DOCSLIB.ORG

Regular Polyhedra in N Dimensions David Vogan

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Regular Polyhedra in N Dimensions David Vogan Regular polyhedra in n dimensions David Vogan Introduction Linear algebra Flags Regular polyhedra in n dimensions Reflections Relations Classification David Vogan Department of Mathematics Massachusetts Institute of Technology Tsinghua University, December 11, 2014 Regular polyhedra Outline in n dimensions David Vogan Introduction Introduction Linear algebra Flags Reflections Ideas from linear algebra Relations Classification Flags in polyhedra Reflections and relations Relations satisfied by reflection symmetries Presentation and classification Regular polyhedra The talk in one line in n dimensions David Vogan Introduction Want to understand the possibilities for a regular Linear algebra Flags polyhedron Pn of dimension n. Reflections Schläfli symbol is string fm1;:::; mn−1g. Relations Classification Meaning of m1: two-dimensional faces are regular m1-gons. Equivalent: m1 edges (“1-faces”) in a fixed 2-face. Meaning of m : fixed vertex ⊂ m 2-faces ⊂ fixed 3-face. 2 . 2 . fixed k − 1-face ⊂ m k + 1-faces ⊂ fixed k + 2-face. k+1 . What are the possible Schläfli symbols, and why do they characterize Pn? Regular polyhedra Dimension two in n dimensions David Vogan Introduction One regular m-gon for every m ≥ 3 Linear algebra Flags m = 3: equilateral triangle Reflections Relations J J Classification s J Schläfli symbol f3g J J s s m = 7: regular heptagon Schläfli symbol f7g Regular polyhedra Dimension three in n dimensions David Vogan Introduction Five regular polyhedra. Linear algebra Flags Reflections Relations Classification tetrahedron Schläfli symbol f3; 3g octahedron Schläfli symbol f3; 4g icosahedron Schläfli symbol f3; 5g Regular polyhedra Dimensions one and zero in n dimensions David Vogan Introduction Linear algebra Flags One regular 1-gon. Reflections interval Schläfli symbol fg Relations Classification Symmetry group: two elements f1; sg There is also just one regular 0-gon: point Schläfli symbol undefined s Symmetry group trivial (zero gens of order 2). Regular polyhedra What’s a regular polyhedron? in n dimensions David Vogan Something really symmetrical. like a square Introduction Linear algebra Flags s s Reflections Relations Classification s s FIX one vertex inside one edge inside square. Two building block symmetries. ss0 s s s −! -s1 s s s s s0 takes red vertex to adj vertex along red edge; s1 takes red edge to adj edge at red vertex. Regular polyhedra More symmetries from building blocks in n dimensions . David Vogan r1 r identity Introduction Linear algebra Flags r r Reflections reflect in x = 0 rs0 r rs1 r reflect in y = x Relations Classification r r r r ◦ ◦ 90 rotation rs0s1 r r s1s0 r 270 rotation r r r r reflect in y = −x rs0s1s0 r rs1s0s1 r reflect in y = 0 Presentation: generators s0; s1; r r r r 2 2 ◦ relations s0 = s1 = 1, rs0s1s0s1 r 180 rotation = s1s0s1s0 s0s1s0s1 = s1s0s1s0. r r Regular polyhedra Understanding all regular polyhedra in n dimensions David Vogan Introduction Linear algebra Define a flag as a chain of faces like vertex ⊂ edge. Flags Reflections Relations Introduce basic symmetries like s0, s1 which change a flag as little as possible. Classification Find a presentation of the symmetry group. Reconstruct the polyhedron from this presentation. Decide which presentations are possible. Regular polyhedra Most of linear algebra in n dimensions David Vogan Introduction V n-diml vec space GL(V ) invertible linear maps. Linear algebra complete flag in V is chain of subspaces F Flags Reflections f0g = V ⊂ V ⊂ · · · ⊂ V ⊂ Vn = V ; dim V = i: 0 1 n−1 i Relations Stabilizer B(F) called Borel subgroup of GL(V ). Classification Example n i V = k , Vi = f(x1;:::; xi ; 0;:::; 0) j xi 2 kg ' k . Stabilizer of this flag is upper triangular matrices. Theorem 1. GL(V ) acts transitively on flags. 