Symmetries

" ... many eminent scholars, endowed with great geometric talent, make a point of never disclosing the simple and direct ideas that guided them, subordinating their elegant results to abstract general theories which often have no application outside the particular case in question. Geometry was becoming a study of algebraic, differential or partial differential equations, thus losing all the charm that comes from its being an art." H. Lebesgue, Ler;ons sur les Constructions Geometriques, Gauthier­ Villars, Paris, 1949. This book is based on lecture courses given to final-year students at the Uni­ versity of Nottingham and to M.Sc. students at the University of the West Indies in an attempt to reverse the process of expurgation of the geometry component from the mathematics curricula of universities. This erosion is in sharp contrast to the situation in research mathematics, where the ideas and methods of geometry enjoy ever-increasing influence and importance. In the other direction, more modern ideas have made a forceful and beneficial impact on the geometry of the ancients in many areas. Thus trigonometry has vastly clarified our concept of angle, calculus has revolutionised the study of plane curves, and group theory has become the language of symmetry.

Contenu

1 Metric Spaces and their Groups.- 1.1 Metric Spaces.- 1.2 Isometries.- 1.3 Isometries of the Real Line.- 1.4 Matters Arising.- 1.5 Symmetry Groups.- 2 Isometries of the Plane.- 2.1 Congruent Triangles.- 2.2 Isometries of Different Types.- 2.3 The Normal Form Theorem.- 2.4 Conjugation of Isometries.- 3 Some Basic Group Theory.- 3.1 Groups.- 3.2 Subgroups.- 3.3 Factor Groups.- 3.4 Semidirect Products.- 4 Products of Reflections.- 4.1 The Product of Two Reflections.- 4.2 Three Reflections.- 4.3 Four or More.- 5 Generators and Relations.- 5.1 Examples.- 5.2 Semidirect Products Again.- 5.3 Change of Presentation.- 5.4 Triangle Groups.- 5.5 Abelian Groups.- 6 Discrete Subgroups of the Euclidean Group.- 6.1 Leonardo's Theorem.- 6.2 A Trichotomy.- 6.3 Friezes and Their Groups.- 6.4 The Classification.- 7 Plane Crystallographic Groups: OP Case.- 7.1 The Crystallographic Restriction.- 7.2 The Parameter n.- 7.3 The Choice of b.- 7.4 Conclusion.- 8 Plane Crystallographic Groups: OR Case.- 8.1 A Useful Dichotomy.- 8.2 The Case n = 1.- 8.3 The Case n = 2.- 8.4 The Case n = 4.- 8.5 The Case n = 3.- 8.6 The Case n = 6.- 9 Tessellations of the Plane.- 9.1 Regular Tessellations.- 9.2 Descendants of (4, 4).- 9.3 Bricks.- 9.4 Split Bricks.- 9.5 Descendants of (3, 6).- 10 Tessellations of the Sphere.- 10.1 Spherical Geometry.- 10.2 The Spherical Excess.- 10.3 Tessellations of the Sphere.- 10.4 The Platonic Solids.- 10.5 Symmetry Groups.- 11 Triangle Groups.- 11.1 The Euclidean Case.- 11.2 The Elliptic Case.- 11.3 The Hyperbolic Case.- 11.4 Coxeter Groups.- 12 Regular Polytopes.- 12.1 The Standard Examples.- 12.2 The Exceptional Types in Dimension Four.- 12.3 Three Concepts and a Theorem.- 12.4 Schläfli's Theorem.- Solutions.- Guide to the Literature.- Index of Notation.