trunc-simplex tilings ... 35

*G 3 (2u,4v,2w) 4 (=2×2) extensions

A01 A01 uu

uu * *

22 *22 22 22 m * * *

0 A0 A0 3 m0 2 m3 2 2 m m3 m m2 2 0 h0

1 1

vv vv * *vv * * vv

r 7 r 6 . 6

. 6 1 . 1

-1 5 m3 5 s

m2 5 s 5 4 4

8 . 4

2* w A3 2* w A3 A2 w A2

0 0 0 0 F (A0 ) G G 2v2w (A0 ) = 2* uv F (A2 ,A3 ) (A2 ,A3 ) =* u- 2-

7 2 m3 m0 r m2 v-

A0 8 5

3 m2 6 m . 2 A 1 2 r 2 + 1 .

v+ v +

w 4 - 1 half m0 2 r1 half turn ext. turn ext. *G 5 (u,2v,2w) 4 (=2×2) extensions

A01 A01

uu uu

* * A * * m0 0 A0 -1 m -1 r 2 r 0 r 2 r m 2 2 0 h0

1 1

1 1 vv vv vv vv

4 4 4

r 4 1 r1 -1 3 m3 3 s

m2 3 s 3 5 5

6 . 5 2 w * A3 2* w A3 A2 A2

0 0 F (A ) u- G (A ) =2u v 0 G0 0 0 F (A2 ,A3 ) (A2 ,A3 ) = v* w 2 2 v- 3

r 3 4 r A0 A 2 A r1 r r r12 3

- 6 w 5 - 1 m0 . w + * m0 v+ 2 v 1 half turn r1 half turn ext. ext.

∗ ∗ Figure 21: F4-3 to Γ3(2u, 4v, 2w) and F4-4 to Γ5(u, 2v, 2w) 36 Moln´ar,Stojanovi´c,Szirmai

22 G 3 (u,4v,2w) 4 (=2×2) extensions

A1 m1

3 simpler 3

22u

22u r vv 3 h r

vv 3 h *

* 3

3 2 3 m0 m0 1 1 *vv vv *vv

* vv 4

4 *

4 4 m 5

m . 1 m 5 2 m 1 7 3 m2 . 6 6 7 7 2 w * 2 w A2 A3 * A2 A3

0 - 0 0 0 F (A0 ,A1) u G (A ,A - G (A ,A 0 1) = u* v F (A2 ,A3) v 2 3) = 2* vw r 3 h 7 A1 r3 4 2.ext.

3 h h half turn

A 2 3 A 0 3 4 h 5 A2 m1

1 v+ m v+ - 6 . - 1 w m1 v half 2.ext. turn 3

22u r vv 3 h

3 3

1 1

*vv vv *

4 4 m h . 1 . . 2 6 6 6

2* w A2 A3

22 Figure 22: F4-5 to Γ3(u, 4v, 2w) Non-fundamental trunc-simplex tilings ... 37

22 G 4 (u,2v,w) 4 (=2×2) extensions

A1 m1

simpler 3 22u

22u

r3 h r

vv 3 h vv

3 2 3 m0 m

1 0

1 1 vv 1

vv vv vv

4 4

4 4 r 5 r

m . 1 1 5 2 . m3 m 6 6 2 . . . 6 6 6 22w 22w A2 A3 A2 A3

0 0 0 0 F (A ,A1) u- G - G 0 (A0 ,A1) = 2uv F (A2 ,A3) v (A2 ,A3) = 2vw

r3 4 . h 6 A1 r3 half turn

3 A2 3 2.ext. 2 5

A0 h 4 h h r1

1 A3 . 6 v+ v+ - -

1 v r1 w half 2.ext. turn 3

22u

r3 h vv

3 3

1 1

4 4 r1 h . . . 2 6 6 6

22w A2 A3

22 Figure 23: F4-6 to Γ4(u, 4v, w) 38 Moln´ar,Stojanovi´c,Szirmai

The distance between the two base planes d(a0(a0), a3(a3)) = d(H,J)(d is the hyperbolic distance function) will be double of the height of packing hyperball at most.

