Flippable Tilings of Constant Curvature Surfaces
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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. -
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). -
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. -
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]. -
Notes on Convex Sets, Polytopes, Polyhedra, Combinatorial Topology, Voronoi Diagrams and Delaunay Triangulations
Notes on Convex Sets, Polytopes, Polyhedra, Combinatorial Topology, Voronoi Diagrams and Delaunay Triangulations Jean Gallier and Jocelyn Quaintance Department of Computer and Information Science University of Pennsylvania Philadelphia, PA 19104, USA e-mail: [email protected] April 20, 2017 2 3 Notes on Convex Sets, Polytopes, Polyhedra, Combinatorial Topology, Voronoi Diagrams and Delaunay Triangulations Jean Gallier Abstract: Some basic mathematical tools such as convex sets, polytopes and combinatorial topology, are used quite heavily in applied fields such as geometric modeling, meshing, com- puter vision, medical imaging and robotics. This report may be viewed as a tutorial and a set of notes on convex sets, polytopes, polyhedra, combinatorial topology, Voronoi Diagrams and Delaunay Triangulations. It is intended for a broad audience of mathematically inclined readers. One of my (selfish!) motivations in writing these notes was to understand the concept of shelling and how it is used to prove the famous Euler-Poincar´eformula (Poincar´e,1899) and the more recent Upper Bound Theorem (McMullen, 1970) for polytopes. Another of my motivations was to give a \correct" account of Delaunay triangulations and Voronoi diagrams in terms of (direct and inverse) stereographic projections onto a sphere and prove rigorously that the projective map that sends the (projective) sphere to the (projective) paraboloid works correctly, that is, maps the Delaunay triangulation and Voronoi diagram w.r.t. the lifting onto the sphere to the Delaunay diagram and Voronoi diagrams w.r.t. the traditional lifting onto the paraboloid. Here, the problem is that this map is only well defined (total) in projective space and we are forced to define the notion of convex polyhedron in projective space. -
Polyhedral Models of the Projective Plane
Bridges 2018 Conference Proceedings Polyhedral Models of the Projective Plane Paul Gailiunas 25 Hedley Terrace, Gosforth, Newcastle, NE3 1DP, England; [email protected] Abstract The tetrahemihexahedron is the only uniform polyhedron that is topologically equivalent to the projective plane. Many other polyhedra having the same topology can be constructed by relaxing the conditions, for example those with faces that are regular polygons but not transitive on the vertices, those with planar faces that are not regular, those with faces that are congruent but non-planar, and so on. Various techniques for generating physical realisations of such polyhedra are discussed, and several examples of different types described. Artists such as Max Bill have explored non-orientable surfaces, in particular the Möbius strip, and Carlo Séquin has considered what is possible with some models of the projective plane. The models described here extend the range of possibilities. The Tetrahemihexahedron A two-dimensional projective geometry is characterised by the pair of axioms: any two distinct points determine a unique line; any two distinct lines determine a unique point. There are projective geometries with a finite number of points/lines but more usually the projective plane is considered as an ordinary Euclidean plane plus a line “at infinity”. By the second axiom any line intersects the line “at infinity” in a single point, so a model of the projective plane can be constructed by taking a topological disc (it is convenient to think of it as a hemisphere) and identifying opposite points. The first known analytical surface matching this construction was discovered by Jakob Steiner in 1844 when he was in Rome, and it is known as the Steiner Roman surface. -
Visualization of Regular Maps
Symmetric Tiling of Closed Surfaces: Visualization of Regular Maps Jarke J. van Wijk Dept. of Mathematics and Computer Science Technische Universiteit Eindhoven [email protected] Abstract The first puzzle is trivial: We get a cube, blown up to a sphere, which is an example of a regular map. A cube is 2×4×6 = 48-fold A regular map is a tiling of a closed surface into faces, bounded symmetric, since each triangle shown in Figure 1 can be mapped to by edges that join pairs of vertices, such that these elements exhibit another one by rotation and turning the surface inside-out. We re- a maximal symmetry. For genus 0 and 1 (spheres and tori) it is quire for all solutions that they have such a 2pF -fold symmetry. well known how to generate and