2. Stabilizer of one flag is isomorphic to group of invertible upper triangular matrices. Regular polyhedra Rest of linear algebra in n dimensions David Vogan Fix integers d = (0 = d0 < d1 < ··· < dr = n) Introduction Partial flag of type d is chain of subspaces G Linear algebra Flags W0 ⊂ W1 ⊂ · · · ⊂ Vr−1 ⊂ Wr ; dim Wj = dj : Reflections Stabilizer P(G) is a parabolic subgroup of GL(V ). Relations Theorem Classification Fix a complete flag (0 = V0 ⊂ · · · ⊂ Vn = V ), and consider the n − 1 partial flags Gp = (V0 ⊂ · · · ⊂ Vbp ⊂ · · · ⊂ Vn) 1 ≤ p ≤ n − 1 obtained by omitting one proper subspace. 1. GL(V ) is generated by the n − 1 subgroups P(Gp). 2. P(Gp) is isomorphic to block upper-triangular matrices with a single 2 × 2 block. So build all linear transformations from two by two matrices and upper triangular matrices. Regular polyhedra Flags in n dimensions David Vogan Suppose Pn compact n-diml convex polyhedron. Introduction A (complete) flag F in P is a chain Linear algebra Flags ⊂ ⊂ · · · ⊂ ; = P0 P1 Pn dim Pi i Reflections with Pi−1 a face of Pi . Relations Classification Example ¨H ¨H ¨ HH ¨¨ HH H¨ s H s H ¨¨ H ¨¨ s H¨ s s H¨ s Two flags in two-dimls P. Symmetrys group (generated by reflections in x and y axes) is transitive on edges, not transitive on flags. Definition P regular if symmetry group acts transitively on flags. Regular polyhedra Adjacent flags in n dimensions David Vogan F = (P0 ⊂ P1 ⊂ · · · ⊂ Pn); dim Pi = i Introduction complete flag in n-diml compact convex polyhedron. Linear algebra Flags 0 0 0 0 A flag F = (P0 ⊂ P1 ⊂ · · · ⊂ Pn) is i-adjacent to F if Reflections 0 0 Pj = Pj for all j 6= i, and Pi 6= Pi . Relations Classification @ @ @ @ s@ s@ s@ s@ s s s s @ @ s@ s@ s s s s Three flags adjacent to F, i = 0; 1; 2. 0 0 F 0: move vertex P0 only. F 1: move edge P1 only. 0 F 2: move face P2 only. There is exactly one F 0 i-adjacent to F (each i = 0; 1;:::; n − 1). Symmetry doesn’t matter for this! Regular polyhedra Stabilizing a flag in n dimensions David Vogan Lemma Introduction Suppose F = (P0 ⊂ P1 ⊂ · · · ) is a complete flag in Linear algebra Flags n-dimensional compact convex polyhedron Pn. Any affine Reflections map T preserving F acts trivially on Pn. Relations Proof. Induction on n. If n = −1, Pn = ; and result is true. Classification Suppose n ≥ 0 and the the result is known for n − 1. Write pn = center of mass of Pn. Since center of mass is preserved by affine transformations, Tpn = pn. By inductive hypothesis, T acts trivially on n − 1-diml affine span(Pn−1) spanned by Pn−1. Easy to see that pn 2= span(Pn−1), so pn and (n − 1)-diml span(Pn−1) must generate n-diml span(Pn). Since T trivial on gens, trivial on span(Pn). Q.E.D. Compactness matters; result fails for P1 = [0; 1). Regular polyhedra Symmetries and flags in n dimensions David Vogan From now on P is a compact convex regular polyhedron n Introduction with fixed flag Linear algebra F = (P0 ⊂ P1 ⊂ · · · ⊂ Pn); dim Pi = i Flags Reflections Write pi = center of mass of Pi . Relations Theorem Classification There is exactly one symmetry w of Pn for each complete flag G, characterized by wF = G. Corollary 0 Define Fi = unique flag (i)-adj to F (0 ≤ i < n). There is 0 a unique symmetrys i of Pn characterized bys i (F) = Fi . It satisfies 0 2 1. si (Fi ) = F, si = 1. 