A 3 A3

Q J E 0 1 u u b b 2 2 2 0 b1 v b v 2 b F K b 03 B D 13 A 03 2 A2

H H 2 2 2 B02 F L L 12 2 w w A A 0 C 1 A0 C A1 b3 b3

Figure 24: Simple and double truncated complete orthoschemes (without the ab- 0 1 2 3 j j solute) in standard coordinate simplex b b b b = A0A1A2A3,(Aib ) = δi with 01 π 12 π 23 π parameters β = u , β = v , β = w . Sketchy pictures also with half-turn axis F03F12 in case u = w

The Coxeter-Schläflisymbol of the complete orthoscheme tiling T generated by re- i flections in the planes b (i ∈ {0, 1, 2, 3}) and aj (j = 0, 3) of a complete orthoscheme O. To every scheme there is a corresponding symmetric 4 × 4 matrix (bij) where bii = 1 and, for i 6= j ∈ {0, 1, 2, 3}, bij equals to cos(π − βij) = − cos βij for all dihedral angles βij between the faces bi,bj of O. For example, in formula (6.1) we see the Coxeter-Schläflimatrix with parameters 01 π 12 π 23 π (u, v, w), i.e. β = u , β = v , β = w .  π  1 − cos u 0 0 π π ij i j − cos u 1 − cos v 0  (b ) = (hb , b i) :=  π π  . (6.1)  0 − cos v 1 − cos w  π 0 0 − cos w 1 This 3-dimensional complete (truncated or frustum) orthoscheme O = O(u, v, w) and its reflection group Gu¯v¯w¯ will be described in Fig. 24,1,8,17,20 and by the symmetric Coxeter-Schläflimatrix (bij) in formula (6.1), furthermore by its inverse matrix (aij) in formula (6.2). ij −1 (aij) = (b ) = hai, aji :=  2 π 2 π π 2 π π π π π π  sin w − cos v cos u sin w cos u cos v cos u cos v cos w 1 π 2 π 2 π π π π (6.2)  cos u sin w sin w cos v cos w cos v  =  π π π 2 π π 2 π  , B  cos u cos v cos v sin u cos w sin u  π π π π π π 2 π 2 π 2 π cos u cos v cos w cos w cos v cos w sin u sin u − cos v Non-fundamental trunc-simplex tilings ... 39 where π π π π π π B = det(bij) = sin2 sin2 − cos2 < 0, i.e. sin sin − cos < 0. u w v u w v In the following the volume of O(u, v, w) is derived by the next Theorem of R. Keller- hals ( [2], by the ideas of N. I. Lobachevsky): Theorem 3. (R. Kellerhals) The volume of a three-dimensional hyperbolic complete orthoscheme O = O(u, v, w) ⊂ H3 can be expressed with the essential angles β01 = π 12 π 23 π ij π u , β = v , β = w , (0 ≤ β ≤ 2 ) in the following form:

1 π Vol(O) = {L(β01 + θ) − L(β01 − θ) + L( + β12 − θ)+ 4 2 π π + L( − β12 − θ) + L(β23 + θ) − L(β23 − θ) + 2L( − θ)}, 2 2 π where θ ∈ (0, 2 ) is deﬁned by: p cos2 β12 − sin2 β01 sin2 β23 tan θ = , cos β01 cos β23

x and where L(x) := − R log |2 sin t|dt denotes the Lobachevsky function (in J. Mil- 0 nor’s interpretation). The hypersphere (or equidistant surface) is a quadratic surface at a constant distance from a plane (base plane) in both halfspaces. The inﬁnite body of the hypersphere, containing the base plane, is called hyperball. The half hyperball (i.e., the part of the hyperball lying on one side of its base h plane) with distance h to a base plane β is denoted by H+. The volume of the h intersection of H+(A) and the right prism with base a 2-polygon A ⊂ β, can be determined by the classical formula of J. Bolyai. 1 2h Vol(Hh (A)) = Area(A) k sinh + 2h . (6.3) + 4 k

q −1 3 The constant k = K is the natural length unit in H , where K denotes the constant negative sectional curvature. In the following we may assume that k = 1. The distance d of two proper points X(x) and Y (y) is calculated by the formula −h x, yi cosh d = . (6.4) ph x, xih y, yi

6.1 Congruent hyperballs to a simply truncated orthoscheme, Family 1 The maximal series of family F1 is characterized by the simplex tiling whose group 43¯ m series is 24 Γ(u) with previous crystallographic notation from [10], and now with 40 Molnár,Stojanović,Szirmai the new orbifold notation ∗233Γ(u). It has the trivial extension by the reflection a0 =m ¯ 0, as in Sect. 2. The corresponding unique fundamental simplex with parameters (u, v, w)=(3, 3, u) is determined by its Coxeter-Schläflimatrix which is derived by (6.1). Its determi- ij 3 2 π 1 nant is B = det(b ) = 4 sin u − 4 < 0 now 6 < u ∈ N. So, the signature is (+ + +−) and the simplex is realizable in hyperbolic space 3 ij H . The inverse (aij) of the Coxeter-Schläflimatrix (b ) follows by (6.2) providing the distance metric (by the general theory of projective metrics [13]).

