# Transformation Geometry: An Introduction to Symmetry

Springer Science & Business Media, Dec 20, 1996 - Mathematics - 240 pages
Transformation geometry is a relatively recent expression of the successful venture of bringing together geometry and algebra. The name describes an approach as much as the content. Our subject is Euclidean geometry. Essential to the study of the plane or any mathematical system is an under standing of the transformations on that system that preserve designated features of the system. Our study of the automorphisms of the plane and of space is based on only the most elementary high-school geometry. In particular, group theory is not a prerequisite here. On the contrary, this modern approach to Euclidean geometry gives the concrete examples that are necessary to appreciate an introduction to group theory. Therefore, a course based on this text is an excellent prerequisite to the standard course in abstract algebra taken by every undergraduate mathematics major. An advantage of having nb college mathematics prerequisite to our study is that the text is then useful for graduate mathematics courses designed for secondary teachers. Many of the students in these classes either have never taken linear algebra or else have taken it too long ago to recall even the basic ideas. It turns out that very little is lost here by not assuming linear algebra. A preliminary version of the text was written for and used in two courses-one was a graduate course for teachers and the other a sophomore course designed for the prospective teacher and the general mathematics major taking one course in geometry.

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### Contents

 Introduction 1 12 Geometric Notation 3 13 Exercises 5 Properties of Transformations 7 22 Involutions 9 23 Exercises 12 Translations and Halfturns 14 32 Halfturns 17
 102 Frieze Patterns 82 103 Exercises 85 The Seventeen Wallpaper Groups 88 112 Wallpaper Groups and Patterns 92 113 Exercises 111 Tessellations 117 122 Reptiles 126 123 Exercises 132

 33 Exercises 20 Reflections 23 42 Properties of a Reflection 26 43 Exercises 30 Congruence 33 52 Paper Folding Experiments and Rotations 36 53 Exercises 40 The Product of Two Reflections 43 62 Fixed Points and Involutions 47 63 Exercises 50 Even Isometries 52 72 The Dihedral Groups 57 73 Exercises 60 Classification of Plane Isometries 62 82 Leonardos Theorem 66 83 Exercises 68 Equations for Isometries 71 92 Supplementary Exercises Chapters 18 73 93 Exercises 76 The Seven Frieze Groups 78
 Similarities on the Plane 136 132 Equations for Similarities 141 133 Exercises 144 Classical Theorems 147 142 Euler Brianchon Poncelet Feuerbach 156 143 Exercises 164 Affine Transformations 167 152 Linear Transformations 175 153 Exercises 180 Transformations on Threespace 182 162 Similarities on Space 194 163 Exercises 196 Space and Symmetry 198 172 Finite Symmetry Groups on Space 211 173 Exercises 222 Hints and Answers 225 Notation Index 231 Index 233 Copyright