2. si fixes the (n − 1)-diml hyperplane through the n points fp0;:::; pi−1; pbi ; pi+1;:::; png. Regular polyhedra Examples of basic symmetries si in n dimensions s David Vogan −!0 Introduction s s s s Linear algebra p2 p2 Flags e e Reflections p0 p1 p1 p0 Relations e e ss ss e e Classification This is s0, which changes F only in P0, so acts trivially on the line through p1 and p2. s p s p sp s 2 1 2 e e e -p0 p1 p0 s1 e e ss e ss This is s1, which changes F only in P1, so acts trivially on the line through p0 and p2. Regular polyhedra What’s a reflection? in n dimensions David Vogan On vector space V (characteristic not 2), a linear Introduction map s with s2 = 1, dim(−1 eigenspace) = 1. Linear algebra _ Flags −1 eigenspace Ls = span of nonzero vector α 2 V Reflections L = fv 2 V j sv = −vg = span(α_): s Relations ∗ +1 eigenspace Hs = kernel of nonzero α 2 V Classification Hs = fv 2 V j sv = vg = ker(α): hα; vi _ sv = s _ (v) = v − 2 α : (α,α ) hα; α_i Definition of reflection does not mention “orthogonal.” If V has quadratic form h; i identifying V ' V ∗, then s is orthogonal () α is proportional to α_: Regular polyhedra Two reflections in n dimensions hαs; vi _ hαt ; vi _ David Vogan sv = v − 2 _ αs ; tv = v − 2 _ αt : hαs; α i hαt ; α i s t Introduction Assume V = Ls ⊕ Lt ⊕ (Hs \ Ht ). Linear algebra _ _ _ _ Flags Subspace Ls ⊕ Lt has basis fαs ; αt g, cst = 2hαs; αt i=hαs; αs i; Reflections −1 cst 1 0 −1 + cst cts cst s = ; t = ; st = : Relations 0 1 cts −1 cts −1 Classification det(st) = 1; tr(st) = −2 + cst cts; −1 −1 st has eigenvalues z; z ; z + z = cst cts − 2: −1 ±iθ −1 z; z = e ; θ = cos (−1 + cst cts=2)): Proposition −1 th Suppose −1 + cst cts=2 = ζ + ζ for a primitive m root ζ. Then st is a rotation of order m in the plane Ls ⊕ Lt . Otherwise st has infinite order. So 1. m = 2 if and only if cst = cts = 0; 2.
Load more

Recommended publications
  • 1 Lifts of Polytopes
    Lecture 5: Lifts of polytopes and non-negative rank CSE 599S: Entropy optimality, Winter 2016 Instructor: James R. Lee Last updated: January 24, 2016 1 Lifts of polytopes 1.1 Polytopes and inequalities Recall that the convex hull of a subset X n is defined by ⊆ conv X λx + 1 λ x0 : x; x0 X; λ 0; 1 : ( ) f ( − ) 2 2 [ ]g A d-dimensional convex polytope P d is the convex hull of a finite set of points in d: ⊆ P conv x1;:::; xk (f g) d for some x1;:::; xk . 2 Every polytope has a dual representation: It is a closed and bounded set defined by a family of linear inequalities P x d : Ax 6 b f 2 g for some matrix A m d. 2 × Let us define a measure of complexity for P: Define γ P to be the smallest number m such that for some C s d ; y s ; A m d ; b m, we have ( ) 2 × 2 2 × 2 P x d : Cx y and Ax 6 b : f 2 g In other words, this is the minimum number of inequalities needed to describe P. If P is full- dimensional, then this is precisely the number of facets of P (a facet is a maximal proper face of P). Thinking of γ P as a measure of complexity makes sense from the point of view of optimization: Interior point( methods) can efficiently optimize linear functions over P (to arbitrary accuracy) in time that is polynomial in γ P . ( ) 1.2 Lifts of polytopes Many simple polytopes require a large number of inequalities to describe.