The optimal hyperball height is hp = d(a0, m0) = d(C,A1) for packings and hc = d(H,A3) is the optimal covering distance where

s −a a + a2 1 − 3 sin2 π h = cosh d(C,A ) = 00 11 01 = 4 4 u , p 1 p 2 1 2 π a00a11 (−a01 + a00a11) 4 − sin u

s −a a + a2 1 1 − sin2 π + 1 cos2 π h = cosh d(H,A ) = 00 33 03 = 2 4 w 16 u c 3 p 2 1 1 2 π a00a33 (−a03 + a00a33) 2 4 − sin u 1 π 1 π 2 sin u 4 cos u equivalent to sinh d(C,A1) = and sinh d(H,A3) = . q 1 2 π q 1 1 2 π 4 − sin u 2 4 − sin u

Then we get the densities δ (packing), ∆ (covering), respectively. E.g.

Area(A) 1 [sinh 2h + 2h ] δ(O(3, 3, u)) = 4 p p , Vol(O)

π π π where Area(A) = − − . 2 3 u The Vol(O) can be calculated by Theorem 3. In this case the maximal packing density is ≈ 0.82251 with u = 7 (see in Table 1p). The optimal covering density of Family 1 is ≈ 1.33093 for u = 7 in Table 1c.

Table 1p, (3, 3, u), 6 < u ∈ N h u h Vol(O) Vol(H+(A)) δ 7 0.78871 0.08856 0.07284 0.82251 8 0.56419 0.10721 0.08220 0.76673 9 0.45320 0.11825 0.08474 0.71663 ...... 20 0.16397 0.14636 0.06064 0.41431 ...... 50 0.06325 0.15167 0.02918 0.19240 ...... 100 0.03147 0.15241 0.01549 0.10165 u → ∞ 0 0.15266 0 0 Non-fundamental trunc-simplex tilings ... 41

Table 1c, (3, 3, u) 6 < u ∈ N h u h Vol(O) Vol(H+(A)) ∆ 7 1.06739 0.08856 0.11787 1.33093 8 0.89198 0.10721 0.15304 1.42747 9 0.81696 0.11825 0.17882 1.51225 ...... 20 0.68136 0.14636 0.29213 1.99596 ...... 50 0.66193 0.15167 0.35361 2.33146 ...... 100 0.65934 0.15241 0.37580 2.46566 u → ∞ 0.65848 0.15266 0.39911 2.61438

By completing Tables 1p and 1c we can read the optimal hyperball packings and coverings with their densities for other extensions in F1 as well. E.g. to 233Γ(2u) 2× we get these for 2u = 8 with δ =≈ 0.76673, ∆ ≈ 1.42747. To Γ3(3u) we get these 33 for 3u = 9 with δ ≈ 0.71663, ∆ ≈ 1.51225. To Γ5(4u) we get these for 4u = 8 again with δ ≈ 0.76673, ∆ ≈ 1.42747. The argument is by our convention that the parameters of groups above in parentheses are just the m23 entries of the D−matrix function, 2m23 is the number of subsimplices around simplex edges. Mostow’s rigidity principle for co-compact hyperbolic space groups are also used (see e.g. [27]).

6.2 On hyperball packings and coverings in a doubly truncated orthoscheme, Family 2

In this case we consider a doubly truncated orthoscheme with Schläfli symbol 1 1 1 {u, v, w = u},( u + v < 2 , 3 ≤ u, v ∈ N) whose fundamental domain is ij CHLA1A2EJQ in Fig. 24.b. Its Coxeter Schläflimatrix (b ) and its inverse (aij) can be derived by formulas (6.1) and (6.2). h Volume formula of half hyperball H+(Ai)(i = 0, 3) of height h can be calculated 1 1 1 by (6.3) where Area(A ) = Area(A ) = π − − according to the above 0 3 2 u¯ v¯ orthoscheme. Both polar planes assigne hyperspheres that are congruent with each other there- 0 3 hi fore the heights h = h of optimal hyperballs Hi (i = 0, 3) can be computed by Fig. 24.b.

