Snubs, Alternated Facetings, & Stott-Coxeter-Dynkin Diagrams
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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. -
On the Archimedean Or Semiregular Polyhedra
ON THE ARCHIMEDEAN OR SEMIREGULAR POLYHEDRA Mark B. Villarino Depto. de Matem´atica, Universidad de Costa Rica, 2060 San Jos´e, Costa Rica May 11, 2005 Abstract We prove that there are thirteen Archimedean/semiregular polyhedra by using Euler’s polyhedral formula. Contents 1 Introduction 2 1.1 RegularPolyhedra .............................. 2 1.2 Archimedean/semiregular polyhedra . ..... 2 2 Proof techniques 3 2.1 Euclid’s proof for regular polyhedra . ..... 3 2.2 Euler’s polyhedral formula for regular polyhedra . ......... 4 2.3 ProofsofArchimedes’theorem. .. 4 3 Three lemmas 5 3.1 Lemma1.................................... 5 3.2 Lemma2.................................... 6 3.3 Lemma3.................................... 7 4 Topological Proof of Archimedes’ theorem 8 arXiv:math/0505488v1 [math.GT] 24 May 2005 4.1 Case1: fivefacesmeetatavertex: r=5. .. 8 4.1.1 At least one face is a triangle: p1 =3................ 8 4.1.2 All faces have at least four sides: p1 > 4 .............. 9 4.2 Case2: fourfacesmeetatavertex: r=4 . .. 10 4.2.1 At least one face is a triangle: p1 =3................ 10 4.2.2 All faces have at least four sides: p1 > 4 .............. 11 4.3 Case3: threefacesmeetatavertes: r=3 . ... 11 4.3.1 At least one face is a triangle: p1 =3................ 11 4.3.2 All faces have at least four sides and one exactly four sides: p1 =4 6 p2 6 p3. 12 4.3.3 All faces have at least five sides and one exactly five sides: p1 =5 6 p2 6 p3 13 1 5 Summary of our results 13 6 Final remarks 14 1 Introduction 1.1 Regular Polyhedra A polyhedron may be intuitively conceived as a “solid figure” bounded by plane faces and straight line edges so arranged that every edge joins exactly two (no more, no less) vertices and is a common side of two faces. -
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 . -
Five-Vertex Archimedean Surface Tessellation by Lanthanide-Directed Molecular Self-Assembly
Five-vertex Archimedean surface tessellation by lanthanide-directed molecular self-assembly David Écijaa,1, José I. Urgela, Anthoula C. Papageorgioua, Sushobhan Joshia, Willi Auwärtera, Ari P. Seitsonenb, Svetlana Klyatskayac, Mario Rubenc,d, Sybille Fischera, Saranyan Vijayaraghavana, Joachim Reicherta, and Johannes V. Bartha,1 aPhysik Department E20, Technische Universität München, D-85478 Garching, Germany; bPhysikalisch-Chemisches Institut, Universität Zürich, CH-8057 Zürich, Switzerland; cInstitute of Nanotechnology, Karlsruhe Institute of Technology, D-76344 Eggenstein-Leopoldshafen, Germany; and dInstitut de Physique et Chimie des Matériaux de Strasbourg (IPCMS), CNRS-Université de Strasbourg, F-67034 Strasbourg, France Edited by Kenneth N. Raymond, University of California, Berkeley, CA, and approved March 8, 2013 (received for review December 28, 2012) The tessellation of the Euclidean plane by regular polygons has by five interfering laser beams, conceived to specifically address been contemplated since ancient times and presents intriguing the surface tiling problem, yielded a distorted, 2D Archimedean- aspects embracing mathematics, art, and crystallography. Sig- like architecture (24). nificant efforts were devoted to engineer specific 2D interfacial In the last decade, the tools of supramolecular chemistry on tessellations at the molecular level, but periodic patterns with surfaces have provided new ways to engineer a diversity of sur- distinct five-vertex motifs remained elusive. Here, we report a face-confined molecular architectures, mainly exploiting molecular direct scanning tunneling microscopy investigation on the cerium- recognition of functional organic species or the metal-directed directed assembly of linear polyphenyl molecular linkers with assembly of molecular linkers (5). Self-assembly protocols have terminal carbonitrile groups on a smooth Ag(111) noble-metal sur- been developed to achieve regular surface tessellations, includ- face. -
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. -
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 . -
Symmetric Networks with Geometric Constraints As Models of Visual Illusions