    [Show full text]
  • Regular Polyhedra of Index 2
    REGULAR POLYHEDRA OF INDEX 2 A dissertation presented by Anthony Cutler to The Department of Mathematics In partial fulfillment of the requirements for the degree of Doctor of Philosophy in the field of Mathematics Northeastern University Boston, Massachusetts April, 2009 1 REGULAR POLYHEDRA OF INDEX 2 by Anthony Cutler ABSTRACT OF DISSERTATION Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate School of Arts and Sciences of Northeastern University, April 2009 2 We classify all finite regular polyhedra of index 2, as defined in Section 2 herein. The definition requires the polyhedra to be combinatorially flag transitive, but does not require them to have planar or convex faces or vertex-figures, and neither does it require the polyhedra to be orientable. We find there are 10 combinatorially regular polyhedra of index 2 with vertices on one orbit, and 22 infinite families of combinatorially regular polyhedra of index 2 with vertices on two orbits, where polyhedra in the same family differ only in the relative diameters of their vertex orbits. For each such polyhedron, or family of polyhedra, we provide the underlying map, as well as a geometric diagram showing a representative face for each face orbit, and a verification of the polyhedron’s combinatorial regularity. A self-contained completeness proof is given. Exactly five of the polyhedra have planar faces, which is consistent with a previously known result. We conclude by describing a non-Petrie duality relation among regular polyhedra of index 2, and suggest how it can be extended to other combinatorially regular polyhedra.
    [Show full text]
  • Binary Icosahedral Group and 600-Cell
    Article Binary Icosahedral Group and 600-Cell Jihyun Choi and Jae-Hyouk Lee * Department of Mathematics, Ewha Womans University 52, Ewhayeodae-gil, Seodaemun-gu, Seoul 03760, Korea; [email protected] * Correspondence: [email protected]; Tel.: +82-2-3277-3346 Received: 10 July 2018; Accepted: 26 July 2018; Published: 7 August 2018 Abstract: In this article, we have an explicit description of the binary isosahedral group as a 600-cell. We introduce a method to construct binary polyhedral groups as a subset of quaternions H via spin map of SO(3). In addition, we show that the binary icosahedral group in H is the set of vertices of a 600-cell by applying the Coxeter–Dynkin diagram of H4. Keywords: binary polyhedral group; icosahedron; dodecahedron; 600-cell MSC: 52B10, 52B11, 52B15 1. Introduction The classification of finite subgroups in SLn(C) derives attention from various research areas in mathematics. Especially when n = 2, it is related to McKay correspondence and ADE singularity theory [1]. The list of finite subgroups of SL2(C) consists of cyclic groups (Zn), binary dihedral groups corresponded to the symmetry group of regular 2n-gons, and binary polyhedral groups related to regular polyhedra. These are related to the classification of regular polyhedrons known as Platonic solids. There are five platonic solids (tetrahedron, cubic, octahedron, dodecahedron, icosahedron), but, as a regular polyhedron and its dual polyhedron are associated with the same symmetry groups, there are only three binary polyhedral groups(binary tetrahedral group 2T, binary octahedral group 2O, binary icosahedral group 2I) related to regular polyhedrons.
    [Show full text]
  • Can Every Face of a Polyhedron Have Many Sides ?
    Can Every Face of a Polyhedron Have Many Sides ? Branko Grünbaum Dedicated to Joe Malkevitch, an old friend and colleague, who was always partial to polyhedra Abstract. The simple question of the title has many different answers, depending on the kinds of faces we are willing to consider, on the types of polyhedra we admit, and on the symmetries we require. Known results and open problems about this topic are presented. The main classes of objects considered here are the following, listed in increasing generality: Faces: convex n-gons, starshaped n-gons, simple n-gons –– for n ≥ 3. Polyhedra (in Euclidean 3-dimensional space): convex polyhedra, starshaped polyhedra, acoptic polyhedra, polyhedra with selfintersections. Symmetry properties of polyhedra P: Isohedron –– all faces of P in one orbit under the group of symmetries of P; monohedron –– all faces of P are mutually congru- ent; ekahedron –– all faces have of P the same number of sides (eka –– Sanskrit for "one"). If the number of sides is k, we shall use (k)-isohedron, (k)-monohedron, and (k)- ekahedron, as appropriate. We shall first describe the results that either can be found in the literature, or ob- tained by slight modifications of these. Then we shall show how two systematic ap- proaches can be used to obtain results that are better –– although in some cases less visu- ally attractive than the old ones. There are many possible combinations of these classes of faces, polyhedra and symmetries, but considerable reductions in their number are possible; we start with one of these, which is well known even if it is hard to give specific references for precisely the assertion of Theorem 1.
