DOCSLIB.ORG

Generating Polytopes from Coxeter Groups — Veronica Viken MAT-3907 Master Thesis in Education Year 8-13, 40 SP

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Generating Polytopes from Coxeter Groups — Veronica Viken MAT-3907 Master Thesis in Education Year 8-13, 40 SP Faculty OF Science AND TECHNOLOGY Department OF Mathematics AND Statistics GenerATING POLYTOPES FROM CoXETER GROUPS — VERONICA VikEN MAT-3907 Master THESIS IN Education YEAR 8-13, 40 SP June 2019 This thesis document was typeset using the UiT Thesis LaTEX Template. © 2019 – http://github.com/egraff/uit-thesis “To find out what one is fitted to do and to secure an opportunity to do it is the key to happiness.” –John Dewey AbstrACT The main goal of this thesis is to study finite reflection groups (Coxeter groups) W and to use these to generate polytopes in two and three dimensions. Such polytopes will be generated as the convex hull of the W -orbit through an initial point λ. We prove an efficient recipe for finding the stabilizer of λ, and examine several examples of such polytopes and illustrate how many vertices, edges and faces these polytopes have. At last we will illustrate how this information can be pictorially encoded on the marked Coxeter diagram for λ. AcknoWLEDGEMENTS Five years of studies have come to an end, and when I arrived here for five years ago I did not know anyone. Through the years I have got a lot of good friends that will always have a special place in my heart. Without these people this had not been possible. First of all I would like to thank my supervisor Dennis The for the patience, all guidance and the continuous feedback throughout writing this thesis. Thank you for giving me the opportunity for writing a thesis in mathematics. Further a special thanks to my parents and my grandmother Solveig for always supporting and believing in me no matter what. To my boyfriend Ole Jakob thank you for bearing with me through the last year. You do not know how much I have appreciated your support. Lastly, I want to thank the gang in the office for making the year unforgettable with our entertaining conversations. And to my friends Gunnhild, Isak, Rikke and Tonje for always cheering me up when needed and for all lunch and dinner breaks. Veronica Viken May 2019 Contents Abstract iii Acknowledgements v List of Figures ix List of Tables xi 1 Introduction 1 1.1 Structure of the thesis ..................... 2 2 Coxeter/ Dynkin diagram 3 2.1 Reflections and rotations ................... 3 2.2 Dihedral group ......................... 4 2.3 Root systems .......................... 6 2.4 Cartan matrix ......................... 10 2.5 Coxeter diagrams and Dynkin diagrams ........... 12 2.6 Coxeter group ......................... 14 2.7 Fundamental domain ..................... 16 2.8 Fundamental weights ..................... 17 3 Reflection recipes and stabilizers 21 3.1 Coxeter diagram and Dynkin diagram reflection recipes ... 21 3.2 Stabilizer ............................ 25 4 Polytopes 29 4.1 Definitions ........................... 29 4.2 Our construction of polytopes ................. 32 4.2.1 Polygons ........................ 33 4.2.2 Polyhedra ....................... 37 4.3 Encode equivalence classes in terms of a recipe ....... 38 4.3.1 Edges .......................... 39 4.3.2 Faces .......................... 41 VII VIII CONTENTS Bibliography 47 Appendices 49 A Weyl group orbits 51 List OF FigurES 1.1 The five Platonic solids: tetrahedron, cube, octahedron, do- decahedron and icosahedron. ................. 1 2.1 A triangle with its reflection lines. ............... 5 2.2 The A2 root system. ...................... 6 2.3 The A3 root system with the white nodes as roots. ...... 8 2.4 The B3 root system with white nodes as roots. ........ 9 2.5 An icosadodecahedron where all vertices form the H3 root system. ............................. 