DOCSLIB.ORG

Final Poster

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Final Poster Associating Finite Groups with Dessins d’Enfants Luis Baeza, Edwin Baeza, Conner Lawrence, and Chenkai Wang Abstract Platonic Solids Rotation Group Dn: Regular Convex Polygon Approach Each finite, connected planar graph has an automorphism group G;such Following Magot and Zvonkin, reduce to easier cases using “hypermaps” permutations can be extended to automorphisms of the Riemann sphere φ : P1(C) P1(C), then composing β = φ f where S 2(R) P1(C). In 1984, Alexander Grothendieck, inspired by a result of f : 1( ) ! 1( )isaBely˘ımapasafunctionofeither◦ zn or ' P C P C Gennadi˘ıBely˘ıfrom 1979, constructed a finite, connected planar graph 4 zn/(zn +1)! 2 such that Aut(f ) Z or Aut(f ) D ,respectively. ' n ' n ∆β via certain rational functions β(z)=p(z)/q(z)bylookingatthe inverse image of the interval from 0 to 1. The automorphisms of such a Hypermaps: Rotation Group Zn graph can be identified with the Galois group Aut(β)oftheassociated 1 1 rational function β : P (C) P (C). In this project, we investigate how Rigid Rotations of the Platonic Solids I Wheel/Pyramids (J1, J2) ! w 3 (w +8) restrictive Grothendieck’s concept of a Dessin d’Enfant is in generating all n 2 I φ(w)= 1 1 z +1 64 (w 1) automorphisms of planar graphs. We discuss the rigid rotations of the We have an action : PSL2(C) P (C) P (C). β(z)= : v = n + n, e =2 n, f =2 − n ◦ ⇥ 2 !n 2 4 zn · Platonic solids (the tetrahedron, cube, octahedron, icosahedron, and I Zn = r r =1 and Dn = r, s s = r =(sr) =1 are the rigid I Cupola (J3, J4, J5) dodecahedron), the Archimedean solids, the Catalan solids, and the rotations of the regular convex polygons,with 4w 4(w 2 20w +105)3 I φ(w)= − ⌦ ↵ ⌦ 1 ↵ Rotation Group A4: Tetrahedron 3 2 Johnson solids via explicit Bely˘ımaps. r(z)=⇣ z and s(z)= . (7w 48) (3w 32) (5w +12) n z − − 2 3 3 I Elongated Pyramids (J7, J8, J9) Graphs I A4 = r, s s = r =(sr) =1 are the rigid rotations of the 4(835+872p2) w 4 (w 1)3 (11+8p2) w+1 − tetrahedron,with I φ(w)= 3 1 z (8+9p2) w+1 (8 5p2) w 1 ⌦ r(z)=⇣3 z ↵ and s(z)= − . I A (finite) graph is an ordered pair V , E consisting of vertices V and 2 z +1 −⇥ − ⇤ 2 3 4 I Diminished Trapezohedron⇥ ⇤ ⇥ ⇤ edges E. I S4 = r, s s = r =(sr) =1 are the rigid rotations of the 4(835 872p2) w 4 (w 1)3 (11 8p2) w +1 v = V =thenumberofvertices octahedron and the cube,with φ(w)= − − − | | I 3 e = E =thenumberofedges ⌦ ⇣4 + z ↵ 1 z (8 9p2) w +1 (8 + 5p2) w 1 r(z)= and s(z)= − . − ⇥ − ⇤ | | ⇣4 z 1+z f = F =thenumberoffaces − ⇥ ⇤ ⇥ ⇤ | | 2 3 5 I A5 = r, s s = r =(sr) =1 are the rigid rotations of the Hypermaps: Rotation Group Dn I A connected graph is a graph where, given any pair of vertices z1 and icosahedron and the dodecahedron,with 3 z ,onecantraverseapathofedgesfromonetotheother. 64 z3 z3 1 2 ⌦ 4 ↵ 4 ⇣5 + ⇣5 ⇣5 z ⇣5 + ⇣5 z β(z)= − : v =4+6, e =2 6, f =4 r(z)= − and s(z)= − . 