## Transformation Geometry: An Introduction to SymmetryTransformation 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 |

231 | |

233 | |

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### Common terms and phrases

2-center AABC affine transformation Algebra Archimedean solid Cartesian plane Cayley table center of symmetry Ceva's Theorem Chapter collineation concurrent congruent cube cyclic group directed angle elements equilateral Euler Exercises Figure finite group fixed point following theorem form a group frieze group frieze pattern geometry glide reflection group of isometries H H H halfturn Hence hexagon identity images integer inverse involution involutory isometry that fixes line of symmetry line with equation mathematics midpoint monohedral tiling n-gon ninepoint circle nonidentity rotation nonidentity translation odd isometries parallel lines parallelogram pattern having symmetry perpendicular bisector Platonic solids point of symmetry point Q polyhedron prototile Prove or disprove reflection with axis regular polygons rotary reflection rotation group set of points sides similarity square stretch Suppose symmetry group tessellation tetrahedron three noncollinear points three reflections tiles the plane triangle unique unit cell vertex vertices wallpaper group wallpaper pattern