present regular maps, the Platonic Puzzle 2 to 4 are not difficult, and lead to other examples: a dodec- solids are a familiar example. We present a method for the gener- ahedron; a beach ball, also known as a hosohedron [Coxeter 1989]; ation of space models of regular maps for genus 2 and higher. The and a torus, obtained by making a 6 × 6 checkerboard, followed by method is based on a generalization of the method for tori. Shapes matching opposite sides. The next puzzles are more challenging. with the proper genus are derived from regular maps by tubifica- tion: edges are replaced by tubes. Tessellations are produced us- ing group theory and hyperbolic geometry. The main results are a generic procedure to produce such tilings, and a collection of in- triguing shapes and images. -
Geometric Point-Circle Pentagonal Geometries from Moore Graphs
Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 11 (2016) 215–229 Geometric point-circle pentagonal geometries from Moore graphs Klara Stokes ∗ School of Engineering Science, University of Skovde,¨ 54128 Skovde¨ Sweden Milagros Izquierdo Department of Mathematics, Linkoping¨ University, 58183 Linkoping¨ Sweden Received 27 December 2014, accepted 13 October 2015, published online 15 November 2015 Abstract We construct isometric point-circle configurations on surfaces from uniform maps. This gives one geometric realisation in terms of points and circles of the Desargues configura- tion in the real projective plane, and three distinct geometric realisations of the pentagonal geometry with seven points on each line and seven lines through each point on three distinct dianalytic surfaces of genus 57. We also give a geometric realisation of the latter pentago- nal geometry in terms of points and hyperspheres in 24 dimensional Euclidean space. From these, we also obtain geometric realisations in terms of points and circles (or hyperspheres) of pentagonal geometries with k circles (hyperspheres) through each point and k −1 points on each circle (hypersphere). Keywords: Uniform map, equivelar map, dessin d’enfants, configuration of points and circles Math. Subj. Class.: 05B30, 05B45, 14H57, 14N20, 30F10, 30F50, 51E26 1 Introduction A compact Klein surface S is a surface (possibly with boundary and non-orientable) en- dowed with a dianalytic structure, that is, the transition maps are holomorphic or antiholo- morphic (the conjugation z ! z is allowed). If the surface S admits analytic structure and is closed, then the surface is a Riemann surface. -
Geometry in Design Geometrical Construction in 3D Forms by Prof
D’source 1 Digital Learning Environment for Design - www.dsource.in Design Course Geometry in Design Geometrical Construction in 3D Forms by Prof. Ravi Mokashi Punekar and Prof. Avinash Shide DoD, IIT Guwahati Source: http://www.dsource.in/course/geometry-design 1. Introduction 2. Golden Ratio 3. Polygon - Classification - 2D 4. Concepts - 3 Dimensional 5. Family of 3 Dimensional 6. References 7. Contact Details D’source 2 Digital Learning Environment for Design - www.dsource.in Design Course Introduction Geometry in Design Geometrical Construction in 3D Forms Geometry is a science that deals with the study of inherent properties of form and space through examining and by understanding relationships of lines, surfaces and solids. These relationships are of several kinds and are seen in Prof. Ravi Mokashi Punekar and forms both natural and man-made. The relationships amongst pure geometric forms possess special properties Prof. Avinash Shide or a certain geometric order by virtue of the inherent configuration of elements that results in various forms DoD, IIT Guwahati of symmetry, proportional systems etc. These configurations have properties that hold irrespective of scale or medium used to express them and can also be arranged in a hierarchy from the totally regular to the amorphous where formal characteristics are lost. The objectives of this course are to study these inherent properties of form and space through understanding relationships of lines, surfaces and solids. This course will enable understanding basic geometric relationships, Source: both 2D and 3D, through a process of exploration and analysis. Concepts are supported with 3Dim visualization http://www.dsource.in/course/geometry-design/in- of models to understand the construction of the family of geometric forms and space interrelationships. -
Chiral Polyhedra in 3-Dimensional Geometries and from a Petrie-Coxeter Construction