10 2.6 The B2 root system. ...................... 17 2.7 The B2 root system with its reflection hyperplanes, the W - orbit to x0 and the fundamental domain indicated in blue. 17 2.8 The simple roots and the fundamental weights in A2. .... 18 2.9 The B3 root system with its simple roots, fundamental weights and fundamental domain indicated with the red arrows. .. 19 3.1 The W ¹H3º-orbit through λ3. ................. 23 3.2 The W ¹B4º-orbit through λ1 ................. 25 4.1 Two convex polygons, where one of them is also regular. .. 30 4.2 The cube and its dual, namely the octahedron. ........ 31 3 4.3 Two regular triangles generated by W ¹I2 º. .......... 35 3 4.4 The non-regular polygon P¹λ1 + 2λ2º generated by W ¹I2 º. .. 35 3 4.5 The regular polygon P¹λ1 + λ2º generated by W ¹I2 º. ..... 36 4.6 The regular polygon P¹λ1 + λ2º generated by W ¹A1 × A1º. .. 36 4.7 The four edges through λ in P¹λ1 + λ2º generated by W ¹A3º. 43 4.8 The polyhedron P¹λ1 + λ3º generated by W ¹A3º. ....... 44 4.9 The polyhedron P¹λ1 + λ2º generated by W ¹H3º. ....... 45 4.10 The polyhedron P¹λ1 + λ3º generated by W ¹B3º. ....... 46 A.1 The W ¹A3º-orbits through λ1, λ2 and λ3. ........... 51 A.2 The W ¹B3º-orbits through λ1, λ2 and λ3. ........... 52 A.3 The W ¹H3º-orbit through λ1. ................. 53 A.4 The W ¹H3º-orbit through λ2. ................. 54 IX List OF TABLES 2.1 The complete classification of all connected Coxeter diagrams. 13 2.2 The complete classification of all connected Dynkin diagrams. 14 2.3 The order of the connected Coxeter groups. ......... 16 4.1 The number of vertices, edges and faces in the five Platonic solids. ............................. 31 4.2 Platonic solids encoded by marked Coxeter diagrams. .... 38 XI 1 INTRODUCTION In this thesis we will use finite reflection groups (Coxeter groups) to generate two and three-dimensional polytopes. Such groups act as the symmetries of these objects. Polytopes in two dimensions are polygons, which are geometrical figures such as squares, triangles, pentagons and so on. The polytopes in three dimensions are called polyhedra, and examples of polyhedra are the five Platonic solids: Figure 1.1: The five Platonic solids: tetrahedron, cube, octahedron, dodecahedron and icosahedron. These are all well known, and pictorially and abstractly we want to figure out how many vertices, edges and faces that more general polyhedra consist of. For the three dimensional polyhedra we will use Zometool to build models of these geometrical figures [Hart, 2001]. Abstractly, we will examine the action of Coxeter groups on different initial points λ, and then get a recipe to encode the marked Coxeter diagrams. CHAPTER I NTRODUCT I ON In the Norwegian school geometric figures in two and three dimensions such as squares, triangles, parallelograms, rhombus, rectangles, cubes, tetrahedrons, prisms and pyramids are an important part of the curriculum. After upper elementary school, students shall be able to explore and describe different properties of such two and three dimensional figures. An important property of these figures is their symmetry group. They shall also be able to reflect points in R2 through the x-axis and the y-axis. So the geometrical figures we generate, how we generate them by reflections and the symmetry group are relevant for the Norwegian school. StructurE OF THE THESIS Chapter 2 introduces reflections, finite groups generated by reflections (Coxeter groups), and Coxeter diagrams and Dynkin diagrams. Chapter 3 we will get pictorial reflection recipes expressed in terms of Coxeter diagrams and the Dynkin diagrams. We also find a recipe for finding the stabilizer for a given initial point λ. Chapter 4 we first will define polytopes. Then we will apply Coxeter groups to an initial point λ to generate polytopes in two and three dimension. A last we will encode the marked Coxeter diagram to figure out what kind of polytope we have. 2 CoXeter/ Dynkin DIAGRAM . ReflECTIONS AND ROTATIONS Let ¹V ; h·; ·iº be a finite-dimensional (real) inner product space. A reflection in V is a linear transformation in V that sends any vector λ 2 V to its mirror image with respect to a hyperplane P (passing through the origin). Equivalently, given a nonzero vector α 2 V (in the orthogonal complement P?) define a reflection in V by hλ; αi S ¹λº = λ − 2 α: (2.1.1) α hα; αi ? 