Regular Polyhedra Through Time
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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 . -
The Geometry of Nim Arxiv:1109.6712V1 [Math.CO] 30
The Geometry of Nim Kevin Gibbons Abstract We relate the Sierpinski triangle and the game of Nim. We begin by defining both a new high-dimensional analog of the Sierpinski triangle and a natural geometric interpretation of the losing positions in Nim, and then, in a new result, show that these are equivalent in each finite dimension. 0 Introduction The Sierpinski triangle (fig. 1) is one of the most recognizable figures in mathematics, and with good reason. It appears in everything from Pascal's Triangle to Conway's Game of Life. In fact, it has already been seen to be connected with the game of Nim, albeit in a very different manner than the one presented here [3]. A number of analogs have been discovered, such as the Menger sponge (fig. 2) and a three-dimensional version called a tetrix. We present, in the first section, a generalization in higher dimensions differing from the more typical simplex generalization. Rather, we define a discrete Sierpinski demihypercube, which in three dimensions coincides with the simplex generalization. In the second section, we briefly review Nim and the theory behind optimal play. As in all impartial games (games in which the possible moves depend only on the state of the game, and not on which of the two players is moving), all possible positions can be divided into two classes - those in which the next player to move can force a win, called N-positions, and those in which regardless of what the next player does the other player can force a win, called P -positions. -
Notes Has Not Been Formally Reviewed by the Lecturer
6.896 Topics in Algorithmic Game Theory February 22, 2010 Lecture 6 Lecturer: Constantinos Daskalakis Scribe: Jason Biddle, Debmalya Panigrahi NOTE: The content of these notes has not been formally reviewed by the lecturer. It is recommended that they are read critically. In our last lecture, we proved Nash's Theorem using Broweur's Fixed Point Theorem. We also showed Brouwer's Theorem via Sperner's Lemma in 2-d using a limiting argument to go from the discrete combinatorial problem to the topological one. In this lecture, we present a multidimensional generalization of the proof from last time. Our proof differs from those typically found in the literature. In particular, we will insist that each step of the proof be constructive. Using constructive arguments, we shall be able to pin down the complexity-theoretic nature of the proof and make the steps algorithmic in subsequent lectures. In the first part of our lecture, we present a framework for the multidimensional generalization of Sperner's Lemma. A Canonical Triangulation of the Hypercube • A Legal Coloring Rule • In the second part, we formally state Sperner's Lemma in n dimensions and prove it using the following constructive steps: Colored Envelope Construction • Definition of the Walk • Identification of the Starting Simplex • Direction of the Walk • 1 Framework Recall that in the 2-dimensional case we had a square which was divided into triangles. We had also defined a legal coloring scheme for the triangle vertices lying on the boundary of the square. Now let us extend those concepts to higher dimensions. 1.1 Canonical Triangulation of the Hypercube We begin by introducing the n-simplex as the n-dimensional analog of the triangle in 2 dimensions as shown in Figure 1. -
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 -
Contractions of Polygons in Abstract Polytopes
Contractions of Polygons in Abstract Polytopes by Ilya Scheidwasser B.S. in Computer Science, University of Massachusetts Amherst M.S. in Mathematics, Northeastern 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 March 31, 2015 Dissertation directed by Egon Schulte Professor of Mathematics Acknowledgements First, I would like to thank my advisor, Professor Egon Schulte. From the first class I took with him in the first semester of my Masters program, Professor Schulte was an engaging, clear, and kind lecturer, deepening my appreciation for mathematics and always more than happy to provide feedback and answer extra questions about the material. Every class with him was a sincere pleasure, and his classes helped lead me to the study of abstract polytopes. As my advisor, Professor Schulte provided me with invaluable assistance in the creation of this thesis, as well as career advice. For all the time and effort Professor Schulte has put in to my endeavors, I am greatly appreciative. I would also like to thank my dissertation committee for taking time out of their sched- ules to provide me with feedback on this thesis. In addition, I would like to thank the various instructors I've had at Northeastern over the years for nurturing my knowledge of and interest in mathematics. I would like to thank my peers and classmates at Northeastern for their company and their assistance with my studies, and the math department's Teaching Committee for the privilege of lecturing classes these past several years. -
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. -
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). -
COXETER GROUPS (Unfinished and Comments Are Welcome)
