Transformation Geometry: An Introduction to Symmetry

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Springer Science & Business Media, Apr 13, 1982 - Mathematics - 237 pages
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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

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