Symmetries

This is a popular topic so there are many textbooks in this area. However, this book is written in a style that is down-to-earth and fun to read so it will also appeal to those such as mechanical engineers, architects, physicists etc. who need to understand three-dimensional geometry, symmetry and trigonometry, but who are put off by the abstract treatment usual in other textbooks.

Includes the derivation and classification of the 17 plane crystallographic groups (i. e. the 17 possible basic wallpaper patterns) Contains numerous exercises, most with solutions, and suggestions for background, alternative and further reading

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.