Author | Johnson, D. L. author |
---|---|
Title | Symmetries [electronic resource] / by D. L. Johnson |
Imprint | London : Springer London : Imprint: Springer, 2001 |
Connect to | http://dx.doi.org/10.1007/978-1-4471-0243-4 |
Descript | XI, 198 p. online resource |
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 Leonardoโs 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 Schlรคfliโs Theorem -- Solutions -- Guide to the Literature -- Index of Notation