Contents

1.  Metric Spaces and their Groups ............................  1

     1.1  Metric Spaces............................................  1

     -1.2-  Isometries ...............................................-4

     1.3  Isometries of the Real Line ................................  5

     1.4  Matters Arising ..........................................  7

     1.5  Symmetry Groups........................................ 10

2.  IsometriesofthePlane.....................................  15

     2.1  Congruent Triangles ...................................... 15

     2.2  IsometriesofDifferentTypes............................... 18

     2.3  The Normal Form Theorem................................ 20

     2.4  Conjugationoflsometries ................................. 21

3.  Some Basic Group Theory.................................. 27

     3.1  Groups.................................................. 28

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     3.2  Subgroups ............................................... 50

     3.3  Factor Groups ........................................... 33

     3.4  Semidirect Products ...................................... 36

4.  Products of Reflections ..................................... 45

     4.1  The Product of Two Reflections............................ 45

     4.2  Three Reflections......................................... 47

     4.3  Four or More ............................................ 50

5.  Generators and Relations................................... 55

     5.1  Examples................................................ 56

     5.2  Semidirect Products Again ................................ 60

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~                             I                    ~       ~                                                             I                I                               I              _     II    I       I    I             I                        I             II I          III          ~                          ~                           I                           I                                                      I            I                    I

     15.3  Change of Presentation.................................... 615

     ,5.4  Triangle Groups.......................................... 6[)

     15.5  Abelian Groups .......................................... 70

6.  Discrete Subgroups of the Euclidean Group ................ 7[)

     -15.-1- -Leonardo's Theorem ......................................-~Su

     6.2 ATrichotomy............................................ 81

     6.3  Friezes and Their Groups.................................. 83

     6.,1  The Classification ........................................ 815

7   Plane Crystallographic Groups' OP Case            89

    7.1  The Crystallographic Restriction ........................... 80

     7.2  TheParametern......................................... Ol

     7.3  The Choice of b .......................................... 02

     7.4  Conclusion .............................................. 04

8.  Plane Crystallographic Groups: OR Case................... 97

      -8.-1--A Useful----Dichotomy ......................................--97

     8.2  The Case n - 1 .......................................... 100

     83  The Case n - 2                              100

     8 4  The Case n - 4                              101

     8 5  The Case n - 3                              102

     8 6  The Case n - 6                              104

9.  Tessellations of the Plane................................... 107

     O.1  Regular Tessellations...................................... 107

      9.2  Descendants of (4, 4) ..................................... 110

      9.3  Bricks................................................... 112

      9.4  Split Bricks.............................................. 113

      9.5  Descendants of (3, 6) ..................................... 116

10. Tessellations of the Sphere.................................. 123

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      10.1 Spherical Geometry....................................... 123

      10.2 The Spherical Excess ..................................... 125

      -1--0.3 Tessellations of- the--Sphere.................................-1--28

      1-0.-4 The-Platonic Solids.......................................-1-~30

      10.5 Symmetry Groups........................................ 133

 11. Triangle Groups ............................................ 139

      11.1 The Euclidean Case....................................... 140

      11.2 The Elliptic Case......................................... 142

      11.3 The Hyperbolic Case...................................... 144

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     11.4 Coxeter Groups . ......................................... 146

12. Regular Polytopes.......................................... 155

     12.1 The Standard Examples................................... 156

     12.2 The Exceptional Types in Dimension Four................... 158

     12.8 Three Concepts and a Theorem ............................ 160

     12.4 Schliifli's Theorem ........................................ 1og

Solutions ....................................................... 167

Guide to the Literature......................................... 187

Biblio~'aphy.................................................... 180

Index of Notation .............................................. 1Ol

Index........................................................... 105