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 Schlafli's Theorem
Solutions
Guide to the Literature
Bibliography
Index of Notation
Index
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