Separators of Simple Polytopes
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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. -
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. -
Archimedean Solids
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln MAT Exam Expository Papers Math in the Middle Institute Partnership 7-2008 Archimedean Solids Anna Anderson University of Nebraska-Lincoln Follow this and additional works at: https://digitalcommons.unl.edu/mathmidexppap Part of the Science and Mathematics Education Commons Anderson, Anna, "Archimedean Solids" (2008). MAT Exam Expository Papers. 4. https://digitalcommons.unl.edu/mathmidexppap/4 This Article is brought to you for free and open access by the Math in the Middle Institute Partnership at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in MAT Exam Expository Papers by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. Archimedean Solids Anna Anderson In partial fulfillment of the requirements for the Master of Arts in Teaching with a Specialization in the Teaching of Middle Level Mathematics in the Department of Mathematics. Jim Lewis, Advisor July 2008 2 Archimedean Solids A polygon is a simple, closed, planar figure with sides formed by joining line segments, where each line segment intersects exactly two others. If all of the sides have the same length and all of the angles are congruent, the polygon is called regular. The sum of the angles of a regular polygon with n sides, where n is 3 or more, is 180° x (n – 2) degrees. If a regular polygon were connected with other regular polygons in three dimensional space, a polyhedron could be created. In geometry, a polyhedron is a three- dimensional solid which consists of a collection of polygons joined at their edges. The word polyhedron is derived from the Greek word poly (many) and the Indo-European term hedron (seat). -
A Fun Approach to Teaching Geometry and Inspiring Creativity
Proceedings of Bridges 2013: Mathematics, Music, Art, Architecture, Culture A Fun Approach to Teaching Geometry and Inspiring Creativity Ioana Browne, Michael Browne, Mircea Draghicescu*, Cristina Draghicescu, Carmen Ionescu ITSPHUN LLC 2023 NW Lovejoy St. Portland, OR 97209, USA [email protected] Abstract ITSPHUNTM geometric shapes can be easily combined to create an infinite number of structures both decorative and functional. We introduce this system of shapes, and discuss its use for teaching mathematics. At the workshop, we will provide a large number of pieces of various shapes, demonstrate their use, and give all participants the opportunity to build their own creations at the intersection of art and mathematics. Introduction Connecting pieces of paper by sliding them together along slits or cuts is a well-known method of building 3D objects ([2], [3], [4]). Based on this idea we developed a system of shapes1 that can be used to build arbitrarily complex structures. Among the geometric objects that can be built are Platonic, Archimedean, and Johnson solids, prisms and antiprisms, many nonconvex polyhedra, several space-filling tessellations, etc. Guided by an user’s imagination and creativity, simple objects can then be connected to form more complex constructs. ITSPHUN pieces made of various materials2 can be used for creative play, for making functional and decorative objects such as jewelry, lamps, furniture, and, in classroom, for teaching mathematical concepts.3 1Patent pending. 2We used many kinds of paper, cardboard, foam, several kinds of plastic, wood and plywood, wood veneer, fabric, and felt. 3In addition to models of geometric objects, ITSPHUN shapes have been used to make birds, animals, flowers, trees, houses, hats, glasses, and much more - see a few examples in the photo gallery at www.itsphun.com. -
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. -
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 . -
Snubs, Alternated Facetings, & Stott-Coxeter-Dynkin Diagrams
Symmetry: Culture and Science Vol. 21, No.4, 329-344, 2010 SNUBS, ALTERNATED FACETINGS, & STOTT-COXETER-DYNKIN DIAGRAMS Dr. Richard Klitzing Non-professional mathematician, (b. Laupheim, BW., Germany, 1966). Address: Kantstrasse 23, 89522 Heidenheim, Germany. E-mail: [email protected] . Fields of interest: Polytopes, geometry, mathematical quasi-crystallography. Publications: (2002) Axial Symmetrical Edge-Facetings of Uniform Polyhedra. In: Symmetry: Cult. & Sci., vol. 13, nos. 3- 4, pp. 241-258. (2001) Convex Segmentochora. In: Symmetry: Cult. & Sci., vol. 11, nos. 1-4, pp. 139-181. (1996) Reskalierungssymmetrien quasiperiodischer Strukturen. [Symmetries of Rescaling for Quasiperiodic Structures, Ph.D. Dissertation, in German], Eberhard-Karls-Universität Tübingen, or Hamburg, Verlag Dr. Kovač, ISBN 978-3-86064-428- 7. – (Former ones: mentioned therein.) Abstract: The snub cube and the snub dodecahedron are well-known polyhedra since the days of Kepler. Since then, the term “snub” was applied to further cases, both in 3D and beyond, yielding an exceptional species of polytopes: those do not bow to Wythoff’s kaleidoscopical construction like most other Archimedean polytopes, some appear in enantiomorphic pairs with no own mirror symmetry, etc. However, they remained stepchildren, since they permit no real one-step access. Actually, snub polytopes are meant to be derived secondarily from Wythoffians, either from omnitruncated or from truncated polytopes. – This secondary process herein is analysed carefully and extended widely: nothing bars a progress from vertex alternation to an alternation of an arbitrary class of sub-dimensional elements, for instance edges, faces, etc. of any type. Furthermore, this extension can be coded in Stott-Coxeter-Dynkin diagrams as well. -
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. -
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. -
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 : : : : : : : : : : : : : -
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. -
Numerically Stable Form Factor of Any Polygon and Polyhedron
research papers Numerically stable form factor of any polygon and polyhedron ISSN 1600-5767 Joachim Wuttke* Forschungszentrum Ju¨lich GmbH, Ju¨lich Centre for Neutron Science (JCNS) at Heinz Maier-Leibnitz Zentrum (MLZ), Lichtenbergstrasse 1, 85748 Garching, Germany. *Correspondence e-mail: [email protected] Received 15 January 2021 Accepted 11 February 2021 Coordinate-free expressions for the form factors of arbitrary polygons and polyhedra are derived using the divergence theorem and Stokes’s theorem. Apparent singularities, all removable, are discussed in detail. Cancellation near Edited by D. I. Svergun, European Molecular the singularities causes a loss of precision that can be avoided by using series Biology Laboratory, Hamburg, Germany expansions. An important application domain is small-angle scattering by nanocrystals. Keywords: form factors; polyhedra; Fourier shape transform. 1. Introduction 1.1. Overview The term ‘form factor’ has different meanings in science and in engineering. Here, we are concerned with the form factor of a geometric figure as defined in the physical sciences, namely the Fourier transform of the figure’s indicator function, also called the ‘shape transform’ of the figure. This form factor has important applications in the emission, detection and scattering of radiation. Two-dimensional shape transforms are used in the theory of reflector antennas (Lee & Mittra, 1983). The three-dimensional form factors of the sphere and the cylinder go back to Lord Rayleigh (1881). Shapes of three-dimensional nanoparticles are investigated by neutron and X-ray small-angle scattering (Hammouda, 2010). Particles grown on a substrate (Henry, 2005) develop many different shapes, especially polyhedral ones, as observed by grazing-incidence neutron and X-ray small-angle scattering (GISAS, GISANS, GISAXS) (Renaud et al., 2009).