Counting Faces of Polytopes

Total Page:16

File Type:pdf, Size:1020Kb

Counting Faces of Polytopes Counting Faces of Polytopes Carl Lee University of Kentucky James Madison University|March 2015 Carl Lee (UK) Counting Faces of Polytopes James Madison University 1 / 36 Convex Polytopes A convex polytope P is the convex hull of a finite set of points in Rd . Example: Cube Carl Lee (UK) Counting Faces of Polytopes James Madison University 2 / 36 4-Cube. f = (16; 32; 24; 8). Question: What are the possible face-vectors of polytopes? Face-Vectors The face-vector of a d-dimensional polytope is f = (f0; f1;:::; fd−1), where fj is the number of faces of dimension j. Define also f−1 = fd = 1. Example: Cube. f = (8; 12; 6). Carl Lee (UK) Counting Faces of Polytopes James Madison University 3 / 36 Question: What are the possible face-vectors of polytopes? Face-Vectors The face-vector of a d-dimensional polytope is f = (f0; f1;:::; fd−1), where fj is the number of faces of dimension j. Define also f−1 = fd = 1. Example: Cube. f = (8; 12; 6). 4-Cube. f = (16; 32; 24; 8). Carl Lee (UK) Counting Faces of Polytopes James Madison University 3 / 36 Face-Vectors The face-vector of a d-dimensional polytope is f = (f0; f1;:::; fd−1), where fj is the number of faces of dimension j. Define also f−1 = fd = 1. Example: Cube. f = (8; 12; 6). 4-Cube. f = (16; 32; 24; 8). Question: What are the possible face-vectors of polytopes? Carl Lee (UK) Counting Faces of Polytopes James Madison University 3 / 36 Three-Dimensional Polytopes Theorem (Euler's Relation) f0 − f1 + f2 = 2 for convex 3-polytopes. Example: Cube. 8 − 12 + 6 = 2. Carl Lee (UK) Counting Faces of Polytopes James Madison University 4 / 36 Initially χ = 0. Bottom vertex. χ changes by 1 − 0 + 0 = 1. Intermediate vertex with k incident lower edges. χ changes by 1 − k + (k − 1) = 0. Top vertex. If its degree is k, then χ changes by 1 − k + k = 1. Total change in χ is therefore 2. Note: This proof technique generalizes to higher dimensions. Three-Dimensional Polytopes Sketch of proof: Sweep the polytope with a plane in general direction. (Think of immersing in water.) Count vertices, edges, and polygons only when fully swept (under water). Watch how χ = f0 − f1 + f2 changes when the plane hits each vertex. Carl Lee (UK) Counting Faces of Polytopes James Madison University 5 / 36 Bottom vertex. χ changes by 1 − 0 + 0 = 1. Intermediate vertex with k incident lower edges. χ changes by 1 − k + (k − 1) = 0. Top vertex. If its degree is k, then χ changes by 1 − k + k = 1. Total change in χ is therefore 2. Note: This proof technique generalizes to higher dimensions. Three-Dimensional Polytopes Sketch of proof: Sweep the polytope with a plane in general direction. (Think of immersing in water.) Count vertices, edges, and polygons only when fully swept (under water). Watch how χ = f0 − f1 + f2 changes when the plane hits each vertex. Initially χ = 0. Carl Lee (UK) Counting Faces of Polytopes James Madison University 5 / 36 Intermediate vertex with k incident lower edges. χ changes by 1 − k + (k − 1) = 0. Top vertex. If its degree is k, then χ changes by 1 − k + k = 1. Total change in χ is therefore 2. Note: This proof technique generalizes to higher dimensions. Three-Dimensional Polytopes Sketch of proof: Sweep the polytope with a plane in general direction. (Think of immersing in water.) Count vertices, edges, and polygons only when fully swept (under water). Watch how χ = f0 − f1 + f2 changes when the plane hits each vertex. Initially χ = 0. Bottom vertex. χ changes by 1 − 0 + 0 = 1. Carl Lee (UK) Counting Faces of Polytopes James Madison University 5 / 36 Top vertex. If its degree is k, then χ changes by 1 − k + k = 1. Total