3 I Two vertices z1 and z2 are adjacent if there is an edge connecting them. 4 4 − 3 · I Elongated Bipyramid ⇣5 + ⇣5 z + ⇣5 ⇣5 + ⇣5 z +1 8 z +1 A bipartite graph is a graph where the vertices V can be partitioned (J , J , J ) 14 15 16 4 into two disjoint sets B and W such that no two edges z1, z2 B w (32 w 5) 2 Platonic, Archimedean, Catalan, and Johnson Solids Rotation Group S4: Rhombicuboctahedron / Deltoidal I φ(w)= − (respectively, z1, z2 W )areadjacent. (80 w +1)3 2 Icositetrahedron I A planar graph is a graph that can be drawn such that the edges only I A Platonic solid is a regular, convex polyhedron. They are named after intersect at the vertices. Plato (424 BC – 348 BC). Aside from the regular polygons, there are five I Orthobicupola such solids. (J27, J28, J30) The Sphere as The Extended Complex Plane I An Archimedean solid is a convex polyhedron that has a similar I φ(w)= arrangement of nonintersecting regular convex polygons of two or more (w 2.411)4 (w +0.138)4 − di↵erent types arranged in the same way about each vertex with all sides w (w +3.086)3 (w 0.441)4 − the same length. Discovered by Johannes Kepler (1571 – 1630) in 1620, Through stereographic projection, we they are named after Archimedes (287 BC – 212 BC). Aside from the I Elongated Gyrobicupola can establish a bijection between the prisms and antiprisms, there are thirteen such solids. 2 (J36, J37, J39) 4 unit sphere S ( )andtheextended 3 2 R I (w +(0.739+0.223 i)w (0.754+0.034 i) w+(0.002 0.020 i) A Catalan solid is a dual polyhedron to an Archimedean solid. They are − − I φ(w)= 3 4 1 w w (0.041 0.283 i) w 2 (2.004+0.189 i) w+(0.005+0.324 i) complex plane P (C)=C . named after Eug`ene Catalan (1814 – 1894) who discovered them in 1865. ⇥ − − − ⇤ [{1} ⇥ ⇤ ⇥ ⇤ Aside from the bipyramids and trapezohedra, there are thirteen such solids. Theorem (E. Baeza, L. Baeza, C. Lawrence, and C. Wang, I A Johnson solid is a convex polyhedron with regular polygons as faces Define stereographic projection as that map from the unit sphere to 2014) but which is not Platonic or Archimedean. They are named after Norman the complex plane. Johnson (1930) who discovered them in 1966. There are ninety-two There are explicit Bely˘ımaps β for S 2(R) ⇠ P1(C) Johnson solids. ! I Wheel/Pyramids (J1, J2) I Elongated Pyramids (J7, J8, J9) 2x 2y x2+y 2 1 u + iv − u, v, w = x2+y 2+1, x2+y 2+1, x2+y 2+1 x + iy= I Cupola (J3, J4, J5) I Diminished Trapezohedron 7! 1 w Proposition (Wushi Goldring, 2012) Rotation Group A5: Truncated Icosahedron / Pentakis − which have rotation group Aut(β) Zn;and ⇣ ⌘ Dodecahedron ' Let %(w)bearationalfunction.Thecomposition% β is also a Bely˘ı Bely˘ımaps ◦ map for every Bely˘ımap β : X P1(C)ifandonlyif% is a Bely˘ımap I Elongated Orthobicupola (J35, J38) ! I Bipyramid (J12, J13) Let β : X 1( )beameromorphicfunctiononaRiemannsurfaceX . which sends the set 0, 1, to itself. I Elongated Gyrobicupola (J36, J37, P C I Elongated Bipyramid (J , J , J ) ! 