2 Note that Sα ¹λº = λ if λ 2 P and Sα ¹λº = −λ if λ 2 P . Since Sα = 1, then a reflection is its own inverse, so its order is two. Also note that hSα ¹λº; Sα ¹µºi = hλ; µi for all λ; µ 2 V , i.e. Sα is an orthogonal transformation. The orthogonal transformations in V are denoted by O¹V º = fT : V ! V linear : hTv;Twi = hv;wi v;w 2 V g; 8 and any T 2 O¹V º satisfies det¹T º = ±1. Note that O¹V º is a group, and it contains the subgroup SO¹V º = fT 2 O¹V º : det¹T º = 1g: We refer to any T 2 SO¹V º as a rotation. When V = Rn, we use the notation O¹nº and SO¹nº respectively. In particular, O¹nº consists of orthogonal matrices CHAPTER COXETER/ DYNK I ND I AGRAM A satisfying AAT = AT A = I, and the columns of A form an orthonormal basis of Rn. cos θ For example, consider a matrix in O¹2º. Its first column is a unit vector sin θ , while the second column is unit and perpendicular to the first. Thus, we have two possibilities: cos θ sin θ cos θ − sin θ A = B = : sin θ − cos θ sin θ cos θ Note that det ¹Bº = 1 and it represents a counterclockwise rotation of the plane about the origin through an angle of θ. On the other hand, det ¹Aº = −1, θ θ cos ¹ 2 º − sin ¹ 2 º and A has eigenvalues 1 and −1 with eigenvectors θ and θ sin ¹ 2 º cos ¹ 2 º respectively.
Load more

Recommended publications
  • COXETER GROUPS (Unfinished and Comments Are Welcome)
    COXETER GROUPS (Unfinished and comments are welcome) Gert Heckman Radboud University Nijmegen [email protected] October 10, 2018 1 2 Contents Preface 4 1 Regular Polytopes 7 1.1 ConvexSets............................ 7 1.2 Examples of Regular Polytopes . 12 1.3 Classification of Regular Polytopes . 16 2 Finite Reflection Groups 21 2.1 NormalizedRootSystems . 21 2.2 The Dihedral Normalized Root System . 24 2.3 TheBasisofSimpleRoots. 25 2.4 The Classification of Elliptic Coxeter Diagrams . 27 2.5 TheCoxeterElement. 35 2.6 A Dihedral Subgroup of W ................... 39 2.7 IntegralRootSystems . 42 2.8 The Poincar´eDodecahedral Space . 46 3 Invariant Theory for Reflection Groups 53 3.1 Polynomial Invariant Theory . 53 3.2 TheChevalleyTheorem . 56 3.3 Exponential Invariant Theory . 60 4 Coxeter Groups 65 4.1 Generators and Relations . 65 4.2 TheTitsTheorem ........................ 69 4.3 The Dual Geometric Representation . 74 4.4 The Classification of Some Coxeter Diagrams . 77 4.5 AffineReflectionGroups. 86 4.6 Crystallography. .. .. .. .. .. .. .. 92 5 Hyperbolic Reflection Groups 97 5.1 HyperbolicSpace......................... 97 5.2 Hyperbolic Coxeter Groups . 100 5.3 Examples of Hyperbolic Coxeter Diagrams . 108 5.4 Hyperbolic reflection groups . 114 5.5 Lorentzian Lattices . 116 3 6 The Leech Lattice 125 6.1 ModularForms ..........................125 6.2 ATheoremofVenkov . 129 6.3 The Classification of Niemeier Lattices . 132 6.4 The Existence of the Leech Lattice . 133 6.5 ATheoremofConway . 135 6.6 TheCoveringRadiusofΛ . 137 6.7 Uniqueness of the Leech Lattice . 140 4 Preface Finite reflection groups are a central subject in mathematics with a long and rich history. The group of symmetries of a regular m-gon in the plane, that is the convex hull in the complex plane of the mth roots of unity, is the dihedral group of order 2m, which is the simplest example of a reflection Dm group.