change in χ is therefore 2. Note: This proof technique generalizes to higher dimensions. Three-Dimensional Polytopes Sketch of proof: Sweep the polytope with a plane in general direction. (Think of immersing in water.) Count vertices, edges, and polygons only when fully swept (under water). Watch how χ = f0 − f1 + f2 changes when the plane hits each vertex. Initially χ = 0. Bottom vertex. χ changes by 1 − 0 + 0 = 1. Intermediate vertex with k incident lower edges. χ changes by 1 − k + (k − 1) = 0. Carl Lee (UK) Counting Faces of Polytopes James Madison University 5 / 36 Total change in χ is therefore 2. Note: This proof technique generalizes to higher dimensions. Three-Dimensional Polytopes Sketch of proof: Sweep the polytope with a plane in general direction. (Think of immersing in water.) Count vertices, edges, and polygons only when fully swept (under water). Watch how χ = f0 − f1 + f2 changes when the plane hits each vertex. Initially χ = 0. Bottom vertex. χ changes by 1 − 0 + 0 = 1. Intermediate vertex with k incident lower edges. χ changes by 1 − k + (k − 1) = 0. Top vertex. If its degree is k, then χ changes by 1 − k + k = 1. Carl Lee (UK) Counting Faces of Polytopes James Madison University 5 / 36 Note: This proof technique generalizes to higher dimensions. Three-Dimensional Polytopes Sketch of proof: Sweep the polytope with a plane in general direction. (Think of immersing in water.) Count vertices, edges, and polygons only when fully swept (under water). Watch how χ = f0 − f1 + f2 changes when the plane hits each vertex. Initially χ = 0. Bottom vertex. χ changes by 1 − 0 + 0 = 1. Intermediate vertex with k incident lower edges. χ changes by 1 − k + (k − 1) = 0. Top vertex. If its degree is k, then χ changes by 1 − k + k = 1. Total change in χ is therefore 2. Carl Lee (UK) Counting Faces of Polytopes James Madison University 5 / 36 Three-Dimensional Polytopes Sketch of proof: Sweep the polytope with a plane in general direction. (Think of immersing in water.) Count vertices, edges, and polygons only when fully swept (under water). Watch how χ = f0 − f1 + f2 changes when the plane hits each vertex. Initially χ = 0. Bottom vertex. χ changes by 1 − 0 + 0 = 1. Intermediate vertex with k incident lower edges. χ changes by 1 − k + (k − 1) = 0. Top vertex. If its degree is k, then χ changes by 1 − k + k = 1. Total change in χ is therefore 2. Note: This proof technique generalizes to higher dimensions. Carl Lee (UK) Counting Faces of Polytopes James Madison University 5 / 36 f0; f1; f2 are positive integers What else? Theorem (Steinitz) A positive integer vector (f0; f1; f2) is the face-vector of a 3-polytope if and only if the following conditions hold. f0 − f1 + f2 = 2, f0 ≤ 2f2 − 4, and f2 ≤ 2f0 − 4. Three-Dimensional Polytopes Other necessary conditions: Carl Lee (UK) Counting Faces of Polytopes James Madison University 6 / 36 What else? Theorem (Steinitz) A positive integer vector (f0; f1; f2) is the face-vector of a 3-polytope if and only if the following conditions hold. f0 − f1 + f2 = 2, f0 ≤ 2f2 − 4, and f2 ≤ 2f0 − 4. Three-Dimensional Polytopes Other necessary conditions: f0; f1; f2 are positive integers Carl Lee (UK) Counting Faces of Polytopes James Madison University 6 / 36 Theorem (Steinitz) A positive integer vector (f0; f1; f2) is the face-vector of a 3-polytope if and only if the following conditions hold. f0 − f1 + f2 = 2, f0 ≤ 2f2 − 4, and f2 ≤ 2f0 − 4. Three-Dimensional Polytopes Other necessary conditions: f0; f1; f2 are positive integers What else? Carl Lee (UK) Counting Faces of Polytopes James Madison University 6 / 36 Three-Dimensional Polytopes Other necessary conditions: f0; f1; f2 are positive integers What else? Theorem (Steinitz) A positive integer vector (f0; f1; f2) is the face-vector of a 3-polytope if and only if the following conditions hold. f0 − f1 + f2 = 2, f0 ≤ 2f2 − 4, and f2 ≤ 2f0 − 4. Carl Lee (UK) Counting Faces of Polytopes James Madison University 6 / 36 Three-Dimensional Polytopes Carl Lee (UK) Counting Faces of Polytopes James Madison University 7 / 36 We don't know! But there are some partial results. Four-Dimensional Polytopes What is the characterization of face-vectors of 4-polytopes? Carl Lee (UK) Counting Faces of Polytopes James Madison University 8 / 36 But there are some partial results. Four-Dimensional Polytopes What is the characterization of face-vectors of 4-polytopes? We don't know! Carl Lee (UK) Counting Faces of Polytopes James Madison University 8 / 36 Four-Dimensional Polytopes What is the characterization of face-vectors of 4-polytopes? We don't know! But there are some partial results. Carl Lee (UK) Counting Faces of Polytopes James Madison University 8 / 36 Early proofs (pre-Poincar´e)relied upon the implicit or unproven assumption of \shellability," not established until 1970 by Bruggesser and Mani. Gr¨unbaumdeveloped a \sweeping-like" proof. d-Dimensional Polytopes Theorem (Euler-Poincar´eFormula) For every d-polytope, d−1 X d fj = 1 − (−1) j=0 Carl Lee (UK) Counting Faces of Polytopes James Madison University 9 / 36 Gr¨unbaumdeveloped a \sweeping-like" proof. d-Dimensional Polytopes Theorem (Euler-Poincar´eFormula) For every d-polytope, d−1 X d fj = 1 − (−1) j=0 Early proofs (pre-Poincar´e)relied upon the implicit or unproven assumption of \shellability," not established until 1970 by Bruggesser and Mani. Carl Lee (UK) Counting Faces of Polytopes James Madison University 9 / 36 d-Dimensional Polytopes Theorem (Euler-Poincar´eFormula) For every d-polytope, d−1 X d fj = 1 − (−1) j=0 Early proofs (pre-Poincar´e)relied upon the implicit or unproven assumption of \shellability," not established until 1970 by Bruggesser and Mani.
Recommended publications
  • On the Circuit Diameter Conjecture
    On the Circuit Diameter Conjecture Steffen Borgwardt1, Tamon Stephen2, and Timothy Yusun2 1 University of Colorado Denver [email protected] 2 Simon Fraser University {tamon,tyusun}@sfu.ca Abstract. From the point of view of optimization, a critical issue is relating the combina- torial diameter of a polyhedron to its number of facets f and dimension d. In the seminal paper of Klee and Walkup [KW67], the Hirsch conjecture of an upper bound of f − d was shown to be equivalent to several seemingly simpler statements, and was disproved for unbounded polyhedra through the construction of a particular 4-dimensional polyhedron U4 with 8 facets. The Hirsch bound for bounded polyhedra was only recently disproved by Santos [San12]. We consider analogous properties for a variant of the combinatorial diameter called the circuit diameter. In this variant, the walks are built from the circuit directions of the poly- hedron, which are the minimal non-trivial solutions to the system defining the polyhedron. We are able to prove that circuit variants of the so-called non-revisiting conjecture and d-step conjecture both imply the circuit analogue of the Hirsch conjecture. For the equiva- lences in [KW67], the wedge construction was a fundamental proof technique. We exhibit why it is not available in the circuit setting, and what are the implications of losing it as a tool. Further, we show the circuit analogue of the non-revisiting conjecture implies a linear bound on the circuit diameter of all unbounded polyhedra – in contrast to what is known for the combinatorial diameter. Finally, we give two proofs of a circuit version of the 4-step conjecture.