The maximal height for optimal congruent hyperball packing is hp = min {d(A2, a3), d(F03, a3)}. In the above doubly truncated orthoscheme d(A2, a3) = d(A2,Q ∼ a23a3 − a33a2), d(F03, a3) = d(a0 + a3, a33a0 − a03a3 ∼ J), (as in [13]). Thus,

s −a a + a2 a2 cosh (d(A , a )) = 33 22 23 = 1 − 23 , 2 3 p 2 2 a a a22 (a33a23 − a23a33) 22 33 42 Moln´ar,Stojanovi´c,Szirmai and v r u 2 π π ! a33 − a03 u1 cos cos cosh (d(F , a )) = = t 1 + u¯ v¯ . 03 3 2a 2 2 π 2 π 33 cos v¯ − sin u¯ The density of optimal hyperball packing will be determined, by the following formula: Area(A)(sinh 2h + 2h ) π 1 − 1 − 1 (sinh 2h + 2h ) δ(O(u, v, u)) = p p = 2 u¯ v¯ p p . (6.5) 2Vol(O) 2Vol(O) For the minimal covering hyperball height:

hc = d(F12, a0) = d(F12, a3) = d(a1 + a2, a1 + a2 + ca3),

−(a13 + a23) where a13 + a23 + ca33 = 0, i.e. c = . Thus, by a11 = a22, we get a33

1 2 − 2 (a22 + a12) − (a13 + a23) a33 cosh (d(F12, a3) = r . 1 2 2 (a22 + a12) 2a22 + 2a12 − (a13 + a23) a33 The covering density ∆(O(u, v, u)) can be deﬁned similarly to the packing density (see (6.5)). V ol(O(u, v, u)) can be calculated by Theorem 3. The maximal volume sum of the hyperball pieces lying in O(u, v, u) can be computed by the formulas (6.1-4) and by the above described computation method for each given possible parameters u, v, u. Therefore, the maximal packing density and the thinnest covering density of the congruent hyperball packings and coverings can be computed for each possible parameters. We see the optimal packing density for F2 is ≈ 0, 81335 with u = w = 7, v = 3 in Table 2p and the top minimal covering density is ≈ 1, 26869 with u = 7, v = 3 in Table 2c. Table 2p, (u, v, u) P h {u, v, u} hp V ol(O(u, v, u)) i=0,3 V ol(H (Ai)) δ(O(u, v, u)) {7, 3, 7} 1.23469 0.38325 0.31172 0.81335 {6, 4, 6} 0.69217; 0.55557 0.42610 0.76696 {8, 3, 8} 0.94946 0.44383 0.33794 0.76143 {8, 4, 8} 0.56419 0.64328 0.49322 0.76673 {5, 4, 5} 0.88055 0.46190 0.36007 0.77955 {4, 5, 4} 0.80846 0.43062 0.31702 0.73620 {4, 6, 4} 0.57311 0.50192 0.33516 0.66775 {3, 7, 3} 0.98399 0.27899 0.20481 0.73411

Table 2c, (u, v, u) P h {u, v, u} hc V ol(O(u, v, u)) i=0,3 V ol(H (Ai)) δ(O(u, v, u)) {7, 3, 7} 1.49903 0.38325 0.48607 1.26829 {6, 4, 6} 1.01481 0.55557 0.75523 1.35938 {8, 3, 8} 1.26595 0.44383 0.57470 1.29487 {5, 4, 5} 1.19095 0.46190 0.60856 1.31751 {8, 4, 8} 0.89198 0.64328 0.91826 1.42747 {4, 5, 4} 1.16974 0.43062 0.58741 1.36411 {4, 6, 4} 0.99583 0.50192 0.73137 1.45714 {3, 7, 3} 1.36406 0.27899 0.38699 1.38713 Non-fundamental trunc-simplex tilings ... 43

By completing Tables 2p and 2c we can determine the other densities in Family ∗22 2. E.g. to Γ2(2u, 2v) (i.e. u = 4, v = 3) we get δ ≈ 0.76143. ∆ ≈ 1.29487. Or to Γ58(2u, 4v) we get {u, v, w} = {6, 4, 6} for optimal arrangement, δ ≈ 0.76696. ∆ ≈ 1.35938.

