Regular Polytopes
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Geometry of Generalized Permutohedra
Geometry of Generalized Permutohedra by Jeffrey Samuel Doker A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, Berkeley Committee in charge: Federico Ardila, Co-chair Lior Pachter, Co-chair Matthias Beck Bernd Sturmfels Lauren Williams Satish Rao Fall 2011 Geometry of Generalized Permutohedra Copyright 2011 by Jeffrey Samuel Doker 1 Abstract Geometry of Generalized Permutohedra by Jeffrey Samuel Doker Doctor of Philosophy in Mathematics University of California, Berkeley Federico Ardila and Lior Pachter, Co-chairs We study generalized permutohedra and some of the geometric properties they exhibit. We decompose matroid polytopes (and several related polytopes) into signed Minkowski sums of simplices and compute their volumes. We define the associahedron and multiplihe- dron in terms of trees and show them to be generalized permutohedra. We also generalize the multiplihedron to a broader class of generalized permutohedra, and describe their face lattices, vertices, and volumes. A family of interesting polynomials that we call composition polynomials arises from the study of multiplihedra, and we analyze several of their surprising properties. Finally, we look at generalized permutohedra of different root systems and study the Minkowski sums of faces of the crosspolytope. i To Joe and Sue ii Contents List of Figures iii 1 Introduction 1 2 Matroid polytopes and their volumes 3 2.1 Introduction . .3 2.2 Matroid polytopes are generalized permutohedra . .4 2.3 The volume of a matroid polytope . .8 2.4 Independent set polytopes . 11 2.5 Truncation flag matroids . 14 3 Geometry and generalizations of multiplihedra 18 3.1 Introduction . -
Two Poset Polytopes
Discrete Comput Geom 1:9-23 (1986) G eometrv)i.~.reh, ~ ( :*mllmlati~ml © l~fi $1~ter-Vtrlq New Yorklu¢. t¢ Two Poset Polytopes Richard P. Stanley* Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139 Abstract. Two convex polytopes, called the order polytope d)(P) and chain polytope <~(P), are associated with a finite poset P. There is a close interplay between the combinatorial structure of P and the geometric structure of E~(P). For instance, the order polynomial fl(P, m) of P and Ehrhart poly- nomial i(~9(P),m) of O(P) are related by f~(P,m+l)=i(d)(P),m). A "transfer map" then allows us to transfer properties of O(P) to W(P). In particular, we transfer known inequalities involving linear extensions of P to some new inequalities. I. The Order Polytope Our aim is to investigate two convex polytopes associated with a finite partially ordered set (poset) P. The first of these, which we call the "order polytope" and denote by O(P), has been the subject of considerable scrutiny, both explicit and implicit, Much of what we say about the order polytope will be essentially a review of well-known results, albeit ones scattered throughout the literature, sometimes in a rather obscure form. The second polytope, called the "chain polytope" and denoted if(P), seems never to have been previously considered per se. It is a special case of the vertex-packing polytope of a graph (see Section 2) but has many special properties not in general valid or meaningful for graphs. -
Abstract Regular Polytopes
ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS Abstract Regular Polytopes PETER MMULLEN University College London EGON SCHULTE Northeastern University PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street, Cambridge, United Kingdom CAMBRIDGE UNIVERSITY PRESS The Edinburgh Building, Cambridge CB2 2RU, UK 40 West 20th Street, New York, NY 10011-4211, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia Ruiz de Alarc´on 13, 28014 Madrid, Spain Dock House, The Waterfront, Cape Town 8001, South Africa http://www.cambridge.org c Peter McMullen and Egon Schulte 2002 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2002 Printed in the United States of America Typeface Times New Roman 10/12.5 pt. System LATEX2ε [] A catalog record for this book is available from the British Library. Library of Congress Cataloging in Publication Data McMullen, Peter, 1955– Abstract regular polytopes / Peter McMullen, Egon Schulte. p. cm. – (Encyclopedia of mathematics and its applications) Includes bibliographical references and index. ISBN 0-521-81496-0 1. Polytopes. I. Schulte, Egon, 1955– II. Title. III. Series. QA691 .M395 2002 516.35 – dc21 2002017391 ISBN 0 521 81496 0 hardback Contents Preface page xiii 1 Classical Regular Polytopes 1 1A The Historical Background 1 1B Regular Convex Polytopes 7 1C Extensions -
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. -
Uniform Polychora
BRIDGES Mathematical Connections in Art, Music, and Science Uniform Polychora Jonathan Bowers 11448 Lori Ln Tyler, TX 75709 E-mail: [email protected] Abstract Like polyhedra, polychora are beautiful aesthetic structures - with one difference - polychora are four dimensional. Although they are beyond human comprehension to visualize, one can look at various projections or cross sections which are three dimensional and usually very intricate, these make outstanding pieces of art both in model form or in computer graphics. Polygons and polyhedra have been known since ancient times, but little study has gone into the next dimension - until recently. Definitions A polychoron is basically a four dimensional "polyhedron" in the same since that a polyhedron is a three dimensional "polygon". To be more precise - a polychoron is a 4-dimensional "solid" bounded by cells with the following criteria: 1) each cell is adjacent to only one other cell for each face, 2) no subset of cells fits criteria 1, 3) no two adjacent cells are corealmic. If criteria 1 fails, then the figure is degenerate. The word "polychoron" was invented by George Olshevsky with the following construction: poly = many and choron = rooms or cells. A polytope (polyhedron, polychoron, etc.) is uniform if it is vertex transitive and it's facets are uniform (a uniform polygon is a regular polygon). Degenerate figures can also be uniform under the same conditions. A vertex figure is the figure representing the shape and "solid" angle of the vertices, ex: the vertex figure of a cube is a triangle with edge length of the square root of 2. -
