The main object of study for this book is geometry, with group theory providing an appropriate language in which to express geometrical ideas. Key features include: An overview of the preliminaries from group theory and geometry Coverage of the discrete subgroups of the Euclidean group A clear and complete derivation and classification of the 17 plane crystallographic groups Tessellations of various spaces (they are constructed, described and classified) A brief introduction to hyperbolic geometry. Each chapter contains a number of exercises, most with solutions, and suggestions for background, alternative and further reading. The author's accessible and down-to-earth approach make this an ideal introduction for readers in the second or third year of a mathematics undergraduate course. It is also recommended for mechanical engineers, architects, physicists and crystallographers needing an understanding of 3-dimensional geometry, symmetry and trigonometry.
Author(s): D.L. Johnson
Series: Springer Undergraduate Mathematics Series
Edition: Corrected
Publisher: Springer
Year: 2004
Language: English
Pages: 216
Cover......Page 1
Symmetries......Page 2
Copyright......Page 3
Preface......Page 6
Contents......Page 8
1.1 Metric Spaces......Page 12
1.2 Isometries......Page 15
1.3 Isometries of the Real Line......Page 16
1.4 Matters Arising......Page 18
1.5 Symmetry Groups......Page 21
2.1 Congruent Triangles......Page 26
2.2 Isometries of Different Types......Page 29
2.3 The Normal Form Theorem......Page 31
2.4 Conjugation of Isometries......Page 32
3. Some Basic Group Theory......Page 38
3.1 Groups......Page 39
3.2 Subgroups......Page 41
3.3 Factor Groups......Page 44
3.4 Semidirect Products......Page 47
4.1 The Product of Two Reflections......Page 56
4.2 Three Reflections......Page 58
4.3 Four or More......Page 61
5. Generators and Relations......Page 66
5.1 Examples......Page 67
5.2 Semidirect Products Again......Page 71
5.3 Change of Presentation......Page 76
5.4 Triangle Groups......Page 80
5.5 Abelian Groups......Page 81
6. Discrete Subgroups of the Euclidean Group......Page 90
6.1 Leonardo's Theorem......Page 91
6.2 A Trichotomy......Page 92
6.3 Friezes and Their Groups......Page 94
6.4 The Classification......Page 96
7.1 The Crystallographic Restriction......Page 100
7.2 The Parameter n......Page 102
7.3 The Choice of b......Page 103
7.4 Conclusion......Page 105
8.1 A Useful Dichotomy......Page 108
8.3 The Case n = 2......Page 111
8.4 The Case n = 4......Page 112
8.5 The Case n = 3......Page 113
8.6 The Case n = 6......Page 115
9.1 Regular Tessellations......Page 118
9.2 Descendants of (4, 4)......Page 121
9.3 Bricks......Page 123
9.4 Split Bricks......Page 124
9.5 Descendants of (3, 6)......Page 127
10.1 Spherical Geometry......Page 134
10.2 The Spherical Excess......Page 136
10.3 Tessellations of the Sphere......Page 139
10.4 The Platonic Solids......Page 141
10.5 Symmetry Groups......Page 144
11. Triangle Groups......Page 150
11.1 The Euclidean Case......Page 151
11.2 The Elliptic Case......Page 153
11.3 The Hyperbolic Case......Page 155
11.4 Coxeter Groups......Page 157
12. Regular Polytopes......Page 166
12.1 The Standard Examples......Page 167
12.2 The Exceptional Types in Dimension Four......Page 169
12.3 Three Concepts and a Theorem......Page 171
12.4 Schlafli's Theorem......Page 174
Solutions......Page 178
Guide to the Literature......Page 198
Bibliography......Page 200
Index of Notation......Page 202
Index......Page 206