6.3 Packings with congruent hyperballs in doubly truncated orthoscheme, Family 3-4

In this subsection we consider congruent hyperball packings to truncated orthoscheme tilings. Both polar planes assign hyperballs with equal height (see hi Fig. 24.b). It is clear, that the height of optimal hyperballs Hi (i = 0, 3) is

hp(u, v, w) = min{d(H,J)/2, d(Q, A2), d(C,A1)}, where u, v, w are integer parameters. In this case the volume sum of the hyperball pieces lying in the orthoscheme has to divide with the volume of trunc-orthoscheme O(u, v, w) as usual. Segments A1C, A2Q and JH can be determined by the ma- chinery of the projective metrics (see subsection 6.1 or [13]). The volume of the orthoscheme O(u, v, w) can be determined by Theorem 3. We note here, that the role of the parameters u and w is symmetrical, therefore we can assume that u > w in F3 or u < w in F4.

Table 3p, (u, v, w = 3), congruent hyperballs P h 1 {u, v, w} h V ol(O(u, v, w)) i=0,3 V ol(H (Ai)) δ (O(u, v, w)) {4, 7, 3} 0.59710 0.39274 0.27700 0.70529 {5, 7, 3} 0.41812; 0.43216 0.25203 0.58320 ...... {50, 7, 3} 0.03492 0.49140 0.03962 0.08062 ...... {4, 8, 3} 0.56419 0.42885 0.32881 0.76673 {5, 8, 3} 0.67409 0.47536 0.26747 0.56266 ...... {50, 8, 3} 0.03405 0.52378 0.04245 0.08105 ...... {4, 9, 3} 0.46841 0.45130 0.30800 0.68247 {5, 9, 3} 0.39083 0.48771 0.31589 0.64771 ...... {50, 9, 3} 0.03348 0.54384 0.04466 0.08212 44 Moln´ar,Stojanovi´c,Szirmai

Table 4p, (u, v, w), congruent hyperballs P h 1 {u, v, w} h V ol(O(u, v, w)) i=0,3 V ol(H (Ai)) δ (O(u, v, w)) {7, 3, 8} 0.93100 0.41326 0.25726 0.62251 {7, 3, 9} 0.76734 0.43171 0.23355 0.54099 ...... {7, 3, 50} 0.11380 0.49016 0.06121 0.12488 ...... {8, 3, 9} 0.78366 0.46266 0.29474 0.63704 {8, 3, 10} 0.67409 0.47536 0.26747 0.56266 ...... {8, 3, 50} 0.11668 0.52248 0.06935 0.13274 ...... {5, 4, 6} 0.73969 0.50747 0.37287 0.73476 {5, 4, 7} 0.59326 0.53230 0.32974 0.61947 ...... {5, 4, 50} 0.07206 0.59291 0.06350 0.10710 ...... {4, 5, 5} 0.69129 0.49789 0.38284 0.76893 {4, 5, 6} 0.53064 0.52971 0.33597 0.63426 ...... {4, 5, 50} 0.05502 0.59318 0.05710 0.096256 ...... {4, 6, 5} 0.50625 0.55992 0.37558 0.67078 {4, 6, 6} 0.48121 0.58850 0.40850 0.69414 ...... {4, 6, 50} 0.05138 0.64697 0.06409 0.09906

Figure 25: The least dense congruent hyperball covering arrangement to parameters {7, 3, 7} with density ≈ 1.26829 and the corresponding Coxeter orthoscheme in Euclidean B-C-K model. Non-fundamental trunc-simplex tilings ... 45