S S symmetry Article Symmetric Networks with Geometric Constraints as Models of Visual Illusions Ian Stewart 1,*,† and Martin Golubitsky 2,† 1 Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK 2 Department of Mathematics, Ohio State University, Columbus, OH 43210, USA; [email protected] * Correspondence: [email protected] † These authors contributed equally to this work. Received: 17 May 2019; Accepted: 13 June 2019; Published: 16 June 2019 Abstract: Multistable illusions occur when the visual system interprets the same image in two different ways. We model illusions using dynamic systems based on Wilson networks, which detect combinations of levels of attributes of the image. In most examples presented here, the network has symmetry, which is vital to the analysis of the dynamics. We assume that the visual system has previously learned that certain combinations are geometrically consistent or inconsistent, and model this knowledge by adding suitable excitatory and inhibitory connections between attribute levels. We first discuss 4-node networks for the Necker cube and the rabbit/duck illusion. The main results analyze a more elaborate model for the Necker cube, a 16-node Wilson network whose nodes represent alternative orientations of specific segments of the image. Symmetric Hopf bifurcation is used to show that a small list of natural local geometric consistency conditions leads to alternation between two global percepts: cubes in two different orientations. The model also predicts brief transitional states in which the percept involves impossible rectangles analogous to the Penrose triangle. A tristable illusion generalizing the Necker cube is modelled in a similar manner. -
Mating of Polyhedrato Evince an Order Ofthe All-Space-Filling Periodic Honeycombs
International Journal of u- and e- Service, Science and Technology Vol.9, No. 6 (2016), pp.179-192 http://dx.doi.org/10.14257/ijunesst.2016.9.6.17 Mating of Polyhedrato Evince an Order ofthe All-Space-Filling Periodic Honeycombs Robert C. Meurant Director Emeritus, Institute of Traditional Studies; Executive Director Education & Research, Harrisco-Enco [email protected] Abstract Earlier discovery and presentation of an inherent order of the regular and semi- regular polyhedra that displays three interrelated classes, together with consideration of the honeycombs, has led me to posit the existence of a coherent and integral metapattern that should relate the various all-space-filling periodical honeycombs, which I advance in an earlier paper that should be read in conjunction with thispaper, as they form part of a series.Here, I approach the periodic polyhedral honeycombs byexploring how pairs of polyhedra regularly combine or mate, whether proximally or distally, along the √1, √2 and √3 axes of their reference cubic and tetrahedral lattices. This is first performed for pairs of what I elsewhere term the Great Enablers (GEs), the positive and negative tetrahedra and truncated tetrahedra; secondly, for pairs of GEs and the Primary Polytopes (PPs); and thirdly, for pairs of PPs. This reveals that these three forms of mating, GE:GE, GE:PP and PP:PP, correlate with the three symmetry groups {2,3,3|2,3,3}, {2,3,3|2,3,4} and {2,3,4|2,3,4}, respectively, of the periodical honeycombs.These matings typically occur in naturallyoccurring pairs along each axis, so in general, a PP mates with just two PPs, though in certain cases one of these is the same as the original. -
Local Symmetry Preserving Operations on Polyhedra
Local Symmetry Preserving Operations on Polyhedra Pieter Goetschalckx Submitted to the Faculty of Sciences of Ghent University in fulfilment of the requirements for the degree of Doctor of Science: Mathematics. Supervisors prof. dr. dr. Kris Coolsaet dr. Nico Van Cleemput Chair prof. dr. Marnix Van Daele Examination Board prof. dr. Tomaž Pisanski prof. dr. Jan De Beule prof. dr. Tom De Medts dr. Carol T. Zamfirescu dr. Jan Goedgebeur © 2020 Pieter Goetschalckx Department of Applied Mathematics, Computer Science and Statistics Faculty of Sciences, Ghent University This work is licensed under a “CC BY 4.0” licence. https://creativecommons.org/licenses/by/4.0/deed.en In memory of John Horton Conway (1937–2020) Contents Acknowledgements 9 Dutch summary 13 Summary 17 List of publications 21 1 A brief history of operations on polyhedra 23 1 Platonic, Archimedean and Catalan solids . 23 2 Conway polyhedron notation . 31 3 The Goldberg-Coxeter construction . 32 3.1 Goldberg ....................... 32 3.2 Buckminster Fuller . 37 3.3 Caspar and Klug ................... 40 3.4 Coxeter ........................ 44 4 Other approaches ....................... 45 References ............................... 46 2 Embedded graphs, tilings and polyhedra 49 1 Combinatorial graphs .................... 49 2 Embedded graphs ....................... 51 3 Symmetry and isomorphisms . 55 4 Tilings .............................. 57 5 Polyhedra ............................ 59 6 Chamber systems ....................... 60 7 Connectivity .......................... 62 References