Designing Modular Sculpture Systems
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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. -
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. -
Petrie Schemes
Canad. J. Math. Vol. 57 (4), 2005 pp. 844–870 Petrie Schemes Gordon Williams Abstract. Petrie polygons, especially as they arise in the study of regular polytopes and Coxeter groups, have been studied by geometers and group theorists since the early part of the twentieth century. An open question is the determination of which polyhedra possess Petrie polygons that are simple closed curves. The current work explores combinatorial structures in abstract polytopes, called Petrie schemes, that generalize the notion of a Petrie polygon. It is established that all of the regular convex polytopes and honeycombs in Euclidean spaces, as well as all of the Grunbaum–Dress¨ polyhedra, pos- sess Petrie schemes that are not self-intersecting and thus have Petrie polygons that are simple closed curves. Partial results are obtained for several other classes of less symmetric polytopes. 1 Introduction Historically, polyhedra have been conceived of either as closed surfaces (usually topo- logical spheres) made up of planar polygons joined edge to edge or as solids enclosed by such a surface. In recent times, mathematicians have considered polyhedra to be convex polytopes, simplicial spheres, or combinatorial structures such as abstract polytopes or incidence complexes. A Petrie polygon of a polyhedron is a sequence of edges of the polyhedron where any two consecutive elements of the sequence have a vertex and face in common, but no three consecutive edges share a commonface. For the regular polyhedra, the Petrie polygons form the equatorial skew polygons. Petrie polygons may be defined analogously for polytopes as well. Petrie polygons have been very useful in the study of polyhedra and polytopes, especially regular polytopes. -
Uniform Panoploid Tetracombs
Uniform Panoploid Tetracombs George Olshevsky TETRACOMB is a four-dimensional tessellation. In any tessellation, the honeycells, which are the n-dimensional polytopes that tessellate the space, Amust by definition adjoin precisely along their facets, that is, their ( n!1)- dimensional elements, so that each facet belongs to exactly two honeycells. In the case of tetracombs, the honeycells are four-dimensional polytopes, or polychora, and their facets are polyhedra. For a tessellation to be uniform, the honeycells must all be uniform polytopes, and the vertices must be transitive on the symmetry group of the tessellation. Loosely speaking, therefore, the vertices must be “surrounded all alike” by the honeycells that meet there. If a tessellation is such that every point of its space not on a boundary between honeycells lies in the interior of exactly one honeycell, then it is panoploid. If one or more points of the space not on a boundary between honeycells lie inside more than one honeycell, the tessellation is polyploid. Tessellations may also be constructed that have “holes,” that is, regions that lie inside none of the honeycells; such tessellations are called holeycombs. It is possible for a polyploid tessellation to also be a holeycomb, but not for a panoploid tessellation, which must fill the entire space exactly once. Polyploid tessellations are also called starcombs or star-tessellations. Holeycombs usually arise when (n!1)-dimensional tessellations are themselves permitted to be honeycells; these take up the otherwise free facets that bound the “holes,” so that all the facets continue to belong to two honeycells. In this essay, as per its title, we are concerned with just the uniform panoploid tetracombs. -
On the Application of the Honeycomb Conjecture to the Bee's Honeycomb
On the Application of the Honeycomb Conjecture to the Bee’s Honeycomb∗y Tim Räzz July 29, 2014 Abstract In a recent paper, Aidan Lyon and Mark Colyvan have proposed an explanation of the structure of the bee’s honeycomb based on the mathematical Honeycomb Conjecture. This explanation has instantly become one of the standard examples in the philosophical debate on mathematical explanations of physical phenomena. In this critical note, I argue that the explanation is not scientifically adequate. The reason for this is that the explanation fails to do justice to the essen- tially three-dimensional structure of the bee’s honeycomb. Contents 1 Introduction 2 2 Lyon’s and Colyvan’s Explanation 2 3 Baker: A Philosophical Motivation 3 4 Why the Explanation Fails 4 4.1 The Explanation is Incomplete . .4 4.2 The Honeycomb Conjecture Is (Probably) Irrelevant . .7 ∗Acknowledgements: I am grateful to Mark Colyvan, Matthias Egg, Michael Esfeld, Martin Gasser, Marion Hämmerli, Michael Messerli, Antoine Muller, Christian Sachse, Tilman Sauer, Raphael Scholl, two anonymous referees, and the participants of the phi- losophy of science research seminar in the fall of 2012 in Lausanne for discussions and comments on previous drafts of this paper. The usual disclaimer applies. This work was supported by the Swiss National Science Foundation, grants (100011-124462/1), (100018- 140201/1) yThis is a pre-copyedited, author-produced PDF of an article accepted for publication in Philosophia Mathematica following peer review. The definitive publisher-authenticated version, Räz, T. (2013): On the Application of the Honeycomb Conjecture to the Bee’s Honeycomb. -
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 . -
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. -
