Using Geometry to Teach Group Theory
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Frieze Groups Tara Mccabe Denis Mortell
Frieze Groups Tara McCabe Denis Mortell Introduction Our chosen topic is frieze groups. A frieze group is an infinite discrete group of symmetries, i.e. the set of the geometric transformations that are composed of rigid motions and reflections that preserve the pattern. A frieze pattern is a two-dimensional design that repeats in one direction. A study on these groups was published in 1969 by Coxeter and Conway. Although they had been studied by Gauss prior to this, Gauss did not publish his studies. Instead, they were found in his private notes years after his death. Visual Perspective Numerical Perspective What second-row numbers validate a frieze pattern? (F) There are seven types of groups. Each of these are the A frieze pattern is an array of natural numbers, displayed in a symmetry group of a frieze pattern. They have been lattice such that the top and bottom lines are composed only of The second-row numbers that determine the pattern are not illustrated below using the Conway notation and 1’s and for each unit diamond, random. Instead, these numbers have a peculiar property. corresponding diagrams: the rule (bc) − (ad) = 1 holds. Theorem I Fifth being a translation I The first, and the one There is a bijection between the valid which must occur in with vertical reflection, Here is an example of a frieze pattern: sequences for frieze patterns and all others is glide reflection and 180◦ rotation, or a the number of triangles adjacent to the Translation, or a vertices of a triangulated polygon. [2] ‘hop’. -
[Math.GT] 9 Oct 2020 Symmetries of Hyperbolic 4-Manifolds
Symmetries of hyperbolic 4-manifolds Alexander Kolpakov & Leone Slavich R´esum´e Pour chaque groupe G fini, nous construisons des premiers exemples explicites de 4- vari´et´es non-compactes compl`etes arithm´etiques hyperboliques M, `avolume fini, telles M G + M G que Isom ∼= , ou Isom ∼= . Pour y parvenir, nous utilisons essentiellement la g´eom´etrie de poly`edres de Coxeter dans l’espace hyperbolique en dimension quatre, et aussi la combinatoire de complexes simpliciaux. C¸a nous permet d’obtenir une borne sup´erieure universelle pour le volume minimal d’une 4-vari´et´ehyperbolique ayant le groupe G comme son groupe d’isom´etries, par rap- port de l’ordre du groupe. Nous obtenons aussi des bornes asymptotiques pour le taux de croissance, par rapport du volume, du nombre de 4-vari´et´es hyperboliques ayant G comme le groupe d’isom´etries. Abstract In this paper, for each finite group G, we construct the first explicit examples of non- M M compact complete finite-volume arithmetic hyperbolic 4-manifolds such that Isom ∼= G + M G , or Isom ∼= . In order to do so, we use essentially the geometry of Coxeter polytopes in the hyperbolic 4-space, on one hand, and the combinatorics of simplicial complexes, on the other. This allows us to obtain a universal upper bound on the minimal volume of a hyperbolic 4-manifold realising a given finite group G as its isometry group in terms of the order of the group. We also obtain asymptotic bounds for the growth rate, with respect to volume, of the number of hyperbolic 4-manifolds having a finite group G as their isometry group. -
Transformation Geometry — Math 331
Transformation Geometry — Math 331 March 8, 2004 Conjugation of Isometries Definition. If f and g are invertible affine transformations of Rn, the conjugate of f by g is the transformation g ◦ f ◦ g−1. When the context is clear, one sometimes abbreviates notation for conjugation by writing g · f = g ◦ f ◦ g−1 . One views this as g “acting on” f. That is, an affine transformation of Rn not only describes a motion of Rn but also a motion on the set of affine transformations of Rn: the action of the set of affine transformations on itself by conjugation. Theorem 1. If f(x) = Ux + v is an affine transformation of Rn and τ(x) = x + a is translation by the vector a, then f · τ is translation by the vector Ua. Proof. f −1(x) = U −1x − U −1v, and, therefore y = τ ◦ f −1(x) = U −1x − U −1v + a. Then (f · τ)(x) = Uy + v = x + Ua. Warning. Although when a translation is conjugated by an arbitrary affine transformation, the result is a translation, it is not true that the result of conjugating an isometry by an affine transformation is always an isometry. For example, if f is the simple linear transformation (x, y) 7→ (2x, y) of R2 and ρ is rotation through the angle π/2 about the origin, i.e., (x, y) 7→ (−y, x), then f · ρ is the linear map (x, y) 7→ (−2y, x/2), which is not an isometry. On the other hand, it is clear that the result of conjugating an isometry by an isometry is an isometry. -
Sorting by Symmetry
