Christopher Glass the Pythagopod
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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. -
Systematic Narcissism
Systematic Narcissism Wendy W. Fok University of Houston Systematic Narcissism is a hybridization through the notion of the plane- tary grid system and the symmetry of narcissism. The installation is based on the obsession of the physical reflection of the object/subject relation- ship, created by the idea of man-made versus planetary projected planes (grids) which humans have made onto the earth. ie: Pierre Charles L’Enfant plan 1791 of the golden section projected onto Washington DC. The fostered form is found based on a triacontahedron primitive and devel- oped according to the primitive angles. Every angle has an increment of 0.25m and is based on a system of derivatives. As indicated, the geometry of the world (or the globe) is based off the planetary grid system, which, according to Bethe Hagens and William S. Becker is a hexakis icosahedron grid system. The formal mathematical aspects of the installation are cre- ated by the offset of that geometric form. The base of the planetary grid and the narcissus reflection of the modular pieces intersect the geometry of the interfacing modular that structures the installation BACKGROUND: Poetics: Narcissism is a fixation with oneself. The Greek myth of Narcissus depicted a prince who sat by a reflective pond for days, self-absorbed by his own beauty. Mathematics: In 1978, Professors William Becker and Bethe Hagens extended the Russian model, inspired by Buckminster Fuller’s geodesic dome, into a grid based on the rhombic triacontahedron, the dual of the icosidodeca- heron Archimedean solid. The triacontahedron has 30 diamond-shaped faces, and possesses the combined vertices of the icosahedron and the dodecahedron. -
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). -
Shape Skeletons Creating Polyhedra with Straws
Shape Skeletons Creating Polyhedra with Straws Topics: 3-Dimensional Shapes, Regular Solids, Geometry Materials List Drinking straws or stir straws, cut in Use simple materials to investigate regular or advanced 3-dimensional shapes. half Fun to create, these shapes make wonderful showpieces and learning tools! Paperclips to use with the drinking Assembly straws or chenille 1. Choose which shape to construct. Note: the 4-sided tetrahedron, 8-sided stems to use with octahedron, and 20-sided icosahedron have triangular faces and will form sturdier the stir straws skeletal shapes. The 6-sided cube with square faces and the 12-sided Scissors dodecahedron with pentagonal faces will be less sturdy. See the Taking it Appropriate tool for Further section. cutting the wire in the chenille stems, Platonic Solids if used This activity can be used to teach: Common Core Math Tetrahedron Cube Octahedron Dodecahedron Icosahedron Standards: Angles and volume Polyhedron Faces Shape of Face Edges Vertices and measurement Tetrahedron 4 Triangles 6 4 (Measurement & Cube 6 Squares 12 8 Data, Grade 4, 5, 6, & Octahedron 8 Triangles 12 6 7; Grade 5, 3, 4, & 5) Dodecahedron 12 Pentagons 30 20 2-Dimensional and 3- Dimensional Shapes Icosahedron 20 Triangles 30 12 (Geometry, Grades 2- 12) 2. Use the table and images above to construct the selected shape by creating one or Problem Solving and more face shapes and then add straws or join shapes at each of the vertices: Reasoning a. For drinking straws and paperclips: Bend the (Mathematical paperclips so that the 2 loops form a “V” or “L” Practices Grades 2- shape as needed, widen the narrower loop and insert 12) one loop into the end of one straw half, and the other loop into another straw half. -
Rhombic Triacontahedron
Rhombic Triacontahedron Figure 1 Rhombic Triacontahedron. Vertex labels as used for the corresponding vertices of the 120 Polyhedron. COPYRIGHT 2007, Robert W. Gray Page 1 of 11 Encyclopedia Polyhedra: Last Revision: August 5, 2007 RHOMBIC TRIACONTAHEDRON Figure 2 Icosahedron (red) and Dodecahedron (yellow) define the rhombic Triacontahedron (green). Figure 3 “Long” (red) and “short” (yellow) face diagonals and vertices. COPYRIGHT 2007, Robert W. Gray Page 2 of 11 Encyclopedia Polyhedra: Last Revision: August 5, 2007 RHOMBIC TRIACONTAHEDRON Topology: Vertices = 32 Edges = 60 Faces = 30 diamonds Lengths: 15+ ϕ = 2 EL ≡ Edge length of rhombic Triacontahedron. ϕ + 2 EL = FDL ≅ 0.587 785 252 FDL 2ϕ 3 − ϕ = DVF ϕ 2 FDL ≡ Long face diagonal = DVF ≅ 1.236 067 977 DVF ϕ 2 = EL ≅ 1.701 301 617 EL 3 − ϕ 1 FDS ≡ Short face diagonal = FDL ≅ 0.618 033 989 FDL ϕ 2 = DVF ≅ 0.763 932 023 DVF ϕ 2 2 = EL ≅ 1.051 462 224 EL ϕ + 2 COPYRIGHT 2007, Robert W. Gray Page 3 of 11 Encyclopedia Polyhedra: Last Revision: August 5, 2007 RHOMBIC TRIACONTAHEDRON DFVL ≡ Center of face to vertex at the end of a long face diagonal 1 = FDL 2 1 = DVF ≅ 0.618 033 989 DVF ϕ 1 = EL ≅ 0.850 650 808 EL 3 − ϕ DFVS ≡ Center of face to vertex at the end of a short face diagonal 1 = FDL ≅ 0.309 016 994 FDL 2 ϕ 1 = DVF ≅ 0.381 966 011 DVF ϕ 2 1 = EL ≅ 0.525 731 112 EL ϕ + 2 ϕ + 2 DFE = FDL ≅ 0.293 892 626 FDL 4 ϕ ϕ + 2 = DVF ≅ 0.363 271 264 DVF 21()ϕ + 1 = EL 2 ϕ 3 − ϕ DVVL = FDL ≅ 0.951 056 516 FDL 2 = 3 − ϕ DVF ≅ 1.175 570 505 DVF = ϕ EL ≅ 1.618 033 988 EL COPYRIGHT 2007, Robert W. -