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. -
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. -
Coloring Uniform Honeycombs
Bridges 2009: Mathematics, Music, Art, Architecture, Culture Coloring Uniform Honeycombs Glenn R. Laigo, [email protected] Ma. Louise Antonette N. De las Peñas, [email protected] Mathematics Department, Ateneo de Manila University Loyola Heights, Quezon City, Philippines René P. Felix, [email protected] Institute of Mathematics, University of the Philippines Diliman, Quezon City, Philippines Abstract In this paper, we discuss a method of arriving at colored three-dimensional uniform honeycombs. In particular, we present the construction of perfect and semi-perfect colorings of the truncated and bitruncated cubic honeycombs. If G is the symmetry group of an uncolored honeycomb, a coloring of the honeycomb is perfect if the group H consisting of elements that permute the colors of the given coloring is G. If H is such that [ G : H] = 2, we say that the coloring of the honeycomb is semi-perfect . Background In [7, 9, 12], a general framework has been presented for coloring planar patterns. Focus was given to the construction of perfect colorings of semi-regular tilings on the hyperbolic plane. In this work, we will extend the method of coloring two dimensional patterns to obtain colorings of three dimensional uniform honeycombs. There is limited literature on colorings of three-dimensional honeycombs. We see studies on colorings of polyhedra; for instance, in [17], a method of coloring shown is by cutting the polyhedra and laying it flat to produce a pattern on a two-dimensional plane. In this case, only the faces of the polyhedra are colored. In [6], enumeration problems on colored patterns on polyhedra are discussed and solutions are obtained by applying Burnside's counting theorem. -
Regular Totally Separable Sphere Packings Arxiv:1506.04171V1
Regular Totally Separable Sphere Packings Samuel Reid∗ September 5, 2018 Abstract The topic of totally separable sphere packings is surveyed with a focus on regular constructions, uniform tilings, and contact number problems. An enumeration of all 2 3 4 regular totally separable sphere packings in R , R , and R which are based on convex uniform tessellations, honeycombs, and tetracombs, respectively, is presented, as well d as a construction of a family of regular totally separable sphere packings in R that is not based on a convex uniform d-honeycomb for d ≥ 3. Keywords: sphere packings, hyperplane arrangements, contact numbers, separability. MSC 2010 Subject Classifications: Primary 52B20, Secondary 14H52. 1 Introduction In the 1940s, P. Erd¨osintroduced the notion of a separable set of domains in the plane, which gained the attention of H. Hadwiger in [1]. G.F. T´othand L.F. T´othextended this notion to totally separable domains and proved the densest totally separable arrangement of congruent copies of a domain is given by a lattice packing of the domains generated by the side-vectors of a parallelogram of least area containing a domain [2]. Totally separable domains are also mentioned by G. Kert´eszin [3], where it is proved that a cube of volume V contains a totally separable set of N balls of radius r with V ≥ 8Nr3. Further results and references regarding separability can be found in a manuscript of J. Pach and G. Tardos [4]. arXiv:1506.04171v1 [math.MG] 6 Jun 2015 This manuscript continues the study of separability in the context of regular unit sphere packings, i.e., infinite sets of unit spheres 1 [ d−1 P = xi + S i=1 ∗University of Calgary, Centre for Computational and Discrete Geometry (Department of Math- ematics & Statistics), and Thangadurai Group (Department of Chemistry), Calgary, AB, Canada. -
Polyhedra and Packings from Hyperbolic Honeycombs
Polyhedra and packings from hyperbolic honeycombs Martin Cramer Pedersena,1 and Stephen T. Hydea aDepartment of Applied Mathematics, Research School of Physics and Engineering, Australian National University, Canberra ACT 2601, Australia Edited by Robion C. Kirby, University of California, Berkeley, CA, and approved May 25, 2018 (received for review November 29, 2017) We derive more than 80 embeddings of 2D hyperbolic honey- Disc Packings and Triangular Patterns combs in Euclidean 3 space, forming 3-periodic infinite polyhedra The density of 2D hard disc packing is characterized by the ratio with cubic symmetry. All embeddings are “minimally frustrated,” of the total area of the packed objects to the area of the embed- formed by removing just enough isometries of the (regular, ding space. Thus, the hexagonal “penny packing” of equal discs but unphysical) 2D hyperbolic honeycombs f3, 7g, f3, 8g, f3, 9g, 2 realizes the maximal packing density in the plane, E (24, 25). f3, 10g, and f3, 12g to allow embeddings in Euclidean 3 space. Dense packings of equal discs on the surface of the 2 sphere, Nearly all of these triangulated “simplicial polyhedra” have sym- 2, are more subtle, since the sphere’s finite area means that metrically identical vertices, and most are chiral. The most sym- S optimal solutions depend on the disc radius. The formal defini- metric examples include 10 infinite “deltahedra,” with equilateral tion of packing density for equal discs in the third homogeneous triangular faces, 6 of which were previously unknown and some 2 of which can be described as packings of Platonic deltahedra. We 2D space—the hyperbolic plane, H —is also complicated by the describe also related cubic crystalline packings of equal hyper- nature of that space.