Sorting by Symmetry Patterns along a Line Bob Burn Sorting patterns by Symmetry: Patterns along a Line 1 Sorting patterns by Symmetry: Patterns along a Line Sorting by Symmetry Patterns along a Line 1 Parallel mirrors ................................................................................................... 4 2 Translations ........................................................................................................ 7 3 Friezes ................................................................................................................ 9 4 Friezes with vertical reflections ....................................................................... 11 5 Friezes with half-turns ..................................................................................... 13 6 Friezes with a horizontal reflection .................................................................. 16 7 Glide-reflection ................................................................................................ 18 8 Fixed lines ........................................................................................................ 20 9 Friezes with half-turn centres not on reflection axes ....................................... 22 10 Friezes with all possible symmetries ............................................................... 26 11 Generators ........................................................................................................ 28 12 The family of friezes ....................................................................................... -
II.3 : Measurement Axioms
I I.3 : Measurement axioms A large — probably dominant — part of elementary Euclidean geometry deals with questions about linear and angular measurements , so it is clear that we must discuss the basic properties of these concepts . Linear measurement In elementary geometry courses , linear measurement is often viewed using the lengths of line segments . We shall view it somewhat differently as giving the (shortest) distance 2 3 between two points . Not surprisingly , in RRR and RRR we define this distance by the usual Pythagorean formula already given in Section I.1. In the synthetic approach , one may view distance as a primitive concept given formally by a function d ; given two points X, Y in the plane or space , the distance d (X, Y) is assumed to be a nonnegative real number , and we further assume that distances have the following standard properties : Axiom D – 1 : The distance d (X, Y) is equal to zero if and only if X = Y. Axiom D – 2 : For all X and Y we have d (X, Y) = d (Y, X). Obviously it is also necessary to have some relationships between an abstract notion of distance and the other undefined concepts introduced thus far ; namely , lines and planes . Intuitively we expect a geometrical line to behave just like the real number line , and the following axiom formulated by G. D. Birkhoff (1884 – 1944) makes this idea precise : Axiom D – 3 ( Ruler Postulate ) : If L is an arbitrary line , then there is a 1 – 1 correspondence between the points of L and the real numbers RRR such that if the points X and Y on L correspond to the real numbers x and y respectively , then we have d (X, Y) = |x – y |. -
Sufficient Conditions for Periodicity of a Killing Vector Field Walter C
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 38, Number 3, May 1973 SUFFICIENT CONDITIONS FOR PERIODICITY OF A KILLING VECTOR FIELD WALTER C. LYNGE Abstract. Let X be a complete Killing vector field on an n- dimensional connected Riemannian manifold. Our main purpose is to show that if X has as few as n closed orbits which are located properly with respect to each other, then X must have periodic flow. Together with a known result, this implies that periodicity of the flow characterizes those complete vector fields having all orbits closed which can be Killing with respect to some Riemannian metric on a connected manifold M. We give a generalization of this characterization which applies to arbitrary complete vector fields on M. Theorem. Let X be a complete Killing vector field on a connected, n- dimensional Riemannian manifold M. Assume there are n distinct points p, Pit' " >Pn-i m M such that the respective orbits y, ylt • • • , yn_x of X through them are closed and y is nontrivial. Suppose further that each pt is joined to p by a unique minimizing geodesic and d(p,p¡) = r¡i<.D¡2, where d denotes distance on M and D is the diameter of y as a subset of M. Let_wx, • ■ ■ , wn_j be the unit vectors in TVM such that exp r¡iwi=pi, i=l, • ■ • , n— 1. Assume that the vectors Xv, wx, ■• • , wn_x span T^M. Then the flow cptof X is periodic. Proof. Fix /' in {1, 2, •••,«— 1}. We first show that the orbit yt does not lie entirely in the sphere Sj,(r¡¿) of radius r¡i about p. -
UCLA Electronic Theses and Dissertations
