DOCSLIB.ORG

Visualization of Regular Maps Scientific Visualization – Math Vis Vis – Human Computer Interaction – Visual Analytics – Parallel

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
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 Regular Maps • Surface topology, combinatorial group theory, graph theory, algebraic geometry, hyperbolic geometry, physics, chemistry, … Jarke van Wijk TU Eindhoven ACM SIGGRAPH 2009, August 3-7, New Orleans 26 27 Triangle group Regular maps p q r T(p, q, r) = < a, b, c | a2 = b2 = c2 = ( ab ) =( bc ) = ( ca ) = I > Quotient group of triangle group T(p, q , 2): p q 2 a, b, c: reflections edges G(p, q) = < R, S | R = S = ( RS ) = … = I > ab , bc , ca : rotations vertices R: ab , rotation center polygon R3 S : bc , rotation vertex 4 R RS : ca , rotation edge 2 2 2 M M R R S aba R−1 aba R2S b ab b ab=R I a I a bc bc=S N O N O 2 c ca c RS 2 ca= S RS 28 29 Regular maps Symmetries of the cube Regular map : Embedding of a graph in a closed surface, such that topologically • faces are identical • vertices are identical • edges are identical Tiling of closed surface with maximal symmetry • 48–fold symmetry (2 pN = 2x4x6 = 48) 30 31 5 2/2/2016 Symmetric tiling sphere Genus 0: Platonic solids Symmetric tiling : Covering of surface with q= 3 regular polygons {3, 3 } {3, 4} {3, 5} {p, q} : Schläffli symbol p= 4 At each vertex: q p -gons meet {4, 3}: cube 32 {4, 3 } {5, 3} 33 Genus 0: hosohedra Genus 1: tori • hosohedron: faces with two edges • Tile the plane • Define a rhombus (all sides same length) • Project tiling {2, 4 } {2, 9} {2, 32} • Fold rhombus to torus 34 35 Genus 1: {4 , 4} torii Genus 1: {6 , 3} torii 36 37 6 2/2/2016 Genus g ≥ 2 Tiling hyperbolic plane g shape geometry transf. tilings {3, 8} tiling 0 sphere spherical 3D {3,3}, {3,4}, {4,3}, hyperbolic plane rotation {3,5}, {5,3}, {2, n} 1 torus planar 2D {4,4}, {3,6}, {6,3} Euclidean ≥ 2 ? hyperbolic Möbius {3,7}, {4,5}, {5,4}, {4,6}, {6,4}, {5,5}, … 38 39 Tiling hyperbolic plane Regular maps {4, 6} tiling Regular map: hyperbolic plane Cut out part of tiling hyperbolic plane For instance: 6 quads 40 41 Regular maps Regular maps Regular map: Regular map: Cut out part of tiling Cut out part of hyperbolic plane tiling hyperbolic plane For instance: 6 quads, and match edges M. Conder (2006): enumerated all regular maps for g ≤ 101 42 43 7 2/2/2016 Conder’s list Conder’s list R2.1 : Type {3,8}_12 Order 96 mV = 2 mF = 1 Defining relations for automorphism group: R2.2 : Type {4,6}_12 Order 48 mV = 3 mF = 2 [ T^2, R^-3, (R * S)^2, (R * T)^2, (S * T)^2, (R * S^-3)^2 ] Defining relations for automorphism group: R2.2 : Type {4,6}_12 Order 48 mV = 3 mF = 2 [ T^2, R^4, (R * S)^2, (R * S^-1)^2, (R * T)^2, (S * T)^2, S^6 ] Defining relations for automorphism group: [ T^2, R^4, (R * S)^2, (R * S^-1)^2, (R * T)^2, (S * T)^2, S^6 ] R2.3 : Type {4,8}_8 Order 32 mV = 8 mF = 2 Defining relations for automorphism group: [ T^2, R^4, (R * S)^2, (R * S^-1)^2, (R * T)^2, (S * T)^2, S^-2 * R^2 * S^-2 ] R2.4 : Type {5,10}_2 Order 20 mV = 10 mF = 5 Defining relations for automorphism group: • Rg.i: genus g, member i [ T^2, S * R^2 * S, (R, S), (R * T)^2, (S * T)^2, R^-5 ] …… • complete definition topology R101.55 : Type {204,204}_2 Order 816 mV = 204 mF = 204 Self-dual Defining relations for automorphism group: (connectivity) [ T^2, S * R^2 * S, (R, S), (R * T)^2, (S * T)^2, R^92 * S^-1 * R^3 * T * S^2 * T * R^16 * S^-89 * R ] R101.56 : Type {404,404}_2 Order 808 mV = 404 mF = 404 Self-dual Defining relations for automorphism group: [ T^2, S * R^2 * S, (R, S), (R * T)^2, (S * T)^2, R^98 * T * S^2 * T * R^10 * T * R^-3 * T * R^4 * S^-85 ] • No cue on possible 3D geometry Total number of maps in list above: 3378 44 45 Conder’s list A perfect puzzle R2.2 : Type {4,6}_12 Order 48 mV = 3 mF = 2 Defining relations for automorphism group: Find a space model of a regular map: [ T^2, R^4, (R * S)^2, (R * S^-1)^2, (R * T)^2, (S * T)^2, S^6 ] • Easy to understand The challenge: • Scalable Given the complete topology, • Many approaches possible find a space model : an • Fascinating embedding of faces, edges and • Many puzzles to be cracked vertices in 3D space • Few known solutions 46 47 Genus 3 Genus 3 • Helaman Ferguson, 1993 • Carlo Séquin, 2006 • The Eightfold Way • Klein’s surface • Hurwitz genus 3 • 24 heptagons • Klein’s surface • tetrahedral frame • 24 heptagons Carlo Séquin, 2009: many more, found manually by exploiting symmetries 48 49 8 2/2/2016 Work of Carlo Séquin An addictive puzzle “During the last few nights I woke up at 3am with some ideas, and sometimes they worked, and sometimes they evaporated in daylight!” “… , that is why I had to physically Carlo Séquin remove all signs of these puzzles Berkeley from my desk and ’lock them up in a vault’, so that I would not be constantly distracted from the duties that I HAVE to fulfill …” 50 51 Tori (genus 1) ( reprise ) • Tile the plane • Take a torus • Unfold to square • Warp to a rhombus • Project tiling • Map rhombus to torus 52 53 Approach for g ≥ 2 Nice genus g shape? • Tile the hyperbolic plane • Take a nice genus g shape …. …. • Unfold to cut out • Warp to match shape Solid shape with holes? • Project tiling Where to place holes or • Map cut out to nice shape handles to get maximal symmetry? … For g = 6, 13, 17, …? Sphere with handles? 54 55 9 2/2/2016 Tubified regular maps Tubified regular maps • Take a regular map • Take a regular map • Turn edges into tubes • Turn edges into tubes • Remove faces • Remove faces • edges → tubes • edges → tubes • vertices → junctions • vertices → junctions • faces → holes • faces → holes • triangles → ¼ tubes • triangles → ¼ tubes 56 57 Basic idea Solving R2.1{3, 8}, 16 triangles tube project target shapes pattern map regular maps genus 2 warp unfold 58 59 Solving R5.1{3, 8}, 24 octagons Tubified regular maps, reprise tube 1. Take a regular map project 2. Turn edges into tubes 3. Remove faces 4. Map a regular map map genus 5 5. Goto step 2 again unfold and warp 1.2.3.4. 60 61 10 2/2/2016 Results Video • About 50 different space models for regular maps found automatically 62 . 63 Visualization of Regular Maps: The Chase Continues Paris 2014 Jarke J. van Wijk Five years later Eindhoven University of Technology IEEE SciVis, 2014, Paris Results 2014 Visualization of cyclic groups • More generic approach for regular shapes • Subdivide a circle into 18 intervals, – 45 new space models for regular maps found given a circle, subdivided in 12 intervals • New smoothing approach • Source: C 18 – Better quality of models • Target: C 12 (Lots of) details: see paper 66 67 11 2/2/2016 Visualization of cyclic groups Visualization of cyclic groups Source C 18 : Target: C 12 : Source C 18 : Target: C 12 : • C9 × {1, 2} • C6 × {1, 2} • C9 × {1, 2} • C6 × {1, 2} • C6 × {1, 2, 3} • C4 × {1, 2, 3} • C6 × {1, 2, 3} • C4 × {1, 2, 3} • C3 × {1, 2, 3, 4, 5, 6} • C3 × {1, 2, 3, 4} • C3 × {1, 2, 3, 4, 5, 6} • C3 × {1, 2, 3, 4} • C2 × {1, 2, 3, 4, 5, 6, 7, 8, 9} • C2 × {1, 2, 3, 4, 5, 6} • C2 × {1, 2, 3, 4, 5, 6, 7, 8, 9} • C2 × {1, 2, 3, 4, 5, 6} •C6: greatest common subgroup • Match sets: – Source {1, 2, 3} – Target {1, 2}: 68 69 Visualization of cyclic groups Approach • Subdivide a circle into 18 intervals, • Take regular maps.