1 1 14 15 16 J ) We say z X is a critical point if β0(z)=0,andw P (C)isa 39 2 2 I Gyroelongated Bipyramid (J17) critical value if w = β(z)forsomecriticalpointz X .Arational Proposition (Nicolas Magot and Alexander Zvonkin, 2000) I Dipole/Hosohedron 1 2 I Orthobicupola (J27, J28, J30) function β : X P (C)whichhasatmostthreecriticalvalues I Truncated Trapezohedron ! The following % are Bely˘ımaps which send the set 0, 1, to itself. I Gyrobicupola (J29, J31) 0, 1, is called a Bely˘ımap. I Bifrustum/Truncated Bipyramid 1 1 (w 1)2/(4 w)isarectification, which have rotation group Aut(β) D . n Dessin d’Enfant − − 3 ' 8(4 w 1) /(27 w)isatruncation, − Acknowledgements Fix a Bely˘ımap β : X P1(C). Denote the preimages >1/w is a birectification, > ! > 3 2 1 1 1 β 1 >(4 w) /(27 w )isabitruncation, We would like to thank the following people for their support: B = β− 0 W = β− 1 E = β− [0, 1] X P (C) %(w)=> { } { } ! > −2 3 2 2 >4(w w +1)/ 27 w (w 1) is a rhombitruncation, I The National Science Foundation (NSF) and Professor Steve Bell, <> − − 4 2 I Dr. Joel Spira, (w +1)/ 16 w (w 1) is a rhombification, black? white? ? ? − 5 3 > I Andris “Andy” Zoltners, ? ? edges? R? > 7496192 (w + ✓) verticesy verticesy y y > 3 is a snubification, > I Professor Edray Goins, and ⇢ ⇢⇢ >25 (3 + 8 ✓) w 88 w (57 ✓ +64) The bipartite graph ∆β = V , E with vertices V = B W and edges E > − I Dr. Kevin Mugo. [ where ✓2> (7/64) ✓ +1=0. is called Dessin d’Enfant.WeembedthegraphonX in 3-dimensions. :> − Department of Mathematics, 150 North University Street, West Lafayette, IN 47907-2067 http://www.math.purdue.edu/.
Load more

Recommended publications
  • Computational Design Framework 3D Graphic Statics
    Computational Design Framework for 3D Graphic Statics 3D Graphic for Computational Design Framework Computational Design Framework for 3D Graphic Statics Juney Lee Juney Lee Juney ETH Zurich • PhD Dissertation No. 25526 Diss. ETH No. 25526 DOI: 10.3929/ethz-b-000331210 Computational Design Framework for 3D Graphic Statics A thesis submitted to attain the degree of Doctor of Sciences of ETH Zurich (Dr. sc. ETH Zurich) presented by Juney Lee 2015 ITA Architecture & Technology Fellow Supervisor Prof. Dr. Philippe Block Technical supervisor Dr. Tom Van Mele Co-advisors Hon. D.Sc. William F. Baker Prof. Allan McRobie PhD defended on October 10th, 2018 Degree confirmed at the Department Conference on December 5th, 2018 Printed in June, 2019 For my parents who made me, for Dahmi who raised me, and for Seung-Jin who completed me. Acknowledgements I am forever indebted to the Block Research Group, which is truly greater than the sum of its diverse and talented individuals. The camaraderie, respect and support that every member of the group has for one another were paramount to the completion of this dissertation. I sincerely thank the current and former members of the group who accompanied me through this journey from close and afar. I will cherish the friendships I have made within the group for the rest of my life. I am tremendously thankful to the two leaders of the Block Research Group, Prof. Dr. Philippe Block and Dr. Tom Van Mele. This dissertation would not have been possible without my advisor Prof. Block and his relentless enthusiasm, creative vision and inspiring mentorship.
