ISBN-10:
1852332700
ISBN-13:
9781852332709
Pub. Date:
02/11/2004
Publisher:
Springer London
Symmetries / Edition 1

Symmetries / Edition 1

by D.L. Johnson

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Product Details

ISBN-13: 9781852332709
Publisher: Springer London
Publication date: 02/11/2004
Series: Springer Undergraduate Mathematics Series
Edition description: 1st ed. 2001. Corr. 2nd printing 2002
Pages: 198
Product dimensions: 6.10(w) x 9.25(h) x 0.02(d)

Table of Contents

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.

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