    [Show full text]
  • Binary Icosahedral Group and 600-Cell
    Article Binary Icosahedral Group and 600-Cell Jihyun Choi and Jae-Hyouk Lee * Department of Mathematics, Ewha Womans University 52, Ewhayeodae-gil, Seodaemun-gu, Seoul 03760, Korea; [email protected] * Correspondence: [email protected]; Tel.: +82-2-3277-3346 Received: 10 July 2018; Accepted: 26 July 2018; Published: 7 August 2018 Abstract: In this article, we have an explicit description of the binary isosahedral group as a 600-cell. We introduce a method to construct binary polyhedral groups as a subset of quaternions H via spin map of SO(3). In addition, we show that the binary icosahedral group in H is the set of vertices of a 600-cell by applying the Coxeter–Dynkin diagram of H4. Keywords: binary polyhedral group; icosahedron; dodecahedron; 600-cell MSC: 52B10, 52B11, 52B15 1. Introduction The classification of finite subgroups in SLn(C) derives attention from various research areas in mathematics. Especially when n = 2, it is related to McKay correspondence and ADE singularity theory [1]. The list of finite subgroups of SL2(C) consists of cyclic groups (Zn), binary dihedral groups corresponded to the symmetry group of regular 2n-gons, and binary polyhedral groups related to regular polyhedra. These are related to the classification of regular polyhedrons known as Platonic solids. There are five platonic solids (tetrahedron, cubic, octahedron, dodecahedron, icosahedron), but, as a regular polyhedron and its dual polyhedron are associated with the same symmetry groups, there are only three binary polyhedral groups(binary tetrahedral group 2T, binary octahedral group 2O, binary icosahedral group 2I) related to regular polyhedrons.
    [Show full text]
  • Computing Invariants of Hyperbolic Coxeter Groups
    LMS J. Comput. Math. 18 (1) (2015) 754{773 C 2015 Author doi:10.1112/S1461157015000273 CoxIter { Computing invariants of hyperbolic Coxeter groups R. Guglielmetti Abstract CoxIter is a computer program designed to compute invariants of hyperbolic Coxeter groups. Given such a group, the program determines whether it is cocompact or of finite covolume, whether it is arithmetic in the non-cocompact case, and whether it provides the Euler characteristic and the combinatorial structure of the associated fundamental polyhedron. The aim of this paper is to present the theoretical background for the program. The source code is available online as supplementary material with the published article and on the author's website (http://coxiter.rgug.ch). Supplementarymaterialsareavailablewiththisarticle. Introduction Let Hn be the hyperbolic n-space, and let Isom Hn be the group of isometries of Hn. For a given discrete hyperbolic Coxeter group Γ < Isom Hn and its associated fundamental polyhedron P ⊂ Hn, we are interested in geometrical and combinatorial properties of P . We want to know whether P is compact, has finite volume and, if the answer is yes, what its volume is. We also want to find the combinatorial structure of P , namely, the number of vertices, edges, 2-faces, and so on. Finally, it is interesting to find out whether Γ is arithmetic, that is, if Γ is commensurable to the reflection group of the automorphism group of a quadratic form of signature (n; 1). Most of these questions can be answered by studying finite and affine subgroups of Γ, but this involves a huge number of computations.
    [Show full text]
  • Petrie Schemes
    Canad. J. Math. Vol. 57 (4), 2005 pp. 844–870 Petrie Schemes Gordon Williams Abstract. Petrie polygons, especially as they arise in the study of regular polytopes and Coxeter groups, have been studied by geometers and group theorists since the early part of the twentieth century. An open question is the determination of which polyhedra possess Petrie polygons that are simple closed curves. The current work explores combinatorial structures in abstract polytopes, called Petrie schemes, that generalize the notion of a Petrie polygon. It is established that all of the regular convex polytopes and honeycombs in Euclidean spaces, as well as all of the Grunbaum–Dress¨ polyhedra, pos- sess Petrie schemes that are not self-intersecting and thus have Petrie polygons that are simple closed curves. Partial results are obtained for several other classes of less symmetric polytopes. 1 Introduction Historically, polyhedra have been conceived of either as closed surfaces (usually topo- logical spheres) made up of planar polygons joined edge to edge or as solids enclosed by such a surface. In recent times, mathematicians have considered polyhedra to be convex polytopes, simplicial spheres, or combinatorial structures such as abstract polytopes or incidence complexes. A Petrie polygon of a polyhedron is a sequence of edges of the polyhedron where any two consecutive elements of the sequence have a vertex and face in common, but no three consecutive edges share a commonface. For the regular polyhedra, the Petrie polygons form the equatorial skew polygons. Petrie polygons may be defined analogously for polytopes as well. Petrie polygons have been very useful in the study of polyhedra and polytopes, especially regular polytopes.