    [Show full text]
  • Four-Dimensional Regular Polytopes
    faculty of science mathematics and applied and engineering mathematics Four-dimensional regular polytopes Bachelor’s Project Mathematics November 2020 Student: S.H.E. Kamps First supervisor: Prof.dr. J. Top Second assessor: P. Kiliçer, PhD Abstract Since Ancient times, Mathematicians have been interested in the study of convex, regular poly- hedra and their beautiful symmetries. These five polyhedra are also known as the Platonic Solids. In the 19th century, the four-dimensional analogues of the Platonic solids were described mathe- matically, adding one regular polytope to the collection with no analogue regular polyhedron. This thesis describes the six convex, regular polytopes in four-dimensional Euclidean space. The focus lies on deriving information about their cells, faces, edges and vertices. Besides that, the symmetry groups of the polytopes are touched upon. To better understand the notions of regularity and sym- metry in four dimensions, our journey begins in three-dimensional space. In this light, the thesis also works out the details of a proof of prof. dr. J. Top, showing there exist exactly five convex, regular polyhedra in three-dimensional space. Keywords: Regular convex 4-polytopes, Platonic solids, symmetry groups Acknowledgements I would like to thank prof. dr. J. Top for supervising this thesis online and adapting to the cir- cumstances of Covid-19. I also want to thank him for his patience, and all his useful comments in and outside my LATEX-file. Also many thanks to my second supervisor, dr. P. Kılıçer. Furthermore, I would like to thank Jeanne for all her hospitality and kindness to welcome me in her home during the process of writing this thesis.
    [Show full text]
  • Regular Polyhedra Through Time
    Fields Institute I. Hubard Polytopes, Maps and their Symmetries September 2011 Regular polyhedra through time The greeks were the first to study the symmetries of polyhedra. Euclid, in his Elements showed that there are only five regular solids (that can be seen in Figure 1). In this context, a polyhe- dron is regular if all its polygons are regular and equal, and you can find the same number of them at each vertex. Figure 1: Platonic Solids. It is until 1619 that Kepler finds other two regular polyhedra: the great dodecahedron and the great icosahedron (on Figure 2. To do so, he allows \false" vertices and intersection of the (convex) faces of the polyhedra at points that are not vertices of the polyhedron, just as the I. Hubard Polytopes, Maps and their Symmetries Page 1 Figure 2: Kepler polyhedra. 1619. pentagram allows intersection of edges at points that are not vertices of the polygon. In this way, the vertex-figure of these two polyhedra are pentagrams (see Figure 3). Figure 3: A regular convex pentagon and a pentagram, also regular! In 1809 Poinsot re-discover Kepler's polyhedra, and discovers its duals: the small stellated dodecahedron and the great stellated dodecahedron (that are shown in Figure 4). The faces of such duals are pentagrams, and are organized on a \convex" way around each vertex. Figure 4: The other two Kepler-Poinsot polyhedra. 1809. A couple of years later Cauchy showed that these are the only four regular \star" polyhedra. We note that the convex hull of the great dodecahedron, great icosahedron and small stellated dodecahedron is the icosahedron, while the convex hull of the great stellated dodecahedron is the dodecahedron.
    [Show full text]
  • 15 BASIC PROPERTIES of CONVEX POLYTOPES Martin Henk, J¨Urgenrichter-Gebert, and G¨Unterm
    15 BASIC PROPERTIES OF CONVEX POLYTOPES Martin Henk, J¨urgenRichter-Gebert, and G¨unterM. Ziegler INTRODUCTION Convex polytopes are fundamental geometric objects that have been investigated since antiquity. The beauty of their theory is nowadays complemented by their im- portance for many other mathematical subjects, ranging from integration theory, algebraic topology, and algebraic geometry to linear and combinatorial optimiza- tion. In this chapter we try to give a short introduction, provide a sketch of \what polytopes look like" and \how they behave," with many explicit examples, and briefly state some main results (where further details are given in subsequent chap- ters of this Handbook). We concentrate on two main topics: • Combinatorial properties: faces (vertices, edges, . , facets) of polytopes and their relations, with special treatments of the classes of low-dimensional poly- topes and of polytopes \with few vertices;" • Geometric properties: volume and surface area, mixed volumes, and quer- massintegrals, including explicit formulas for the cases of the regular simplices, cubes, and cross-polytopes. We refer to Gr¨unbaum [Gr¨u67]for a comprehensive view of polytope theory, and to Ziegler [Zie95] respectively to Gruber [Gru07] and Schneider [Sch14] for detailed treatments of the combinatorial and of the convex geometric aspects of polytope theory. 15.1 COMBINATORIAL STRUCTURE GLOSSARY d V-polytope: The convex hull of a finite set X = fx1; : : : ; xng of points in R , n n X i X P = conv(X) := λix λ1; : : : ; λn ≥ 0; λi = 1 : i=1 i=1 H-polytope: The solution set of a finite system of linear inequalities, d T P = P (A; b) := x 2 R j ai x ≤ bi for 1 ≤ i ≤ m ; with the extra condition that the set of solutions is bounded, that is, such that m×d there is a constant N such that jjxjj ≤ N holds for all x 2 P .