Load more

Recommended publications
  • 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.
    [Show full text]
  • On Discrete Generalised Triangle Groups
    Proceedings of the Edinburgh Mathematical Society (1995) 38, 397-412 © ON DISCRETE GENERALISED TRIANGLE GROUPS by M. HAGELBERG, C. MACLACHLAN and G. ROSENBERGER (Received 29th October 1993) A generalised triangle group has a presentation of the form where R is a cyclically reduced word involving both x and y. When R=xy, these classical triangle groups have representations as discrete groups of isometries of S2, R2, H2 depending on In this paper, for other words R, faithful discrete representations of these groups in Isom + H3 = PSL(2,C) are considered with particular emphasis on the case /? = [x, y] and also on the relationship between the Euler characteristic x and finite covolume representations. 1991 Mathematics subject classification: 20H15. 1. Introduction In this article, we consider generalised triangle groups, i.e. groups F with a presentation of the form where R(x,y) is a cyclically reduced word in the free product on x,y which involves both x and y. These groups have been studied for their group theoretical interest [8, 7, 13], for topological reasons [2], and more recently for their connections with hyperbolic 3-manifolds and orbifolds [12, 10]. Here we will be concerned with faithful discrete representations p:T-*PSL(2,C) with particular emphasis on the cases where the Kleinian group p(F) has finite covolume. In Theorem 3.2, we give necessary conditions on the group F so that it should admit such a faithful discrete representation of finite covolume. For certain generalised triangle groups where the word R(x,y) has a specified form, faithful discrete representations as above have been constructed by Helling-Mennicke- Vinberg [12] and by the first author [11].
    [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]
  • Arxiv:Math/0306105V2
    A bound for the number of automorphisms of an arithmetic Riemann surface BY MIKHAIL BELOLIPETSKY Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia e-mail: [email protected] AND GARETH A. JONES Faculty of Mathematical Studies, University of Southampton, Southampton, SO17 1BJ e-mail: [email protected] Abstract We show that for every g ≥ 2 there is a compact arithmetic Riemann surface of genus g with at least 4(g − 1) automorphisms, and that this lower bound is attained by infinitely many genera, the smallest being 24. 1. Introduction Schwarz [17] proved that the automorphism group of a compact Riemann surface of genus g ≥ 2 is finite, and Hurwitz [10] showed that its order is at most 84(g − 1). This bound is sharp, by which we mean that it is attained for infinitely many g, and the least genus of such an extremal surface is 3. However, it is also well known that there are infinitely many genera for which the bound 84(g − 1) is not attained. It therefore makes sense to consider the maximal order N(g) of the group of automorphisms of any Riemann surface of genus g. Accola [1] and Maclachlan [14] independently proved that N(g) ≥ 8(g +1). This bound is also sharp, and according to p. 93 of [2], Paul Hewitt has shown that the least genus attaining it is 23. Thus we have the following sharp bounds for N(g) with g ≥ 2: 8(g + 1) ≤ N(g) ≤ 84(g − 1). arXiv:math/0306105v2 [math.GR] 21 Nov 2004 We now consider these bounds from an arithmetic point of view, defining arithmetic Riemann surfaces to be those which are uniformized by arithmetic Fuchsian groups.