    [Show full text]
  • Arxiv:1705.01294V1
    Branes and Polytopes Luca Romano email address: [email protected] ABSTRACT We investigate the hierarchies of half-supersymmetric branes in maximal supergravity theories. By studying the action of the Weyl group of the U-duality group of maximal supergravities we discover a set of universal algebraic rules describing the number of independent 1/2-BPS p-branes, rank by rank, in any dimension. We show that these relations describe the symmetries of certain families of uniform polytopes. This induces a correspondence between half-supersymmetric branes and vertices of opportune uniform polytopes. We show that half-supersymmetric 0-, 1- and 2-branes are in correspondence with the vertices of the k21, 2k1 and 1k2 families of uniform polytopes, respectively, while 3-branes correspond to the vertices of the rectified version of the 2k1 family. For 4-branes and higher rank solutions we find a general behavior. The interpretation of half- supersymmetric solutions as vertices of uniform polytopes reveals some intriguing aspects. One of the most relevant is a triality relation between 0-, 1- and 2-branes. arXiv:1705.01294v1 [hep-th] 3 May 2017 Contents Introduction 2 1 Coxeter Group and Weyl Group 3 1.1 WeylGroup........................................ 6 2 Branes in E11 7 3 Algebraic Structures Behind Half-Supersymmetric Branes 12 4 Branes ad Polytopes 15 Conclusions 27 A Polytopes 30 B Petrie Polygons 30 1 Introduction Since their discovery branes gained a prominent role in the analysis of M-theories and du- alities [1]. One of the most important class of branes consists in Dirichlet branes, or D-branes. D-branes appear in string theory as boundary terms for open strings with mixed Dirichlet-Neumann boundary conditions and, due to their tension, scaling with a negative power of the string cou- pling constant, they are non-perturbative objects [2].
    [Show full text]
  • Regular Polyhedra Through Time
    Fields Institute I. Hubard Polytopes, Maps and their Symmetries September 2011 Regular polyhedra through time The greeks were the first to study the symmetries of polyhedra. Euclid, in his Elements showed that there are only five regular solids (that can be seen in Figure 1). In this context, a polyhe- dron is regular if all its polygons are regular and equal, and you can find the same number of them at each vertex. Figure 1: Platonic Solids. It is until 1619 that Kepler finds other two regular polyhedra: the great dodecahedron and the great icosahedron (on Figure 2. To do so, he allows \false" vertices and intersection of the (convex) faces of the polyhedra at points that are not vertices of the polyhedron, just as the I. Hubard Polytopes, Maps and their Symmetries Page 1 Figure 2: Kepler polyhedra. 1619. pentagram allows intersection of edges at points that are not vertices of the polygon. In this way, the vertex-figure of these two polyhedra are pentagrams (see Figure 3). Figure 3: A regular convex pentagon and a pentagram, also regular! In 1809 Poinsot re-discover Kepler's polyhedra, and discovers its duals: the small stellated dodecahedron and the great stellated dodecahedron (that are shown in Figure 4). The faces of such duals are pentagrams, and are organized on a \convex" way around each vertex. Figure 4: The other two Kepler-Poinsot polyhedra. 1809. A couple of years later Cauchy showed that these are the only four regular \star" polyhedra. We note that the convex hull of the great dodecahedron, great icosahedron and small stellated dodecahedron is the icosahedron, while the convex hull of the great stellated dodecahedron is the dodecahedron.