    [Show full text]
  • On a Remarkable Cube of Pyrite, Carrying Crys- Tallized Gold and Galena of Unusual Habit
    ON A REMARKABLE CUBE OF PYRITE, CARRYING CRYS- TALLIZED GOLD AND GALENA OF UNUSUAL HABIT By JOSEPH E. POGUE Assistant Curator, Division of Mineralogy, U. S. National Museum With One Plate The intergrowth or interpenetration of two or more minerals, especially if these be well crystallized, often shows a certain mutual crystallographic control in the arrangement of the individuals, sug- gestive of interacting molecular forces. Occasionally a crystal upon nearly completing its growth exerts what may be termed "surface affinit}'," in that it seems to attract molecules of composition differ- ent from its own and causes these to crystallize in positions bearing definite crystallographic relations to the host crystal, as evidenced, for example, by the regular arrangement of marcasite on calcite, chalcopyrite on galena, quartz on fluorite, and so on. Of special interest, not only because exhibiting the features mentioned above, but also on account of the unusual development of the individuals and the great beauty of the specimen, is a large cube of pyrite, studded with crystals of native gold and partly covered by plates of galena, acquired some years ago by the U. S. National Museum. This cube measures about 2 inches (51 mm.) along its edge, and is prominently striated, as is often the case with pyrite. It contains something more than 130 crystals of gold attached to its surface, has about one-fourth of its area covered with galena, and upon one face shows an imperfect crystal of chalcopyrite. The specimen came into the possession of the National Museum in 1906 and was ob- tained from the Snettisham District, near Juneau, Southeast Alaska.
    [Show full text]
  • Abstract Shape Synthesis from Linear Combinations of Clelia Curves’
    Putnam, Lance; Todd, Stephen and Latham, William. 2019. ’Abstract Shape Synthesis From Linear Combinations of Clelia Curves’. In: ACM/EG Expressive Symposium 2019. Genoa, Italy 5-6 May 2019. [Conference or Workshop Item] https://research.gold.ac.uk/id/eprint/27622/ The version presented here may differ from the published, performed or presented work. Please go to the persistent GRO record above for more information. If you believe that any material held in the repository infringes copyright law, please contact the Repository Team at Goldsmiths, University of London via the following email address: [email protected]. The item will be removed from the repository while any claim is being investigated. For more information, please contact the GRO team: [email protected] The 8th ACM/EG Expressive Symposium EXPRESSIVE 2019 C. Kaplan, A. Forbes, and S. DiVerdi (Editors) Abstract Shape Synthesis From Linear Combinations of Clelia Curves L. Putnam, S. Todd and W. Latham Computing, Goldsmiths, University of London, United Kingdom Abstract This article outlines several families of shapes that can be produced from a linear combination of Clelia curves. We present parameters required to generate a single curve that traces out a large variety of shapes with controllable axial symmetries. Several families of shapes emerge from the equation that provide a productive means by which to explore the parameter space. The mathematics involves only arithmetic and trigonometry making it accessible to those with only the most basic mathematical background. We outline formulas for producing basic shapes, such as cones, cylinders, and tori, as well as more complex fami- lies of shapes having non-trivial symmetries.
    [Show full text]
  • Platonic and Archimedean Geometries in Multicomponent Elastic Membranes
    Platonic and Archimedean geometries in multicomponent elastic membranes Graziano Vernizzia,1, Rastko Sknepneka, and Monica Olvera de la Cruza,b,c,2 aDepartment of Materials Science and Engineering, Northwestern University, Evanston, IL 60208; bDepartment of Chemical and Biological Engineering, Northwestern University, Evanston, IL 60208; and cDepartment of Chemistry, Northwestern University, Evanston, IL 60208 Edited by L. Mahadevan, Harvard University, Cambridge, MA, and accepted by the Editorial Board February 8, 2011 (received for review August 30, 2010) Large crystalline molecular shells, such as some viruses and fuller- possible to construct a lattice on the surface of a sphere such enes, buckle spontaneously into icosahedra. Meanwhile multi- that each site has only six neighbors. Consequently, any such crys- component microscopic shells buckle into various polyhedra, as talline lattice will contain defects. If one allows for fivefold observed in many organelles. Although elastic theory explains defects only then, according to the Euler theorem, the minimum one-component icosahedral faceting, the possibility of buckling number of defects is 12 fivefold disclinations. In the absence of into other polyhedra has not been explored. We show here that further defects (21), these 12 disclinations are positioned on the irregular and regular polyhedra, including some Archimedean and vertices of an inscribed icosahedron (22). Because of the presence Platonic polyhedra, arise spontaneously in elastic shells formed of disclinations, the ground state of the shell has a finite strain by more than one component. By formulating a generalized elastic that grows with the shell size. When γ > γà (γà ∼ 154) any flat model for inhomogeneous shells, we demonstrate that coas- fivefold disclination buckles into a conical shape (19).