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. -
The Truncated Icosahedron As an Inflatable Ball
https://doi.org/10.3311/PPar.12375 Creative Commons Attribution b |99 Periodica Polytechnica Architecture, 49(2), pp. 99–108, 2018 The Truncated Icosahedron as an Inflatable Ball Tibor Tarnai1, András Lengyel1* 1 Department of Structural Mechanics, Faculty of Civil Engineering, Budapest University of Technology and Economics, H-1521 Budapest, P.O.B. 91, Hungary * Corresponding author, e-mail: [email protected] Received: 09 April 2018, Accepted: 20 June 2018, Published online: 29 October 2018 Abstract In the late 1930s, an inflatable truncated icosahedral beach-ball was made such that its hexagonal faces were coloured with five different colours. This ball was an unnoticed invention. It appeared more than twenty years earlier than the first truncated icosahedral soccer ball. In connection with the colouring of this beach-ball, the present paper investigates the following problem: How many colourings of the dodecahedron with five colours exist such that all vertices of each face are coloured differently? The paper shows that four ways of colouring exist and refers to other colouring problems, pointing out a defect in the colouring of the original beach-ball. Keywords polyhedron, truncated icosahedron, compound of five tetrahedra, colouring of polyhedra, permutation, inflatable ball 1 Introduction Spherical forms play an important role in different fields – not even among the relics of the Romans who inherited of science and technology, and in different areas of every- many ball games from the Greeks. day life. For example, spherical domes are quite com- The Romans mainly used balls composed of equal mon in architecture, and spherical balls are used in most digonal panels, forming a regular hosohedron (Coxeter, 1973, ball games. -
Polyhedral Harmonics
value is uncertain; 4 Gr a y d o n has suggested values of £estrap + A0 for the first three excited 4,4 + 0,1 v. e. states indicate 11,1 v.e. för D0. The value 6,34 v.e. for Dextrap for SO leads on A rough correlation between Dextrap and bond correction by 0,37 + 0,66 for the'valence states of type is evident for the more stable states of the the two atoms to D0 ^ 5,31 v.e., in approximate agreement with the precisely known value 5,184 v.e. diatomic molecules. Thus the bonds A = A and The valence state for nitrogen, at 27/100 F2 A = A between elements of the first short period (with 2D at 9/25 F2 and 2P at 3/5 F2), is calculated tend to have dissociation energy to the atomic to lie about 1,67 v.e. above the normal state, 4S, valence state equal to about 6,6 v.e. Examples that for the iso-electronic oxygen ion 0+ is 2,34 are 0+ X, 6,51; N2 B, 6,68; N2 a, 6,56; C2 A, v.e., and that for phosphorus is 1,05 v.e. above 7,05; C2 b, 6,55 v.e. An increase, presumably due their normal states. Similarly the bivalent states to the stabilizing effect of the partial ionic cha- of carbon, : C •, the nitrogen ion, : N • and racter of the double bond, is observed when the atoms differ by 0,5 in electronegativity: NO X, Silicon,:Si •, are 0,44 v.e., 0,64 v.e., and 0,28 v.e., 7,69; CN A, 7,62 v.e. -
Sink Or Float? Thought Problems in Math and Physics
AMS / MAA DOLCIANI MATHEMATICAL EXPOSITIONS VOL AMS / MAA DOLCIANI MATHEMATICAL EXPOSITIONS VOL 33 33 Sink or Float? Sink or Float: Thought Problems in Math and Physics is a collection of prob- lems drawn from mathematics and the real world. Its multiple-choice format Sink or Float? forces the reader to become actively involved in deciding upon the answer. Thought Problems in Math and Physics The book’s aim is to show just how much can be learned by using everyday Thought Problems in Math and Physics common sense. The problems are all concrete and understandable by nearly anyone, meaning that not only will students become caught up in some of Keith Kendig the questions, but professional mathematicians, too, will easily get hooked. The more than 250 questions cover a wide swath of classical math and physics. Each problem s solution, with explanation, appears in the answer section at the end of the book. A notable feature is the generous sprinkling of boxes appearing throughout the text. These contain historical asides or A tub is filled Will a solid little-known facts. The problems themselves can easily turn into serious with lubricated aluminum or tin debate-starters, and the book will find a natural home in the classroom. ball bearings. cube sink... ... or float? Keith Kendig AMSMAA / PRESS 4-Color Process 395 page • 3/4” • 50lb stock • finish size: 7” x 10” i i \AAARoot" – 2009/1/21 – 13:22 – page i – #1 i i Sink or Float? Thought Problems in Math and Physics i i i i i i \AAARoot" – 2011/5/26 – 11:22 – page ii – #2 i i c 2008 by The -