    [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:1012.2020V1 [Math.CV]
    TRANSITIVITY ON WEIERSTRASS POINTS ZOË LAING AND DAVID SINGERMAN 1. Introduction An automorphism of a Riemann surface will preserve its set of Weier- strass points. In this paper, we search for Riemann surfaces whose automorphism groups act transitively on the Weierstrass points. One well-known example is Klein’s quartic, which is known to have 24 Weierstrass points permuted transitively by it’s automorphism group, PSL(2, 7) of order 168. An investigation of when Hurwitz groups act transitively has been made by Magaard and Völklein [19]. After a section on the preliminaries, we examine the transitivity property on several classes of surfaces. The easiest case is when the surface is hy- perelliptic, and we find all hyperelliptic surfaces with the transitivity property (there are infinitely many of them). We then consider surfaces with automorphism group PSL(2, q), Weierstrass points of weight 1, and other classes of Riemann surfaces, ending with Fermat curves. Basically, we find that the transitivity property property seems quite rare and that the surfaces we have found with this property are inter- esting for other reasons too. 2. Preliminaries Weierstrass Gap Theorem ([6]). Let X be a compact Riemann sur- face of genus g. Then for each point p ∈ X there are precisely g integers 1 = γ1 < γ2 <...<γg < 2g such that there is no meromor- arXiv:1012.2020v1 [math.CV] 9 Dec 2010 phic function on X whose only pole is one of order γj at p and which is analytic elsewhere. The integers γ1,...,γg are called the gaps at p. The complement of the gaps at p in the natural numbers are called the non-gaps at p.
    [Show full text]
  • REFLECTION GROUPS and COXETER GROUPS by Kouver
    REFLECTION GROUPS AND COXETER GROUPS by Kouver Bingham A Senior Honors Thesis Submitted to the Faculty of The University of Utah In Partial Fulfillment of the Requirements for the Honors Degree in Bachelor of Science In Department of Mathematics Approved: Mladen Bestviva Dr. Peter Trapa Supervisor Chair, Department of Mathematics ( Jr^FeraSndo Guevara Vasquez Dr. Sylvia D. Torti Department Honors Advisor Dean, Honors College July 2014 ABSTRACT In this paper we give a survey of the theory of Coxeter Groups and Reflection groups. This survey will give an undergraduate reader a full picture of Coxeter Group theory, and will lean slightly heavily on the side of showing examples, although the course of discussion will be based on theory. We’ll begin in Chapter 1 with a discussion of its origins and basic examples. These examples will illustrate the importance and prevalence of Coxeter Groups in Mathematics. The first examples given are the symmetric group <7„, and the group of isometries of the ^-dimensional cube. In Chapter 2 we’ll formulate a general notion of a reflection group in topological space X, and show that such a group is in fact a Coxeter Group. In Chapter 3 we’ll introduce the Poincare Polyhedron Theorem for reflection groups which will vastly expand our understanding of reflection groups thereafter. We’ll also give some surprising examples of Coxeter Groups that section. Then, in Chapter 4 we’ll make a classification of irreducible Coxeter Groups, give a linear representation for an arbitrary Coxeter Group, and use this complete the fact that all Coxeter Groups can be realized as reflection groups with Tit’s Theorem.
    [Show full text]
  • Conformal Quasicrystals and Holography
    Conformal Quasicrystals and Holography Latham Boyle1, Madeline Dickens2 and Felix Flicker2;3 1Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada, N2L 2Y5 2Department of Physics, University of California, Berkeley, California 94720, USA 3Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Department of Physics, Clarendon Laboratory, Parks Road, Oxford, OX1 3PU, United Kingdom Recent studies of holographic tensor network models defined on regular tessellations of hyperbolic space have not yet addressed the underlying discrete geometry of the boundary. We show that the boundary degrees of freedom naturally live on a novel structure, a conformal quasicrystal, that pro- vides a discrete model of conformal geometry. We introduce and construct a class of one-dimensional conformal quasicrystals, and discuss a higher-dimensional example (related to the Penrose tiling). Our construction permits discretizations of conformal field theories that preserve an infinite discrete subgroup of the global conformal group at the cost of lattice periodicity. I. INTRODUCTION dom [12{24]. Meanwhile, quantum information theory provides a unifying language for these studies in terms of entanglement, quantum circuits, and quantum error A central topic in theoretical physics over the past two correction [25]. decades has been holography: the idea that a quantum These investigations have gradually clarified our un- theory in a bulk space may be precisely dual to another derstanding of the discrete geometry in the bulk. There living on the boundary of that space. The most concrete has been a common expectation, based on an analogy and widely-studied realization of this idea has been the with AdS/CFT [1{3], that TNs living on discretizations AdS/CFT correspondence [1{3], in which a gravitational of a hyperbolic space define a lattice state of a critical theory living in a (d + 1)-dimensional negatively-curved system on the boundary and vice-versa.