UCLA UCLA Electronic Theses and Dissertations Title Shapes of Finite Groups through Covering Properties and Cayley Graphs Permalink https://escholarship.org/uc/item/09b4347b Author Yang, Yilong Publication Date 2017 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California UNIVERSITY OF CALIFORNIA Los Angeles Shapes of Finite Groups through Covering Properties and Cayley Graphs A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by Yilong Yang 2017 c Copyright by Yilong Yang 2017 ABSTRACT OF THE DISSERTATION Shapes of Finite Groups through Covering Properties and Cayley Graphs by Yilong Yang Doctor of Philosophy in Mathematics University of California, Los Angeles, 2017 Professor Terence Chi-Shen Tao, Chair This thesis is concerned with some asymptotic and geometric properties of finite groups. We shall present two major works with some applications. We present the first major work in Chapter 3 and its application in Chapter 4. We shall explore the how the expansions of many conjugacy classes is related to the representations of a group, and then focus on using this to characterize quasirandom groups. Then in Chapter 4 we shall apply these results in ultraproducts of certain quasirandom groups and in the Bohr compactification of topological groups. This work is published in the Journal of Group Theory [Yan16]. We present the second major work in Chapter 5 and 6. We shall use tools from number theory, combinatorics and geometry over finite fields to obtain an improved diameter bounds of finite simple groups. We also record the implications on spectral gap and mixing time on the Cayley graphs of these groups. -
Augmented Reality Mobile Learning System: Study to Improve Psts’ Understanding of Mathematical Development
Short Paper—Augmented Reality Mobile Learning System: Study Study to Improve PSTs’… Augmented Reality Mobile Learning System: Study to Improve PSTs’ Understanding of Mathematical Development https://doi.org/10.3991/ijim.v14i09.12909 Mohammad Faizal Amir(), Novia Ariyanti, Najih Anwar Universitas Muhammadiyah Sidoarjo, Sidoarjo, Indonesia [email protected] Erik Valentino STKIP Bina Insan Mandiri Surabaya, Surabaya, Indonesia Dian Septi Nur Afifah STKIP PGRI Tulungagung, Tulungagung, Indonesia Abstract—Augmented Reality research in mobile learning systems so far have not especially to improve Preservice Students Teachers' (PSTs) geometry understanding in mathematical development. Studies conducted in this study ba- sically use a case study. This research aims to develop a mobile augmented reality system for PSTs learning so that it can be used to improve PSTs understanding of mathematics development includes doing, image-making, an image having, property noticing, formalizing, observing, structuring and inventing. In this de- velopment, PSTs can understand the understanding of the geometry transfor- mation is a translation, reflection, rotation, and dilatation. Keywords—Mobile systems, augmented reality, understanding mathematics, geometry transformation. 1 Introduction In PISA 2015 mathematics assessed as minor domains, providing the opportunity to make comparisons of student performance over time. The results of preliminary field studies, mathematical understanding of geometry Preservice Student Teachers (PSTs) 20 of 36 in Universitas Muhammadiyah Sidoarjo have difficulty in solving mathemat- ical understanding of geometry. The development of learners by understanding participants [1] includes doing, im- age-making, an image having, property noticing, formalizing, observing, structuring, and inventing, linking concepts, ideas, facts, or mathematics procedures. Results of re- search by [2], [6] shows the importance of understanding the development problems of geometry PSTs that many of the teachers or prospective elementary school teachers do iJIM ‒ Vol. -
Structure of Isometry Group of Bilinear Spaces ୋ
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Linear Algebra and its Applications 416 (2006) 414–436 www.elsevier.com/locate/laa Structure of isometry group of bilinear spaces ୋ Dragomir Ž. Ðokovic´ Department of Pure Mathematics, University of Waterloo, Waterloo, Ont., Canada N2L 3G1 Received 5 November 2004; accepted 27 November 2005 Available online 18 January 2006 Submitted by H. Schneider Abstract We describe the structure of the isometry group G of a finite-dimensional bilinear space over an algebra- ically closed field of characteristic not two. If the space has no indecomposable degenerate orthogonal summands of odd dimension, it admits a canonical orthogonal decomposition into primary components and G is isomorphic to the direct product of the isometry groups of the primary components. Each of the latter groups is shown to be isomorphic to the centralizer in some classical group of a nilpotent element in the Lie algebra of that group. In the general case, the description of G is more complicated. We show that G is a semidirect product of a normal unipotent subgroup K with another subgroup which, in its turn, is a direct product of a group of the type described in the previous paragraph and another group H which we can describe explicitly. The group H has a Levi decomposition whose Levi factor is a direct product of several general linear groups of various degrees. We obtain simple formulae for the dimensions of H and K. © 2005 Elsevier Inc. All rights reserved. Keywords: Bilinear space; Isometry group; Asymmetry; Gabriel block; Toeplitz matrix 1. -