THE INVENTION of ATOMIST ICONOGRAPHY 1. Introductory
THE INVENTION OF ATOMIST ICONOGRAPHY Christoph Lüthy Center for Medieval and Renaissance Natural Philosophy University of Nijmegen1 1. Introductory Puzzlement For centuries now, particles of matter have invariably been depicted as globules. These glob- ules, representing very different entities of distant orders of magnitudes, have in turn be used as pictorial building blocks for a host of more complex structures such as atomic nuclei, mole- cules, crystals, gases, material surfaces, electrical currents or the double helixes of DNA. May- be it is because of the unsurpassable simplicity of the spherical form that the ubiquity of this type of representation appears to us so deceitfully self-explanatory. But in truth, the spherical shape of these various units of matter deserves nothing if not raised eyebrows. Fig. 1a: Giordano Bruno: De triplici minimo et mensura, Frankfurt, 1591. 1 Research for this contribution was made possible by a fellowship at the Max-Planck-Institut für Wissenschafts- geschichte (Berlin) and by the Netherlands Organization for Scientific Research (NWO), grant 200-22-295. This article is based on a 1997 lecture. Christoph Lüthy Fig. 1b: Robert Hooke, Micrographia, London, 1665. Fig. 1c: Christian Huygens: Traité de la lumière, Leyden, 1690. Fig. 1d: William Wollaston: Philosophical Transactions of the Royal Society, 1813. Fig. 1: How many theories can be illustrated by a single image? How is it to be explained that the same type of illustrations should have survived unperturbed the most profound conceptual changes in matter theory? One needn’t agree with the Kuhnian notion that revolutionary breaks dissect the conceptual evolution of science into incommensu- rable segments to feel that there is something puzzling about pictures that are capable of illus- 2 THE INVENTION OF ATOMIST ICONOGRAPHY trating diverging “world views” over a four-hundred year period.2 For the matter theories illustrated by the nearly identical images of fig. -
On the Chiral Archimedean Solids Dedicated to Prof
Beitr¨agezur Algebra und Geometrie Contributions to Algebra and Geometry Volume 43 (2002), No. 1, 121-133. On the Chiral Archimedean Solids Dedicated to Prof. Dr. O. Kr¨otenheerdt Bernulf Weissbach Horst Martini Institut f¨urAlgebra und Geometrie, Otto-von-Guericke-Universit¨atMagdeburg Universit¨atsplatz2, D-39106 Magdeburg Fakult¨atf¨urMathematik, Technische Universit¨atChemnitz D-09107 Chemnitz Abstract. We discuss a unified way to derive the convex semiregular polyhedra from the Platonic solids. Based on this we prove that, among the Archimedean solids, Cubus simus (i.e., the snub cube) and Dodecaedron simum (the snub do- decahedron) can be characterized by the following property: it is impossible to construct an edge from the given diameter of the circumsphere by ruler and com- pass. Keywords: Archimedean solids, enantiomorphism, Platonic solids, regular poly- hedra, ruler-and-compass constructions, semiregular polyhedra, snub cube, snub dodecahedron 1. Introduction Following Pappus of Alexandria, Archimedes was the first person who described those 13 semiregular convex polyhedra which are named after him: the Archimedean solids. Two representatives from this family have various remarkable properties, and therefore the Swiss pedagogicians A. Wyss and P. Adam called them “Sonderlinge” (eccentrics), cf. [25] and [1]. Other usual names are Cubus simus and Dodecaedron simum, due to J. Kepler [17], or snub cube and snub dodecahedron, respectively. Immediately one can see the following property (characterizing these two polyhedra among all Archimedean solids): their symmetry group contains only proper motions. There- fore these two polyhedra are the only Archimedean solids having no plane of symmetry and having no center of symmetry. So each of them occurs in two chiral (or enantiomorphic) forms, both having different orientation. -
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.