References

[1] Fejes TóthL., Regular Figures, Macmillian (New York), 1964. [2] Kellerhals R., On the volume of hyperbolic polyhedra, Math. Ann., 285 (1989) 541–569. [3] LuˇcićZ., MolnárE., VasiljevićN., An algorithm for classification of fundamental polygons for a plane discontinuous group, Geom-Sym Conf. 2015: Discrete Geometry and Symmetry 234 (2018) 257–278, DOI: 10.1007/978-3-319-78434- 2 14 [4] Maskit B., On Poincaré’stheorem for fundamental polygons, Adv. in Math., 7 (1971), 219–230. [5] MolnárE., Eine Klasse von hyperbolischen Raumgruppen, Beiträgezur Algebra und Geometrie, 30 (1990), 79–100. [6] MolnárE., Polyhedron complexes with simply transitive group actions and their realizations, Acta Math. Hung., 59 (1-2) (1992), 175–216. [7] MolnárE., Prok I., A polyhedron algorithm for finding space groups, Proceed- ings of Third Int. Conf. On Engineering Graphics and Descriptive Geometry, Vienna 1988, 2, 37–44. [8] MolnárE., Prok I., Classification of solid transitive simplex tilings in simply connected 3-spaces, Part I, Colloquia Math. Soc. JánosBolyai 63. Intuitive Geometry, Szeged (Hungary), 1991. North-Holland (1994), 311–362. [9] MolnárE., Prok I., Szirmai J., Classification of solid transitive simplex tilings in simply connected 3-spaces, Part II, Periodica Math. Hung. 35 (1-2) (1997), 47–94. [10] MolnárE., Prok I., Szirmai J., Classification of tile-transitive 3-simplex tilings and their realizations in homogeneous spaces, Non-Euclidean Geometries, JánosBolyai Memorial Volume, Editors: A. Prékopa and E. Molnár,Mathe- matics and Its Applications, 581, Springer (2006), 321–363. [11] MolnárE., On D−symbols and orbifolds in an algorithmic way, Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia, 58 (2011), 263–276. [12] MolnárE., Szirmai J., Weeks J., 3-simplex tilings, splitting orbifolds and man- ifolds, Symmetry: Culture and Science, 22 (3-4) (2011), 435-458. [13] MolnárE., Szirmai J., Top dense hyperbolic ball packings and coverings for complete Coxeter orthoscheme groups, Publications de l’Institut Mathematique, 103 (117) (2018), 129–146, DOI: 10.2298/PIM1817129M [14] StojanovićM., Some series of hyperbolic space groups, Annales Univ. Sci. Budapest, Sect. Math. 36 (1993), 85–102. 46 Molnár,Stojanović,Szirmai

[15] Stojanovi´c M., Fundamental simplices with outer vertices for hyperbolic groups, Filomat, 24 (1) (2010), 1–19. [16] Stojanovi´cM., Four series of hyperbolic space groups with simplicial domains, and their supergroups, Krag. J.Math., 35 (2) (2011), 303–315.

[17] StojanovićM., Supergroups for six series of hyperbolic simplex groups, Peri- odica Math. Hung., 67 (1) (2013), 115–131. [18] StojanovićM., Coxeter groups as automorphism groups of solid transitive 3- simplex tilings, Filomat, 28 (3) (2014), 557–577, DOI: 10.2298/FIL1403557S [19] Stojanović M., Hyperbolic space groups and their supergroups for fundamental simplex tilings, Acta Math. Hung., 153 (2) (2017), 276–288, DOI: 10.1007/s10474-017-0761-z [20] StojanovićM., Hyperbolic space groups with truncated simplices as fundamental domains, Filomat, 33 (4) (2019), 1107-1116, DOI: 10.2298/FIL1904107S

[21] Szirmai J., The least dense hyperball covering to the regular prism tilings in the hyperbolic n-space, Ann. Mat. Pur. Appl. (2016) 195, 235-248, DOI: 10.1007/s10231-014-0460-0. [22] Szirmai J., Density upper bound of congruent and non-congruent hyperball packings generated by truncated regular simplex tilings, Rendiconti del Circolo Matematico di Palermo Series 2, 67 (2018), 307–322, DOI: 10.1007/s12215- 017-0316-8. [23] Szirmai J., Decomposition method related to saturated hyperball packings, Ars Mathematica Contemporanea, 16 (2019), 349–358, DOI: 10.26493/1855- 3974.1485.0b1.

[24] Szirmai J., Hyperball packings in hyperbolic 3-space, MatematiˇckiVesnik, 70 (3) (2018), 211–221. [25] Szirmai J., Hyperball packings related to cube and octahedron tilings in hyperbolic space, Contributions to Discrete Mathematics (to appear) (2019), arXiv:1803.04948.

[26] Szirmai J., Upper bound of density for packing of congruent hyperballs in hyperbolic 3−space, Submitted manuscript, (2019). [27] Vinberg E. B. (Ed.), Geometry II. Spaces of Constant Curvature Spriger Ver- lag Berlin-Heidelberg-New York-London-Paris-Tokyo-Hong Kong-Barcelona- Budapest, 1993.

[28] Zhuk I. K., Fundamental tetrahedra in Euclidean and Lobachevsky spaces, Soviet Math. Dokl., 270 (3) (1983), 540–543.