Reflexivity of the Automorphism and Isometry Groups of the Suspension of B(H)
journal of functional analysis 159, 568586 (1998) article no. FU983325 Reflexivity of the Automorphism and Isometry Groups of the Suspension of B(H) Lajos Molnar and Mate Gyo ry Institute of Mathematics, Lajos Kossuth University, P.O. Box 12, 4010 Debrecen, Hungary E-mail: molnarlÄmath.klte.hu and gyorymÄmath.klte.hu Received November 30, 1997; revised June 18, 1998; accepted June 26, 1998 The aim of this paper is to show that the automorphism and isometry groups of the suspension of B(H), H being a separable infinite-dimensional Hilbert space, are algebraically reflexive. This means that every local automorphism, respectively, View metadata, citation and similar papers at core.ac.uk brought to you by CORE every local surjective isometry, of C0(R)B(H) is an automorphism, respectively, provided by Elsevier - Publisher Connector a surjective isometry. 1998 Academic Press Key Words: reflexivity; automorphisms; surjective isometries; suspension. 1. INTRODUCTION AND STATEMENT OF THE RESULTS The study of reflexive linear subspaces of the algebra B(H) of all bounded linear operators on the Hilbert space H represents one of the most active research areas in operator theory (see [Had] for a beautiful general view of reflexivity of this kind). In the past decade, similar questions concerning certain important sets of transformations acting on Banach algebras rather than Hilbert spaces have also attracted attention. The originators of the research in this direction are Kadison and Larson. In [Kad], Kadison studied local derivations from a von Neumann algebra R into a dual R-bimodule M. A continuous linear map from R into M is called a local derivation if it agrees with some derivation at each point (the derivations possibly differing from point to point) in the algebra. -
Symmetries in Physics WS 2018/19
Symmetries in Physics WS 2018/19 Dr. Wolfgang Unger Universit¨atBielefeld Preliminary version: January 30, 2019 2 Contents 1 Introduction 5 1.1 The Many Faces of Symmetries . .5 1.1.1 The Notion of Symmetry . .5 1.1.2 Symmetries in Nature . .5 1.1.3 Symmetries in the Arts . .5 1.1.4 Symmetries in Mathematics . .5 1.1.5 Symmetries in Physical Phenomena . .5 1.2 Symmetries of States and Invariants of Natural Laws . .5 1.2.1 Structure of Classical Physics . .5 1.2.2 Structure of Quantum Mechanics . .6 1.2.3 Role of Mathematics . .7 1.2.4 Examples . .7 2 Basics of Group Theory 11 2.1 Axioms and Definitions . 11 2.1.1 Group Axioms . 11 2.1.2 Examples: Numbers . 13 2.1.3 Examples: Permutations . 13 2.1.4 Examples: Matrix Groups . 14 2.1.5 Group Presentation . 14 2.2 Relations Between Group . 15 2.2.1 Homomorphism, Isomorphisms . 15 2.2.2 Automorphism and Conjugation . 16 2.2.3 Symmetry Group . 16 2.2.4 Subgroups and Normal Subgroups, Center . 17 2.2.5 Cosets, Quotient Groups . 20 2.2.6 Conjugacy Class . 21 2.2.7 Direct Products, Semidirect Products . 22 2.3 Finite Groups . 24 2.3.1 Cyclic Groups . 24 2.3.2 Dihedral Group . 24 2.3.3 The Symmetric Group . 25 2.3.4 The Alternating Group . 27 2.3.5 Other Permutation Groups and Wreath Product . 27 2.3.6 Classification of finite simple groups . 28 2.3.7 List of finite groups . -
1 Lecture 4 DISCRETE SUBGROUPS of the ISOMETRY GROUP OF
1 Lecture 4 DISCRETE SUBGROUPS OF THE ISOMETRY GROUP OF THE PLANE AND TILINGS This lecture, just as the previous one, deals with a classification of objects, the original interest in which was perhaps more aesthetic than scientific, and goes back many centuries ago. The objects in question are regular tilings (also called tessellations), i.e., configurations of identical figures that fill up the plane in a regular way. Each regular tiling is a geometry in the sense of Klein; it turns out that, up to isomorphism, there are 17 such geometries; their classification will be obtained by studying the corresponding transformation groups, which are discrete subgroups (see the definition in Section 4.3) of the isometry group of the Euclidean plane. 4.1. Tilings in architecture, art, and science In architecture, regular tilings appear, in particular, as decorative mosaics in the famous Alhambra palace (14th century Spain). Two of these are reproduced below in black and white (see Fig.4.1). Beautiful color photos of the Alhambra mosaics may be found at the website www.alhambra.com ??????? Figure 4.1. Two Alhambra mosaics In art, the famous Dutch artist A.Escher, famous for his “impossible” paintings, used regular tilings as the geometric basis of his wonderful “peri- odic” watercolors. Two of those are shown Fig.4.2. Color reproductions of his work appear on the website www.Escher.com. 2 Fig.4.2. Two periodic watercolors by Escher From the scientific viewpoint, not only regular tilings are important: it is possible to tile the plane by copies of one tile (or two) in an irregular (nonperiodic) way.