    [Show full text]
  • 7 LATTICE POINTS and LATTICE POLYTOPES Alexander Barvinok
    7 LATTICE POINTS AND LATTICE POLYTOPES Alexander Barvinok INTRODUCTION Lattice polytopes arise naturally in algebraic geometry, analysis, combinatorics, computer science, number theory, optimization, probability and representation the- ory. They possess a rich structure arising from the interaction of algebraic, convex, analytic, and combinatorial properties. In this chapter, we concentrate on the the- ory of lattice polytopes and only sketch their numerous applications. We briefly discuss their role in optimization and polyhedral combinatorics (Section 7.1). In Section 7.2 we discuss the decision problem, the problem of finding whether a given polytope contains a lattice point. In Section 7.3 we address the counting problem, the problem of counting all lattice points in a given polytope. The asymptotic problem (Section 7.4) explores the behavior of the number of lattice points in a varying polytope (for example, if a dilation is applied to the polytope). Finally, in Section 7.5 we discuss problems with quantifiers. These problems are natural generalizations of the decision and counting problems. Whenever appropriate we address algorithmic issues. For general references in the area of computational complexity/algorithms see [AB09]. We summarize the computational complexity status of our problems in Table 7.0.1. TABLE 7.0.1 Computational complexity of basic problems. PROBLEM NAME BOUNDED DIMENSION UNBOUNDED DIMENSION Decision problem polynomial NP-hard Counting problem polynomial #P-hard Asymptotic problem polynomial #P-hard∗ Problems with quantifiers unknown; polynomial for ∀∃ ∗∗ NP-hard ∗ in bounded codimension, reduces polynomially to volume computation ∗∗ with no quantifier alternation, polynomial time 7.1 INTEGRAL POLYTOPES IN POLYHEDRAL COMBINATORICS We describe some combinatorial and computational properties of integral polytopes.
    [Show full text]
  • 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]
  • 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]
  • 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]
  • 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]
  • Separators of Simple Polytopes
    Fachbereich Mathematik und Informatik der Freien Universit¨atBerlin Separators of simple polytopes A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Natural Sciences to the Department of Department of Mathematics and Computer Science by Lauri Loiskekoski Berlin 2017 Supervisor: Prof. Dr. G¨unter M. Ziegler Second Examiner: Prof. Dr. Volker Kaibel Date of defense: 21.12.2017 Acknowledgements First and foremost I would like to thank my supervisor G¨unter M. Ziegler for introducing me to this fascinating topic and always having time to discuss. You always knew the right references and had good ideas what to look at. I would like to thank Hao Chen, Arnau Padrol and Raman Sanyal for help- ing me get started on research and and answering my questions on separators and polytopes. I would like to thank my friends and colleague at the villa, in particular Francesco Grande, Katy Beeler, Albert Haase, Tobias Friedl, Philip Brinkmann, Nevena Pali´c,Jean-Philippe Labb´e,Giulia Codenotti, Jorge Alberto Olarte, Hannah Sch¨aferSj¨oberg and Anna Maria Hartkopf. Thanks for navigating the bureaucracy go to Elke Pose. Thanks to Johanna Steinmeyer for the TikZ pictures. I wish to thank the Berlin Mathematical School for funding my research and European Research Council and Osaka University for funding my travels. Finally, I would like to thank my friends in Finland, especially the people of Rissittely for answering all the unconventional questions I have had about academia and life in general. iii iv Contents Acknowledgements . iii 1 Introduction 1 2 Graphs and Polytopes 3 2.1 Polytopes .
