The Geometric Viewpoint: A Survey of Geometries

Cover page......Page 1
Errata......Page 2
Title page......Page 4
Preface......Page 6
Contents......Page 10
1  Euclidean Geometry......Page 14
1.1  Overview and History......Page 15
1.2  Constructions, Congruence, and Parallels: Euclid\'s Approach to Geometry......Page 22
Archimedes......Page 28
1.3  A Critique of Euclid-Modern Axiomatics......Page 35
David Hiibert......Page 37
1.4  Axiomatic Systems, Models, and Metamathematics......Page 43
1.5  Similar Figures......Page 51
1.6  Three-Dimensional Geometry......Page 57
Projects for Chapter 1......Page 70
Suggested Readings......Page 75
Suggested Media......Page 76
2  Analytic Geometry......Page 77
2.1  Overview and History......Page 78
René Descartes......Page 79
2.2  Conics and Locus Problems......Page 83
Pierre Fermat......Page 85
2.3  Further Topics in Analytic Geometry......Page 90
2.4  Curves in Computer-Aided Design......Page 96
2.5  Higher Dimensional Analytic Geometry......Page 103
Gaspard Monge......Page 105
Projects for Chapter 2......Page 111
Suggested Readings......Page 113
Suggested Media......Page 114
3  Non-Euclidean Geometries......Page 115
3.1  Overview and History......Page 116
Nikolai Lobachevsky and János Bolyai......Page 118
Carl Friedrich Gauss......Page 119
Georg Friedrich Bernhard Riemann......Page 120
3.2  Properties of Lines and Omega Triangles......Page 124
3.3  Saccheri Quadrilaterals and Triangles......Page 130
3.4  Area and Hyperbolic Designs......Page 134
3.5  Spherical and Single Elliptic Geomrtries......Page 142
Projects for Chapter 3......Page 146
Suggested Media......Page 147
4  Transformational Geometry......Page 149
4.1  Overview and History......Page 150
4.2  Isometries......Page 155
Felix Klein......Page 161
4.3  Algebraic Representation of Transformations......Page 163
4.4  Similarities and Affine Transformations......Page 171
Sophus Lie......Page 173
4.5  Transformations in Higher Dimensions; Computer-Aided Design......Page 179
4.6  Inversions and the Complex Plane......Page 185
Augustus Möbius......Page 190
Projects for Chapter 4......Page 193
Suggested Readings......Page 195
Suggested Media......Page 196
5  Symmetry......Page 197
5.1  Overview and History......Page 198
5.2  Finite Plane Symmetry Groups......Page 203
5.3  Symmetry in the Plane......Page 207
5.4  Symmetries in Higher Dimensions......Page 221
5.5  Symmetry in Science......Page 224
Marjorie Senechal......Page 227
5.6  Fractals......Page 232
Benoit Mandelbrot......Page 234
Projects for Chapter 5......Page 240
Suggested Readings......Page 241
Suggested Media......Page 242
6  Projective Geometry......Page 243
6.1  Overview and History......Page 244
6.2  Axiomatic Projectire Geometry......Page 249
Jean Victor Poncelet......Page 251
6.3  Analytic Projective Geometry......Page 256
6.4  Projectire Transformations......Page 261
6.5  Subgeometries......Page 265
Arthur Cayley......Page 266
6.6  Projective Space......Page 271
Projects for Chapter 6......Page 278
Suggested Media......Page 279
7  Finite Geometries......Page 281
7.1  Overview and History......Page 282
Leonhard Euler......Page 283
7.2  Affine and Projective Planes......Page 285
7.3  Design Theory......Page 290
Sir Ronald A. Fisher......Page 291
7.4  Finite Analytic Geometry......Page 296
Projects for Chapter 7......Page 301
Suggested Readings......Page 303
Appendix A Definitions, Postulates, Common Notions, and Propositions from Book I of Euclid\'s Elements......Page 305
Selected Answers......Page 315
Index......Page 324