COXETER GROUPS (Unfinished and comments are welcome) Gert Heckman Radboud University Nijmegen [email protected] October 10, 2018 1 2 Contents Preface 4 1 Regular Polytopes 7 1.1 ConvexSets............................ 7 1.2 Examples of Regular Polytopes . 12 1.3 Classification of Regular Polytopes . 16 2 Finite Reflection Groups 21 2.1 NormalizedRootSystems . 21 2.2 The Dihedral Normalized Root System . 24 2.3 TheBasisofSimpleRoots. 25 2.4 The Classification of Elliptic Coxeter Diagrams . 27 2.5 TheCoxeterElement. 35 2.6 A Dihedral Subgroup of W ................... 39 2.7 IntegralRootSystems . 42 2.8 The Poincar´eDodecahedral Space . 46 3 Invariant Theory for Reflection Groups 53 3.1 Polynomial Invariant Theory . 53 3.2 TheChevalleyTheorem . 56 3.3 Exponential Invariant Theory . 60 4 Coxeter Groups 65 4.1 Generators and Relations . 65 4.2 TheTitsTheorem ........................ 69 4.3 The Dual Geometric Representation . 74 4.4 The Classification of Some Coxeter Diagrams . 77 4.5 AffineReflectionGroups. 86 4.6 Crystallography. .. .. .. .. .. .. .. 92 5 Hyperbolic Reflection Groups 97 5.1 HyperbolicSpace......................... 97 5.2 Hyperbolic Coxeter Groups . 100 5.3 Examples of Hyperbolic Coxeter Diagrams . 108 5.4 Hyperbolic reflection groups . 114 5.5 Lorentzian Lattices . 116 3 6 The Leech Lattice 125 6.1 ModularForms ..........................125 6.2 ATheoremofVenkov . 129 6.3 The Classification of Niemeier Lattices . 132 6.4 The Existence of the Leech Lattice . 133 6.5 ATheoremofConway . 135 6.6 TheCoveringRadiusofΛ . 137 6.7 Uniqueness of the Leech Lattice . 140 4 Preface Finite reflection groups are a central subject in mathematics with a long and rich history. The group of symmetries of a regular m-gon in the plane, that is the convex hull in the complex plane of the mth roots of unity, is the dihedral group of order 2m, which is the simplest example of a reflection Dm group. -
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. -
Package 'Hypercube'
Package ‘hypercube’ February 28, 2020 Type Package Title Organizing Data in Hypercubes Version 0.2.1 Author Michael Scholz Maintainer Michael Scholz <[email protected]> Description Provides functions and methods for organizing data in hypercubes (i.e., a multi-dimensional cube). Cubes are generated from molten data frames. Each cube can be manipulated with five operations: rotation (change.dimensionOrder()), dicing and slicing (add.selection(), remove.selection()), drilling down (add.aggregation()), and rolling up (remove.aggregation()). License GPL-3 Encoding UTF-8 Depends R (>= 3.3.0), stats, plotly Imports methods, stringr, dplyr LazyData TRUE RoxygenNote 7.0.2 NeedsCompilation no Repository CRAN Date/Publication 2020-02-28 07:10:08 UTC R topics documented: hypercube-package . .2 add.aggregation . .3 add.selection . .4 as.data.frame.Cube . .5 change.dimensionOrder . .6 Cube-class . .7 Dimension-class . .7 generateCube . .8 importance . .9 1 2 hypercube-package plot,Cube-method . 10 print.Importances . 11 remove.aggregation . 12 remove.selection . 13 sales . 14 show,Cube-method . 14 show,Dimension-method . 15 sparsity . 16 summary . 17 Index 18 hypercube-package Provides functions and methods for organizing data in hypercubes Description This package provides methods for organizing data in a hypercube Each cube can be manipu- lated with five operations rotation (changeDimensionOrder), dicing and slicing (add.selection, re- move.selection), drilling down (add.aggregation), and rolling up (remove.aggregation). Details Package: hypercube -
Hamiltonicity and Combinatorial Polyhedra
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF COMBINATORIAL THEORY, Series B 31, 297-312 (1981) Hamiltonicity and Combinatorial Polyhedra D. NADDEF Laboratoire d’lnformatique et de Mathkmatiques AppliquPes de Grenoble. Grenoble, France AND W. R. PULLEYBLANK* Department of Computer Science, The University of Calgary, Calgary, Alberta T2N lN4, Canada Communicated by the Editors Received April 8, 1980 We say that a polyhedron with O-1 valued vertices is combinatorial if the midpoint of the line joining any pair of nonadjacent vertices is the midpoint of the line joining another pair of vertices. We show that the class of combinatorial polyhedra includes such well-known classes of polyhedra as matching polyhedra, matroid basis polyhedra, node packing or stable set polyhedra and permutation polyhedra. We show the graph of a combinatorial polyhedron is always either a hypercube (i.e., isomorphic to the convex hull of a k-dimension unit cube) or else is hamilton connected (every pair of nodes is the set of terminal nodes of a hamilton path). This imfilies several earlier results concerning special cases of combinatorial polyhedra. 1. INTRODUCTION The graph G(P) of a polyhedron P is the graph whose nodes are the vertices of the polyhedron and which has an edge joining each pair of nodes for which the corresponding vertices of the polyhedron are adjacent, that is, joined by an edge of the polyhedron. Such graphs have been studied since the beginnings of graph theory; in 1857 Sir William Hamilton introduced his “tour of the world” game which consisted of constructing a closed tour passing exactly once through each vertex of the dodecahedron. -
A Tourist Guide to the RCSR
A tourist guide to the RCSR Some of the sights, curiosities, and little-visited by-ways Michael O'Keeffe, Arizona State University RCSR is a Reticular Chemistry Structure Resource available at http://rcsr.net. It is open every day of the year, 24 hours a day, and admission is free. It consists of data for polyhedra and 2-periodic and 3-periodic structures (nets). Visitors unfamiliar with the resource are urged to read the "about" link first. This guide assumes you have. The guide is designed to draw attention to some of the attractions therein. If they sound particularly attractive please visit them. It can be a nice way to spend a rainy Sunday afternoon. OKH refers to M. O'Keeffe & B. G. Hyde. Crystal Structures I: Patterns and Symmetry. Mineral. Soc. Am. 1966. This is out of print but due as a Dover reprint 2019. POLYHEDRA Read the "about" for hints on how to use the polyhedron data to make accurate drawings of polyhedra using crystal drawing programs such as CrystalMaker (see "links" for that program). Note that they are Cartesian coordinates for (roughly) equal edge. To make the drawing with unit edge set the unit cell edges to all 10 and divide the coordinates given by 10. There seems to be no generally-agreed best embedding for complex polyhedra. It is generally not possible to have equal edge, vertices on a sphere and planar faces. Keywords used in the search include: Simple. Each vertex is trivalent (three edges meet at each vertex) Simplicial. Each face is a triangle.