    [Show full text]
  • Are Your Polyhedra the Same As My Polyhedra?
    Are Your Polyhedra the Same as My Polyhedra? Branko Gr¨unbaum 1 Introduction “Polyhedron” means different things to different people. There is very little in common between the meaning of the word in topology and in geometry. But even if we confine attention to geometry of the 3-dimensional Euclidean space – as we shall do from now on – “polyhedron” can mean either a solid (as in “Platonic solids”, convex polyhedron, and other contexts), or a surface (such as the polyhedral models constructed from cardboard using “nets”, which were introduced by Albrecht D¨urer [17] in 1525, or, in a more mod- ern version, by Aleksandrov [1]), or the 1-dimensional complex consisting of points (“vertices”) and line-segments (“edges”) organized in a suitable way into polygons (“faces”) subject to certain restrictions (“skeletal polyhedra”, diagrams of which have been presented first by Luca Pacioli [44] in 1498 and attributed to Leonardo da Vinci). The last alternative is the least usual one – but it is close to what seems to be the most useful approach to the theory of general polyhedra. Indeed, it does not restrict faces to be planar, and it makes possible to retrieve the other characterizations in circumstances in which they reasonably apply: If the faces of a “surface” polyhedron are sim- ple polygons, in most cases the polyhedron is unambiguously determined by the boundary circuits of the faces. And if the polyhedron itself is without selfintersections, then the “solid” can be found from the faces. These reasons, as well as some others, seem to warrant the choice of our approach.
    [Show full text]
  • Reconstructing Detailed Dynamic Face Geometry from Monocular Video
    Reconstructing Detailed Dynamic Face Geometry from Monocular Video Pablo Garrido1 ∗ Levi Valgaerts1 † Chenglei Wu1,2 ‡ Christian Theobalt1 § 1Max Planck Institute for Informatics 2Intel Visual Computing Institute Figure 1: Two results obtained with our method. Left: The input video. Middle: The tracked mesh shown as an overlay. Right: Applying texture to the mesh and overlaying it with the input video using the estimated lighting to give the impression of virtual face make-up. Abstract Links: DL PDF WEB VIDEO 1 Introduction Detailed facial performance geometry can be reconstructed using dense camera and light setups in controlled studios. However, Optical performance capture methods can reconstruct faces of vir- a wide range of important applications cannot employ these ap- tual actors in videos to deliver detailed dynamic face geometry. proaches, including all movie productions shot from a single prin- However, existing approaches are expensive and cumbersome as cipal camera. For post-production, these require dynamic monoc- they can require dense multi-view camera systems, controlled light ular face capture for appearance modification. We present a new setups, active markers in the scene, and recording in a controlled method for capturing face geometry from monocular video. Our ap- studio (Sec. 2.2). At the other end of the spectrum are computer proach captures detailed, dynamic, spatio-temporally coherent 3D vision methods that capture face models from monocular video face geometry without the need for markers. It works under un- (Sec. 2.1). These captured models are extremely coarse, and usually controlled lighting, and it successfully reconstructs expressive mo- only contain sparse collections of 2D or 3D facial landmarks rather tion including high-frequency face detail such as folds and laugh than a detailed 3D shape.