    [Show full text]
  • Three Questions on Graphs of Polytopes
    Three questions on graphs of polytopes Guillermo Pineda-Villavicencio Federation University Australia G. Pineda-Villavicencio (FedUni) Mar 18 1 / 30 Outline 1 A polytope as a combinatorial object 2 First question: Reconstruction of polytopes 3 Second question: Connectivity of cubical polytopes 4 Third question: Linkedness of cubical polytopes G. Pineda-Villavicencio (FedUni) Mar 18 2 / 30 A polytope as a combinatorial object 2 3 4 5 6 7 1 3 5 7 0 1 4 5 2 3 6 7 0 2 4 6 0 1 2 3 6 7 5 7 4 5 1 5 6 7 3 7 4 6 0 4 1 3 0 1 2 6 2 3 0 2 0 1 4 5 5 7 4 1 6 3 0 2 G. Pineda-Villavicencio (FedUni) Mar 18 3 / 30 Reconstruction of polytopes (Dolittle, Nevo, Ugon & Yost) The k-skeleton of a polytope is the set of all its faces of dimension ≤ k. k-skeleton reconstruction: Given the k-skeleton of a polytope, can the face lattice of the polytope be completed? 4 5 6 7 1 3 5 7 0 1 4 5 2 3 6 7 0 2 4 6 0 1 2 3 5 7 4 5 1 5 6 7 3 7 4 6 0 4 1 3 0 1 2 6 2 3 0 2 5 7 4 1 6 3 0 2 G. Pineda-Villavicencio (FedUni) Mar 18 4 / 30 For d ≥ 4 there are pairs of d-polytopes with isomorphic (d − 3)-skeleta: a bipyramid over a (d − 1)-simplex and, a pyramid over the (d − 1)-bipyramid over a (d − 2)-simplex.
    [Show full text]
  • Combinatorial Aspects of Convex Polytopes Margaret M
    Combinatorial Aspects of Convex Polytopes Margaret M. Bayer1 Department of Mathematics University of Kansas Carl W. Lee2 Department of Mathematics University of Kentucky August 1, 1991 Chapter for Handbook on Convex Geometry P. Gruber and J. Wills, Editors 1Supported in part by NSF grant DMS-8801078. 2Supported in part by NSF grant DMS-8802933, by NSA grant MDA904-89-H-2038, and by DIMACS (Center for Discrete Mathematics and Theoretical Computer Science), a National Science Foundation Science and Technology Center, NSF-STC88-09648. 1 Definitions and Fundamental Results 3 1.1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 1.2 Faces : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 1.3 Polarity and Duality : : : : : : : : : : : : : : : : : : : : : : : : : 3 1.4 Overview : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2 Shellings 4 2.1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.2 Euler's Relation : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Line Shellings : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 2.4 Shellable Simplicial Complexes : : : : : : : : : : : : : : : : : : : 5 2.5 The Dehn-Sommerville Equations : : : : : : : : : : : : : : : : : : 6 2.6 Completely Unimodal Numberings and Orientations : : : : : : : 7 2.7 The Upper Bound Theorem : : : : : : : : : : : : : : : : : : : : : 8 2.8 The Lower Bound Theorem : : : : : : : : : : : : : : : : : : : : : 9 2.9 Constructions Using Shellings : : : : : : : : : : : : :
    [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]
  • Gauss-Bonnet for Multi-Linear Valuations
    GAUSS-BONNET FOR MULTI-LINEAR VALUATIONS OLIVER KNILL Abstract. We prove Gauss-Bonnet and Poincar´e-Hopfformulas for multi- linear valuations on finite simple graphs G = (V; E) and answer affirmatively a conjecture of Gr¨unbaum from 1970 by constructing higher order Dehn- Sommerville valuations which vanish for all d-graphs without boundary. A first example of a higher degree valuations was introduced by Wu in 1959. P dim(x) It is the Wu characteristic !(G) = x\y6=; σ(x)σ(y) with σ(x) = (−1) which sums over all ordered intersecting pairs of complete subgraphs of a finite simple graph G. It more generally defines an intersection number !(A; B) = P x\y6=; σ(x)σ(y), where x ⊂ A; y ⊂ B are the simplices in two subgraphs A; B of a given graph. The self intersection number !(G) is a higher order Euler P characteristic. The later is the linear valuation χ(G) = x σ(x) which sums over all complete subgraphs of G. We prove that all these characteristics share the multiplicative property of Euler characteristic: for any pair G; H of finite simple graphs, we have !(G × H) = !(G)!(H) so that all Wu characteristics like Euler characteristic are multiplicative on the Stanley-Reisner ring. The Wu characteristics are invariant under Barycentric refinements and are so com- binatorial invariants in the terminology of Bott. By constructing a curvature P K : V ! R satisfying Gauss-Bonnet !(G) = a K(a), where a runs over all vertices we prove !(G) = χ(G) − χ(δ(G)) which holds for any d-graph G with boundary δG.
    [Show full text]