    [Show full text]
  • 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.
    [Show full text]
  • Arxiv:1403.3190V4 [Gr-Qc] 18 Jun 2014 ‡ † ∗ Stefloig H Oetintr Ftehmloincon Hamiltonian the of [16]
    A curvature operator for LQG E. Alesci,∗ M. Assanioussi,† and J. Lewandowski‡ Institute of Theoretical Physics, University of Warsaw (Instytut Fizyki Teoretycznej, Uniwersytet Warszawski), ul. Ho˙za 69, 00-681 Warszawa, Poland, EU We introduce a new operator in Loop Quantum Gravity - the 3D curvature operator - related to the 3-dimensional scalar curvature. The construction is based on Regge Calculus. We define this operator starting from the classical expression of the Regge curvature, we derive its properties and discuss some explicit checks of the semi-classical limit. I. INTRODUCTION Loop Quantum Gravity [1] is a promising candidate to finally realize a quantum description of General Relativity. The theory presents two complementary descriptions based on the canon- ical and the covariant approach (spinfoams) [2]. The first implements the Dirac quantization procedure [3] for GR in Ashtekar-Barbero variables [4] formulated in terms of the so called holonomy-flux algebra [1]: one considers smooth manifolds and defines a system of paths and dual surfaces over which the connection and the electric field can be smeared. The quantiza- tion of the system leads to the full Hilbert space obtained as the projective limit of the Hilbert space defined on a single graph. The second is instead based on the Plebanski formulation [5] of GR, implemented starting from a simplicial decomposition of the manifold, i.e. restricting to piecewise linear flat geometries. Even if the starting point is different (smooth geometry in the first case, piecewise linear in the second) the two formulations share the same kinematics [6] namely the spin-network basis [7] first introduced by Penrose [8].
    [Show full text]
  • On Some Properties of Distance in TO-Space
    Aksaray University Aksaray J. Sci. Eng. Journal of Science and Engineering Volume 4, Issue 2, pp. 113-126 e-ISSN: 2587-1277 doi: 10.29002/asujse.688279 http://dergipark.gov.tr/asujse http://asujse.aksaray.edu.tr Available online at Research Article On Some Properties of Distance in TO-Space Zeynep Can* Department of Mathematics, Faculty of Science and Letters, Aksaray University, Aksaray 68100, Turkey ▪Received Date: Feb 12, 2020 ▪Revised Date: Dec 21, 2020 ▪Accepted Date: Dec 24, 2020 ▪Published Online: Dec 25, 2020 Abstract The aim of this work is to investigate some properties of the truncated octahedron metric introduced in the space in further studies on metric geometry. With this metric, the 3- dimensional analytical space is a Minkowski geometry which is a non-Euclidean geometry in a finite number of dimensions. In a Minkowski geometry, the unit ball is a certain symmetric closed convex set instead of the usual sphere in Euclidean space. The unit ball of the truncated octahedron geometry is a truncated octahedron which is an Archimedean solid. In this study, 2 first, metric properties of truncated octahedron distance, 푑푇푂, in ℝ has been examined by 3 metric approach. Then, by using synthetic approach some distance formulae in ℝ푇푂, 3- dimensional analytical space furnished with the truncated octahedron metric has been found. Keywords Metric, Convex polyhedra, Truncated octahedron, Distance of a point to a line, Distance of a point to a plane, Distance between two lines *Corresponding Author: Zeynep Can, [email protected], 0000-0003-2656-5555 2017-2020©Published by Aksaray University 113 Zeynep Can (2020).