    [Show full text]
  • 1 More Background on Polyhedra
    CS 598CSC: Combinatorial Optimization Lecture date: 26 January, 2010 Instructor: Chandra Chekuri Scribe: Ben Moseley 1 More Background on Polyhedra This material is mostly from [3]. 1.1 Implicit Equalities and Redundant Constraints Throughout this lecture we will use affhull to denote the affine hull, linspace to be the linear space, charcone to denote the characteristic cone and convexhull to be the convex hull. Recall n that P = x Ax b is a polyhedron in R where A is a m n matrix and b is a m 1 matrix. f j ≤ g × × An inequality aix bi in Ax b is an implicit equality if aix = bi x P . Let I 1; 2; : : : ; m be the index set of≤ all implicit≤ equalities in Ax b. Then we can partition8 2 A into⊆A= fx b= andg A+x b+. Here A= consists of the rows of A with≤ indices in I and A+ are the remaining≤ rows of A. Therefore,≤ P = x A=x = b=;A+x b+ . In other words, P lies in an affine subspace defined by A=x = b=. f j ≤ g Exercise 1 Prove that there is a point x0 P such that A=x0 = b= and A+x0 < b+. 2 Definition 1 The dimension, dim(P ), of a polyhedron P is the maximum number of affinely independent points in P minus 1. n Notice that by definition of dimension, if P R then dim(P ) n, if P = then dim(P ) = 1, and dim(P ) = 0 if and only if P consists of a⊆ single point.
    [Show full text]
  • Convex Polytopes and Tilings with Few Flag Orbits
    Convex Polytopes and Tilings with Few Flag Orbits by Nicholas Matteo B.A. in Mathematics, Miami University M.A. in Mathematics, Miami 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 Abstract of Dissertation The amount of symmetry possessed by a convex polytope, or a tiling by convex polytopes, is reflected by the number of orbits of its flags under the action of the Euclidean isometries preserving the polytope. The convex polytopes with only one flag orbit have been classified since the work of Schläfli in the 19th century. In this dissertation, convex polytopes with up to three flag orbits are classified. Two-orbit convex polytopes exist only in two or three dimensions, and the only ones whose combinatorial automorphism group is also two-orbit are the cuboctahedron, the icosidodecahedron, the rhombic dodecahedron, and the rhombic triacontahedron. Two-orbit face-to-face tilings by convex polytopes exist on E1, E2, and E3; the only ones which are also combinatorially two-orbit are the trihexagonal plane tiling, the rhombille plane tiling, the tetrahedral-octahedral honeycomb, and the rhombic dodecahedral honeycomb. Moreover, any combinatorially two-orbit convex polytope or tiling is isomorphic to one on the above list. Three-orbit convex polytopes exist in two through eight dimensions. There are infinitely many in three dimensions, including prisms over regular polygons, truncated Platonic solids, and their dual bipyramids and Kleetopes. There are infinitely many in four dimensions, comprising the rectified regular 4-polytopes, the p; p-duoprisms, the bitruncated 4-simplex, the bitruncated 24-cell, and their duals.
    [Show full text]
  • Designing Modular Sculpture Systems
    Bridges 2017 Conference Proceedings Designing Modular Sculpture Systems Christopher Carlson Wolfram Research, Inc 100 Trade Centre Drive Champaign, IL, 61820, USA E-mail: [email protected] Abstract We are interested in the sculptural possibilities of closed chains of modular units. One such system is embodied in MathMaker, a set of wooden pieces that can be connected end-to-end to create a fascinating variety of forms. MathMaker is based on a cubic honeycomb. We explore the possibilities of similar systems based on octahedral- tetrahedral, rhombic dodecahedral, and truncated octahedral honeycombs. Introduction The MathMaker construction kit consists of wooden parallelepipeds that can be connected end-to-end to make a great variety of sculptural forms [1]. Figure 1 shows on the left an untitled sculpture by Koos Verhoeff based on that system of modular units [2]. MathMaker derives from a cubic honeycomb. Each unit connects the center of one cubic face to the center of an adjacent face (Figure 1, center). Units connect via square planar faces which echo the square faces of cubes in the honeycomb. The square connecting faces can be matched in 4 ways via 90° rotation, so from any position each adjacent cubic face is accessible. A chain of MathMaker units jumps from cube to adjacent cube, adjacent units never traversing the same cube. A variant of MathMaker called “turned cross mitre” has units whose square cross-sections are rotated 45° about their central axes relative to the standard MathMaker units (Figure 1, right). Although the underlying cubic honeycomb structure is the same, the change in unit yields structures with strikingly different visual impressions.
    [Show full text]