    [Show full text]
  • Platonic Solids Generate Their Four-Dimensional Analogues
    This is a repository copy of Platonic solids generate their four-dimensional analogues. White Rose Research Online URL for this paper: https://eprints.whiterose.ac.uk/85590/ Version: Accepted Version Article: Dechant, Pierre-Philippe orcid.org/0000-0002-4694-4010 (2013) Platonic solids generate their four-dimensional analogues. Acta Crystallographica Section A : Foundations of Crystallography. pp. 592-602. ISSN 1600-5724 https://doi.org/10.1107/S0108767313021442 Reuse Items deposited in White Rose Research Online are protected by copyright, with all rights reserved unless indicated otherwise. They may be downloaded and/or printed for private study, or other acts as permitted by national copyright laws. The publisher or other rights holders may allow further reproduction and re-use of the full text version. This is indicated by the licence information on the White Rose Research Online record for the item. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing [email protected] including the URL of the record and the reason for the withdrawal request. [email protected] https://eprints.whiterose.ac.uk/ 1 Platonic solids generate their four-dimensional analogues PIERRE-PHILIPPE DECHANT a,b,c* aInstitute for Particle Physics Phenomenology, Ogden Centre for Fundamental Physics, Department of Physics, University of Durham, South Road, Durham, DH1 3LE, United Kingdom, bPhysics Department, Arizona State University, Tempe, AZ 85287-1604, United States, and cMathematics Department, University of York, Heslington, York, YO10 5GG, United Kingdom. E-mail: [email protected] Polytopes; Platonic Solids; 4-dimensional geometry; Clifford algebras; Spinors; Coxeter groups; Root systems; Quaternions; Representations; Symmetries; Trinities; McKay correspondence Abstract In this paper, we show how regular convex 4-polytopes – the analogues of the Platonic solids in four dimensions – can be constructed from three-dimensional considerations concerning the Platonic solids alone.
    [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]
  • Introduction to Zome a New Language for Understanding the Structure of Space
    Introduction to Zome A new language for understanding the structure of space Paul Hildebrandt Abstract This paper addresses some basic mathematical principles underlying Zome and the importance of numeracy in education. Zome geometry Zome is a new language for understanding the structure of space. In other words, Zome is “hands-on” math. Math has been called the “queen of the sciences,” and Zome’s simple elegance applies to every discipline represented at the Form conference. Zome balls and struts represent points and lines, and are designed to model the relationship between the numbers 2, 3, and 5 in space in a simple, intuitive way. There’s a relationship between the shape of a strut, its vector in space, its length and the number it represents. Shape – Each Zome strut represents a number. Notice that the blue strut has a cross-section of a golden rectangle -- the long sides are in the Divine Proportion to the short sides, i.e. if the length of the short side is 1 the long side is approximately 1.618. The rectangle has 2 short sides and 2 long sides, has 2-fold symmetry in 2 directions. Everything about the rectangle seems to be related to the number 2. If the blue strut represents the number 2, it makes sense that you can only build a square out of blue struts in Zome: the square has 2x2 edges, 2x2 corners and 2-fold symmetry along 2x2 axes. It seems that everything about the square is related to the number 2. What about the cube? Vector -- It seems logical that you must also build a cube with blue struts, since a cube is made out of squares.
    [Show full text]
  • Tetrahedral Coxeter Groups, Large Group-Actions on 3-Manifolds And
    Tetrahedral Coxeter groups, large group-actions on 3-manifolds and equivariant Heegaard splittings Bruno P. Zimmermann Universit`adegli Studi di Trieste Dipartimento di Matematica e Geoscienze 34127 Trieste, Italy Abstract. We consider finite group-actions on closed, orientable and nonorientable 3-manifolds M which preserve the two handlebodies of a Heegaard splitting of M of some genus g > 1 (maybe interchanging the two handlebodies). The maximal possible order of a finite group-action on a handlebody of genus g > 1 is 12(g − 1) in the orientation-preserving case and 24(g − 1) in general, and the maximal order of a finite group preserving the Heegaard surface of a Heegaard splitting of genus g is 48(g − 1). This defines a hierarchy for finite group-actions on 3-manifolds which we discuss in the present paper; we present various manifolds with an action of type 48(g − 1) for small values of g, and in particular the unique hyperbolic 3-manifold with such an action of smallest possible genus g = 6 (in strong analogy with the Euclidean case of the 3-torus which has such actions for g = 3). 1. A hierarchy for large finite group-actions on 3-manifolds The maximal possible order of a finite group G of orientation-preserving diffeomor- phisms of an orientable handlebody of genus g > 1 is 12(g − 1) ([Z1], [MMZ]); for orientation-reversing finite group-actions on an orientable handlebody and for actions on a nonorientable handlebody, the maximal possible order is 24(g − 1); we will always assume g > 1 in the present paper.