Chiral polyhedra in 3-dimensional geometries and from a Petrie-Coxeter construction Javier Bracho, Isabel Hubard and Daniel Pellicer January 19, 2021 Abstract We study chiral polyhedra in 3-dimensional geometries (Euclidian, hy- perbolic and projective) in a unified manner. This extends to hyperbolic and projective spaces some structural results in the classification of chi- ral polyhedra in Euclidean 3-space given in 2005 by Schulte. Then, we describe a way to produce examples with helical faces based on a clas- 3 sic Petrie-Coxeter construction that yields a new family in S , which is described exhaustively. AMS Subject Classification (2010): 52B15 (Primary), 51G05, 51M20, 52B10 Keywords: chiral polyhedron, projective polyhedron, Petrie-Coxeter construction 1 Introduction Symmetric structures in space have been studied since antiquity. An example of this is the inclusion of the Platonic solids in Euclid's Elements [14]. In particular, highly symmetric polyhedra-like structures have a rich history including convex ones such as the Archimedean solids (see for example [7, Chapter 21]), star- shaped ones such as the Kepler-Poinsot solids [27], and infinite ones such as the Petrie-Coxeter polyhedra [8]. In recent decades a lot of attention has been dedicated to the so-called skeletal polyhedra, where faces are not associated to membranes. The symmetry of a skeletal polyhedron is measured by the number of orbits of flags (triples of incident vertex, edge and face) under the action of the symmetry group. Those with only one flag-orbit are called regular, and in a combinatorial sense they have 1 maximal symmetry by reflections. Chiral polyhedra are those with maximal (combinatorial) symmetry by rotations but none by reflections and have two flag-orbits. -
Visualization of Regular Maps Scientific Visualization – Math Vis Vis – Human Computer Interaction – Visual Analytics – Parallel
2/2/2016 Visualization Eindhoven – information visualization – software visualization – perception – geographic visualization Visualization of Regular Maps scientific visualization – math vis vis – human computer interaction – visual analytics – parallel Jarke J. van Wijk Eindhoven University of Technology JCB 2016, Bordeaux Data : flow fields – trees – graphs – tables – mobile data – events – … Applications : software analysis – business cases – bioinformatics – … Can you draw a Seifert surface? Eindhoven 2004 Huh? Starting MathVis Arjeh Cohen Me Discrete geometry, Visualization algebra Seifert surface Eindhoven 2006 1 2/2/2016 Can you show 364 triangles in 3D? Aberdeenshire Sure! 2000 BC I’ll give you the Arjeh Cohenpattern, you can Me pick a nice shape. Discrete geometry, Visualization algebra Ok! Athens 450 BC Scottish Neolithic carved stone balls 8 12 20 8 12 20? 6 6 4 4 Platonic solids: perfectly symmetric Platonic solids: perfectly symmetric 2 2/2/2016 How to get more faces, How to get more faces, all perfectly symmetric? all perfectly symmetric? Use shapes with holes Aim only at topological symmetry 64 64 #faces? Königsberg 1893 3 2/2/2016 Adolf Hurwitz 1859-1919 max. #faces: 3 28(genus – 1) 7 Genus Faces Genus Faces 3 56 3 56 7 168 7 168 14 364 14 364 … … … … Can you show 364 triangles in 3D? New Orleans Sure! 2009 I’ll give you the three years later Arjeh pattern, you can Me Cohen pick a nice shape. Ok! 4 2/2/2016 The general puzzle Construct space models of regular maps Symmetric Tiling of Closed Surfaces: Visualization of -
Spherical Trihedral Metallo-Borospherenes
Lawrence Berkeley National Laboratory Recent Work Title Spherical trihedral metallo-borospherenes. Permalink https://escholarship.org/uc/item/59z4w76k Journal Nature communications, 11(1) ISSN 2041-1723 Authors Chen, Teng-Teng Li, Wan-Lu Chen, Wei-Jia et al. Publication Date 2020-06-02 DOI 10.1038/s41467-020-16532-x Peer reviewed eScholarship.org Powered by the California Digital Library University of California ARTICLE https://doi.org/10.1038/s41467-020-16532-x OPEN Spherical trihedral metallo-borospherenes ✉ ✉ Teng-Teng Chen 1,5, Wan-Lu Li2,5 , Wei-Jia Chen1, Xiao-Hu Yu 3, Xin-Ran Dong4, Jun Li 2,4 & ✉ Lai-Sheng Wang 1 The discovery of borospherenes unveiled the capacity of boron to form fullerene-like cage structures. While fullerenes are known to entrap metal atoms to form endohedral metallo- fullerenes, few metal atoms have been observed to be part of the fullerene cages. Here we report the observation of a class of remarkable metallo-borospherenes, where metal atoms 1234567890():,; – – are integral parts of the cage surface. We have produced La3B18 and Tb3B18 and probed their structures and bonding using photoelectron spectroscopy and theoretical calculations. – Global minimum searches revealed that the most stable structures of Ln3B18 are hollow – cages with D3h symmetry. The B18-framework in the Ln3B18 cages can be viewed as con- sisting of two triangular B6 motifs connected by three B2 units, forming three shared B10 rings which are coordinated to the three Ln atoms on the cage surface. These metallo- borospherenes represent a new class of unusual geometry that has not been observed in chemistry heretofore.