    [Show full text]
  • Dessins D'enfants and Counting Quasiplatonic Surfaces
    Dessins d'Enfants and Counting Quasiplatonic Surfaces Charles Camacho Oregon State University USTARS at Reed College, Portland, Oregon April 8, 2018 1 The Compact Surface X ... 2 ...with a Bipartite Map... 4 2 6 1 7 5 1 3 7 2 6 3 5 4 3 ...as an Algebraic Curve over Q... 4 2 6 1 7 5 1 3 7 2 6 3 5 4 y 2 = x7 − x 4 ∼ ...with a Uniformizing Fuchsian Group Γ = π1(X )... 6 1 7 7 5 2 6 1 4 3 5 2 3 4 ∼ X = H=Γ 5 ...Contained in a Hyperbolic Triangle Group... 6 1 7 7 5 2 6 1 4 3 5 2 3 4 ∼ X = H=Γ Γ C ∆(n1; n2; n3) 6 ...with the Cyclic Group as Quotient 6 1 7 7 5 2 6 1 4 3 5 2 3 ∼ 4 X = H=Γ Γ C ∆(n1; n2; n3) ∼ ∆(n1; n2; n3)=Γ = Cn, the cyclic group of order n 7 Method: Enumerate all quasiplatonic actions of Cn on surfaces! Main Question How many quasiplatonic surfaces have regular dessins d'enfants with automorphism group Cn? 8 Main Question How many quasiplatonic surfaces have regular dessins d'enfants with automorphism group Cn? Method: Enumerate all quasiplatonic actions of Cn on surfaces! 8 Example Three dessins with order-seven symmetry on the Riemann sphere. Dessins d'Enfants A dessin d'enfant is a pair (X ; D), where X is an orientable, compact surface and D ⊂ X is a finite graph such that 1 D is connected, 2 D is bicolored (i.e., bipartite), 3 X n D is the union of finitely many topological discs, called the faces of D.
    [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 Subgroups of Tetrahedron Groups
    On Subgroups of Tetrahedron Groups Ma. Louise N. De Las Peñas1, Rene P. Felix2 and Glenn R. Laigo3 1Mathematics Department, Ateneo de Manila University, Loyola Heights, Quezon City, Philippines. [email protected] 2Institute of Mathematics, University of the Philippines – Diliman, Diliman, Quezon City, Philippines. [email protected] 3Mathematics Department, Ateneo de Manila University, Loyola Heights, Quezon City, Philippines. [email protected] Abstract. We present a framework to determine subgroups of tetrahedron groups and tetrahedron Kleinian groups, based on tools in color symmetry theory. Mathematics Subject Classification (2000). 20F55, 20F67 Key words. Coxeter tetrahedra, tetrahedron groups, tetrahedron Kleinian groups, color symmetry 1 Introduction In [6, 7] the determination of the subgroups of two-dimensional symmetry groups was facilitated using a geometric approach and applying concepts in color symmetry theory. In particular, these works derived low-indexed subgroups of a triangle group *abc, a group generated by reflections along the sides of a given triangle ∆ with interior angles π/a, π/b and π/c where a, b, c are integers ≥ 2. The elements of *abc are symmetries of the tiling formed by copies of ∆. The subgroups of *abc were calculated from colorings of this given tiling, assuming a corresponding group of color permutations. In this work, we pursue the three dimensional case and extend the problem to determine subgroups of tetrahedron groups, groups generated by reflections along the faces of a Coxeter tetrahedron, which are groups of symmetries of tilings in space by tetrahedra. The study will also address determining subgroups of other types of three dimensional symmetry groups such as the tetrahedron Kleinian groups.
    [Show full text]
  • 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.
    [Show full text]