    [Show full text]
  • Zome and Euler's Theorem Julia Robinson Day Hosted by Google Tom Davis [email protected] April 23, 2008
    Zome and Euler's Theorem Julia Robinson Day Hosted by Google Tom Davis [email protected] http://www.geometer.org/mathcircles April 23, 2008 1 Introduction The Zome system is a construction set based on a set of plastic struts and balls that can be attached together to form an amazing set of mathematically or artistically interesting structures. For information on Zome and for an on-line way to order kits or parts, see: http://www.zometool.com The main Zome strut colors are red, yellow and blue and most of what we’ll cover here will use those as examples. There are green struts that are necessary for building structures with regular tetrahedrons and octahedrons, and almost everything we say about the red, yellow and blue struts will apply to the green ones. The green ones are a little harder to work with (both physically and mathematically) because they have a pentagon-shaped head, but can fit into any pentagonal hole in five different orientations. With the regular red, yellow and blue struts there is only one way to insert a strut into a Zome ball hole. The blue- green struts are not really part of the Zome geometry (because they have the “wrong” length). They are necessary for building a few of the Archimedian solids, like the rhombicuboctahedron and the truncated cuboctahedron. For these exercises, we recommend that students who have not worked with the Zome system before restrict themselves to the blue, yellow and red struts. Students with some prior experience, or who seem to be very fast learners, can try building things (like a regular octahedron or a regular tetrahedron) with some green struts.
    [Show full text]
  • Lesson Plans 1.1
    Lesson Plans 1.1 © 2009 Zometool, Inc. Zometool is a registered trademark of Zometool Inc. • 1-888-966-3386 • www.zometool.com US Patents RE 33,785; 6,840,699 B2. Based on the 31-zone system, discovered by Steve Baer, Zomeworks Corp., USA www zometool ® com Zome System Table of Contents Builds Genius! Zome System in the Classroom Subjects Addressed by Zome System . .5 Organization of Plans . 5 Graphics . .5 Standards and Assessment . .6 Preparing for Class . .6 Discovery Learning with Zome System . .7 Cooperative vs Individual Learning . .7 Working with Diverse Student Groups . .7 Additional Resources . .8 Caring for your Zome System Kit . .8 Basic Concept Plans Geometric Shapes . .9 Geometry Is All Around Us . .11 2-D Polygons . .13 Animal Form . .15 Attention!…Angles . .17 Squares and Rectangles . .19 Try the Triangles . .21 Similar Triangles . .23 The Equilateral Triangle . .25 2-D and 3-D Shapes . .27 Shape and Number . .29 What Is Reflection Symmetry? . .33 Multiple Reflection Symmetries . .35 Rotational Symmetry . .39 What is Perimeter? . .41 Perimeter Puzzles . .43 What Is Area? . .45 Volume For Beginners . .47 Beginning Bubbles . .49 Rhombi . .53 What Are Quadrilaterals? . .55 Trying Tessellations . .57 Tilings with Quadrilaterals . .59 Triangle Tiles - I . .61 Translational Symmetries in Tilings . .63 Spinners . .65 Printing with Zome System . .67 © 2002 by Zometool, Inc. All rights reserved. 1 Table of Contents Zome System Builds Genius! Printing Cubes and Pyramids . .69 Picasso and Math . .73 Speed Lines! . .75 Squashing Shapes . .77 Cubes - I . .79 Cubes - II . .83 Cubes - III . .87 Cubes - IV . .89 Even and Odd Numbers . .91 Intermediate Concept Plans Descriptive Writing .
    [Show full text]