    [Show full text]
  • Building Ideas
    TM Geometiles Building Ideas Patent Pending GeometilesTM is a product of TM www.geometiles.com Welcome to GeometilesTM! Here are some ideas of what you can build with your set. You can use them as a springboard for your imagination! Hints and instructions for making selected objects are in the back of this booklet. Platonic Solids CUBE CUBE 6 squares 12 isosceles triangles OCTAHEDRON OCTAHEDRON 8 equilateral triangles 16 scalene triangles 2 Building Ideas © 2015 Imathgination LLC REGULAR TETRAHEDRA 4 equilateral triangles; 8 scalene triangles; 16 equilateral triangles ICOSAHEDRON DODECAHEDRON 20 equilateral triangles 12 pentagons 3 Building Ideas © 2015 Imathgination LLC Selected Archimedean Solids CUBOCTAHEDRON ICOSIDODECAHEDRON 6 squares, 8 equilateral triangles 20 equilateral triangles, 12 pentagons Miscellaneous Solids DOUBLE TETRAHEDRON RHOMBIC PRISM 12 scalene triangles 8 scalene triangles; 4 rectangles 4 Building Ideas © 2015 Imathgination LLC PENTAGONAL ANTIPRISM HEXAGONAL ANTIPRISM 10 equilateral triangles, 2 pentagons. 24 equilateral triangles STELLA OCTANGULA, OR STELLATED OCTAHEDRON 24 equilateral triangles 5 Building Ideas © 2015 Imathgination LLC TRIRECTANGULAR TETRAHEDRON 12 isosceles triangles, 8 scalene triangles SCALENOHEDRON TRAPEZOHEDRON 8 scalene triangles 16 scalene triangles 6 Building Ideas © 2015 Imathgination LLC Playful shapes FLOWER 12 pentagons, 10 squares, 9 rectagles, 6 scalene triangles 7 Building Ideas © 2015 Imathgination LLC GEMSTONE 8 equilateral triangles, 8 rectangles, 4 isosceles triangles, 8 scalene
    [Show full text]
  • [ENTRY POLYHEDRA] Authors: Oliver Knill: December 2000 Source: Translated Into This Format from Data Given In
    ENTRY POLYHEDRA [ENTRY POLYHEDRA] Authors: Oliver Knill: December 2000 Source: Translated into this format from data given in http://netlib.bell-labs.com/netlib tetrahedron The [tetrahedron] is a polyhedron with 4 vertices and 4 faces. The dual polyhedron is called tetrahedron. cube The [cube] is a polyhedron with 8 vertices and 6 faces. The dual polyhedron is called octahedron. hexahedron The [hexahedron] is a polyhedron with 8 vertices and 6 faces. The dual polyhedron is called octahedron. octahedron The [octahedron] is a polyhedron with 6 vertices and 8 faces. The dual polyhedron is called cube. dodecahedron The [dodecahedron] is a polyhedron with 20 vertices and 12 faces. The dual polyhedron is called icosahedron. icosahedron The [icosahedron] is a polyhedron with 12 vertices and 20 faces. The dual polyhedron is called dodecahedron. small stellated dodecahedron The [small stellated dodecahedron] is a polyhedron with 12 vertices and 12 faces. The dual polyhedron is called great dodecahedron. great dodecahedron The [great dodecahedron] is a polyhedron with 12 vertices and 12 faces. The dual polyhedron is called small stellated dodecahedron. great stellated dodecahedron The [great stellated dodecahedron] is a polyhedron with 20 vertices and 12 faces. The dual polyhedron is called great icosahedron. great icosahedron The [great icosahedron] is a polyhedron with 12 vertices and 20 faces. The dual polyhedron is called great stellated dodecahedron. truncated tetrahedron The [truncated tetrahedron] is a polyhedron with 12 vertices and 8 faces. The dual polyhedron is called triakis tetrahedron. cuboctahedron The [cuboctahedron] is a polyhedron with 12 vertices and 14 faces. The dual polyhedron is called rhombic dodecahedron.