Arxiv:Math/9906062V1 [Math.MG] 10 Jun 1999 Udo Udmna Eerh(Rn 96-01-00166)
Embedding the graphs of regular tilings and star-honeycombs into the graphs of hypercubes and cubic lattices ∗ Michel DEZA CNRS and Ecole Normale Sup´erieure, Paris, France Mikhail SHTOGRIN Steklov Mathematical Institute, 117966 Moscow GSP-1, Russia Abstract We review the regular tilings of d-sphere, Euclidean d-space, hyperbolic d-space and Coxeter’s regular hyperbolic honeycombs (with infinite or star-shaped cells or vertex figures) with respect of possible embedding, isometric up to a scale, of their skeletons into a m-cube or m-dimensional cubic lattice. In section 2 the last remaining 2-dimensional case is decided: for any odd m ≥ 7, star-honeycombs m m {m, 2 } are embeddable while { 2 ,m} are not (unique case of non-embedding for dimension 2). As a spherical analogue of those honeycombs, we enumerate, in section 3, 36 Riemann surfaces representing all nine regular polyhedra on the sphere. In section 4, non-embeddability of all remaining star-honeycombs (on 3-sphere and hyperbolic 4-space) is proved. In the last section 5, all cases of embedding for dimension d> 2 are identified. Besides hyper-simplices and hyper-octahedra, they are exactly those with bipartite skeleton: hyper-cubes, cubic lattices and 8, 2, 1 tilings of hyperbolic 3-, 4-, 5-space (only two, {4, 3, 5} and {4, 3, 3, 5}, of those 11 have compact both, facets and vertex figures). 1 Introduction arXiv:math/9906062v1 [math.MG] 10 Jun 1999 We say that given tiling (or honeycomb) T has a l1-graph and embeds up to scale λ into m-cube Hm (or, if the graph is infinite, into cubic lattice Zm ), if there exists a mapping f of the vertex-set of the skeleton graph of T into the vertex-set of Hm (or Zm) such that λdT (vi, vj)= ||f(vi), f(vj)||l1 = X |fk(vi) − fk(vj)| for all vertices vi, vj, 1≤k≤m ∗This work was supported by the Volkswagen-Stiftung (RiP-program at Oberwolfach) and Russian fund of fundamental research (grant 96-01-00166). -
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. -
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. -
Arxiv:1705.01294V1
Branes and Polytopes Luca Romano email address: [email protected] ABSTRACT We investigate the hierarchies of half-supersymmetric branes in maximal supergravity theories. By studying the action of the Weyl group of the U-duality group of maximal supergravities we discover a set of universal algebraic rules describing the number of independent 1/2-BPS p-branes, rank by rank, in any dimension. We show that these relations describe the symmetries of certain families of uniform polytopes. This induces a correspondence between half-supersymmetric branes and vertices of opportune uniform polytopes. We show that half-supersymmetric 0-, 1- and 2-branes are in correspondence with the vertices of the k21, 2k1 and 1k2 families of uniform polytopes, respectively, while 3-branes correspond to the vertices of the rectified version of the 2k1 family. For 4-branes and higher rank solutions we find a general behavior. The interpretation of half- supersymmetric solutions as vertices of uniform polytopes reveals some intriguing aspects. One of the most relevant is a triality relation between 0-, 1- and 2-branes. arXiv:1705.01294v1 [hep-th] 3 May 2017 Contents Introduction 2 1 Coxeter Group and Weyl Group 3 1.1 WeylGroup........................................ 6 2 Branes in E11 7 3 Algebraic Structures Behind Half-Supersymmetric Branes 12 4 Branes ad Polytopes 15 Conclusions 27 A Polytopes 30 B Petrie Polygons 30 1 Introduction Since their discovery branes gained a prominent role in the analysis of M-theories and du- alities [1]. One of the most important class of branes consists in Dirichlet branes, or D-branes. D-branes appear in string theory as boundary terms for open strings with mixed Dirichlet-Neumann boundary conditions and, due to their tension, scaling with a negative power of the string cou- pling constant, they are non-perturbative objects [2]. -
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. -
On Regular Polytope Numbers
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 131, Number 1, Pages 65{75 S 0002-9939(02)06710-2 Article electronically published on June 12, 2002 ON REGULAR POLYTOPE NUMBERS HYUN KWANG KIM (Communicated by David E. Rohrlich) Abstract. Lagrange proved a theorem which states that every nonnegative integer can be written as a sum of four squares. This result can be generalized as the polygonal number theorem and the Hilbert-Waring problem. In this paper, we shall generalize Lagrange's sum of four squares theorem further. To each regular polytope V in a Euclidean space, we will associate a sequence of nonnegative integers which we shall call regular polytope numbers, and consider the problem of finding the order g(V )ofthesetofregularpolytope numbers associated to V . In 1770, Lagrange proved a theorem which states that every nonnegative integer can be written as a sum of four squares. There are two major generalizations of this beautiful result. One is a horizontal generalization due to Cauchy which is known as the polygonal number theorem, and the other is a higher dimensional generalization which is known as the Hilbert-Waring problem. A nonempty subset A of nonnegative integers is called a basis of order g if g is the minimum number with the property that every nonnegative integer can be written as a sum of g elements in A. Lagrange's sum of four squares can be restated as: the set n2 n =0; 1; 2;::: of nonnegative squares forms a basis of order 4. The polygonalf numbersj are sequencesg of nonnegative integers constructed geometrically from regular polygons.