Transformation Geometry: An Introduction to Symmetry offers a modern approach to Euclidean Geometry. This study of the automorphism groups of the plane and space gives the classical concrete examples that serve as a meaningful preparation for the standard undergraduate course in abstract algebra. The detailed development of the isometries of the plane is based on only the most elementary geometry and is appropriate for graduate courses for secondary teachers.

Inhalt

1 Introduction.- 1.1 Transformations and Collineations.- 1.2 Geometric Notation.- 1.3 Exercises.- 2 Properties of Transformations.- 2.1 Groups of Transformations.- 2.2 Involutions.- 2.3 Exercises.- 3 Translations and Halfturns.- 3.1 Translations.- 3.2 Halfturns.- 3.3 Exercises.- 4 Reflections.- 4.1 Equations for a Reflection.- 4.2 Properties of a Reflection.- 4.3 Exercises.- 5 Congruence.- 5.1 Isometries as Products of Reflections.- 5.2 Paper Folding Experiments and Rotations.- 5.3 Exercises.- 6 The Product of Two Reflections.- 6.1 Translations and Rotations.- 6.2 Fixed Points and Involutions.- 6.3 Exercises.- 7 Even Isometries.- 7.1 Parity.- 7.2 The Dihedral Groups.- 7.3 Exercises.- 8 Classification of Plane Isometries.- 8.1 Glide Reflections.- 8.2 Leonardo's Theorem.- 8.3 Exercises.- 9 Equations for Isometries.- 9.1 Equations.- 9.2 Supplementary Exercises (Chapter 1-8).- 9.3 Exercises.- 10 The Seven Frieze Groups.- 10.1 Frieze Groups.- 10.2 Frieze Patterns.- 10.3 Exercises.- 11 The Seventeen Wallpaper Groups.- 11.1 The Crystallographic Restriction.- 11.2 Wallpaper Groups and Patterns.- 11.3 Exercises.- 12 Tessellations.- 12.1 Tiles.- 12.2 Reptiles.- 12.3 Exercises.- 13 Similarities on the Plane.- 13.1 Classification of Similarities.- 13.2 Equations for Similarities.- 13.3 Exercises.- 14 Classical Theorems.- 14.1 Menelaus, Ceva, Desargues, Pappus, Pascal.- 14.2 Euler, Brianchon, Poncelet, Feuerbach.- 14.3 Exercises.- 15 Affine Transformations.- 15.1 Collineations.- 15.2 Linear Transformations.- 15.3 Exercises.- 16 Transformations on Three-space.- 16.1 Isometries on Space.- 16.2 Similarities on Space.- 16.3 Exercises.- 17 Space and Symmetry.- 17.1 The Platonic Solids.- 17.2 Finite Symmetry Groups on Space.- 17.3 Exercises.- Hints and Answers.- Notation Index.