    [Show full text]
  • Spectral Realizations of Graphs
    SPECTRAL REALIZATIONS OF GRAPHS B. D. S. \DON" MCCONNELL 1. Introduction 2 The boundary of the regular hexagon in R and the vertex-and-edge skeleton of the regular tetrahe- 3 dron in R , as geometric realizations of the combinatorial 6-cycle and complete graph on 4 vertices, exhibit a significant property: Each automorphism of the graph induces a \rigid" isometry of the figure. We call such a figure harmonious.1 Figure 1. A pair of harmonious graph realizations (assuming the latter in 3D). Harmonious realizations can have considerable value as aids to the intuitive understanding of the graph's structure, but such realizations are generally elusive. This note explains and explores a proposition that provides a straightforward way to generate an entire family of harmonious realizations of any graph: A matrix whose rows form an orthogonal basis of an eigenspace of a graph's adjacency matrix has columns that serve as coordinate vectors of the vertices of an harmonious realization of the graph. This is a (projection of a) spectral realization. The hundreds of diagrams in Section 42 illustrate that spectral realizations of graphs with a high degree of symmetry can have great visual appeal. Or, not: they may exist in arbitrarily-high- dimensional spaces, or they may appear as an uninspiring jumble of points in one-dimensional space. In most cases, they collapse vertices and even edges into single points, and are therefore only very rarely faithful. Nevertheless, spectral realizations can often provide useful starting points for visualization efforts. (A basic Mathematica recipe for computing (projected) spectral realizations appears at the end of Section 3.) Not every harmonious realization of a graph is spectral.
    [Show full text]
  • 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
    [Show full text]
  • Paper Models of Polyhedra
    Paper Models of Polyhedra Gijs Korthals Altes Polyhedra are beautiful 3-D geometrical figures that have fascinated philosophers, mathematicians and artists for millennia Copyrights © 1998-2001 Gijs.Korthals Altes All rights reserved . It's permitted to make copies for non-commercial purposes only email: [email protected] Paper Models of Polyhedra Platonic Solids Dodecahedron Cube and Tetrahedron Octahedron Icosahedron Archimedean Solids Cuboctahedron Icosidodecahedron Truncated Tetrahedron Truncated Octahedron Truncated Cube Truncated Icosahedron (soccer ball) Truncated dodecahedron Rhombicuboctahedron Truncated Cuboctahedron Rhombicosidodecahedron Truncated Icosidodecahedron Snub Cube Snub Dodecahedron Kepler-Poinsot Polyhedra Great Stellated Dodecahedron Small Stellated Dodecahedron Great Icosahedron Great Dodecahedron Other Uniform Polyhedra Tetrahemihexahedron Octahemioctahedron Cubohemioctahedron Small Rhombihexahedron Small Rhombidodecahedron S mall Dodecahemiododecahedron Small Ditrigonal Icosidodecahedron Great Dodecahedron Compounds Stella Octangula Compound of Cube and Octahedron Compound of Dodecahedron and Icosahedron Compound of Two Cubes Compound of Three Cubes Compound of Five Cubes Compound of Five Octahedra Compound of Five Tetrahedra Compound of Truncated Icosahedron and Pentakisdodecahedron Other Polyhedra Pentagonal Hexecontahedron Pentagonalconsitetrahedron Pyramid Pentagonal Pyramid Decahedron Rhombic Dodecahedron Great Rhombihexacron Pentagonal Dipyramid Pentakisdodecahedron Small Triakisoctahedron Small Triambic
    [Show full text]