Preface to the Second Edition ................................... v
Preface to the First Edition ................................. xiii
1. Axiomatic Systems and Finite Geometries ...................... 1
1.1. Gaining Perspective ..................................... 1
1.2. Axiomatic Systems ....................................... 2
1.3. Finite Projective Planes ................................ 9
1.4. An Application to Error-Correcting Codes ............... 18
1.5. Desargues' Configurations .............................. 25
1.6. Suggestions for Further Reading ........................ 30
2. Non-Euclidean Geometry ...................................... 33
2.1. Gaining Perspective .................................... 33
2.2. Euclid's Geometry ...................................... 34
2.3. Non-Euclidean Geometry ................................. 47
2.4. Hyperbolic Geometry—Sensed Parallels ................... 51
2.5. Hyperbolic Geometry—Asymptotic Triangles ............... 61
2.6. Hyperbolic Geometry—Saccheri Quadrilaterals ............ 68
2.7. Hyperbolic Geometry—Area of Triangles .................. 74
2.8. Hyperbolic Geometry—Ultraparallels ..................... 80
2.9. Elliptic Geometry ...................................... 84
2.10.Significance of the Discovery of Non-Euclidean
Geometries ............................................. 93
2.11.Suggestions for Further Reading ........................ 93
3. Geometric Transformations of the Euclidean Plane ............ 99
3.1. Gaining Perspective .................................... 99
3.2. Exploring Line and Point Reflections .................. 103
3.3. Exploring Rotations and Finite Symmetry Groups ........ 108
3.4. Exploring Translations and Frieze Pattern
Symmetries ............................................ 116
3.5. An Analytic Model of the Euclidean Plane .............. 121
3.6. Transformations of the Euclidean Plane ................ 129
3.7. Isometries ............................................ 136
3.8. Direct Isometries ..................................... 144
3.9. Indirect Isometries ................................... 154
3.10.Frieze and Wallpaper Patterns ......................... 165
3.11.Exploring Plane Tilings ............................... 173
3.12.Similarity Transformations ............................ 183
3.13.Affine Transformations ................................ 190
3.14.Exploring 3-D Isometries .............................. 198
3.15.Suggestions for Further Reading ....................... 207
4. Projective Geometry ........................................ 213
4.1. Gaining Perspective ................................... 213
4.2. The Axiomatic System and Duality ...................... 214
4.3. Perspective Triangles ................................. 221
4.4. Harmonic Sets ......................................... 223
4.5. Perspectivities and Projectivities .................... 229
4.6. Conies in the Projective Plane ........................ 240
4.7. An Analytic Model for the Projective Plane ............ 250
4.8. The Analytic Form of Projectivities ................... 258
4.9. Cross Ratios .......................................... 264
4.10.Collineations ......................................... 270
4.11.Correlations and Polarities ........................... 283
4.12.Subgeometries of Projective Geometry .................. 298
4.13.Suggestions for Further Reading ....................... 311
5. Chaos to Symmetry: An Introduction to Fractal Geometry ..... 315
5.1. A Chaotic Background .................................. 316
5.2. Need for a New Geometric Language ..................... 334
5.3. Fractal Dimension ..................................... 347
5.4. Iterated Function Systems ............................. 360
5.5. Finally—What Is a Fractal? ............................ 377
5.6. Applications of Fractal Geometry ...................... 380
5.7. Suggestions for Further Reading ....................... 382
Appendices .................................................... 389
A. Euclid's Definitions, Postulates, and the First 30
Propositions of Elements, Book I ........................ 389
В. Hilbert's Axioms for Plane Geometry ..................... 395
С. Birkhoff s Postulates for Euclidean Plane Geometry ...... 399
D. The SMSG Postulates for Euclidean Geometry .............. 401
E. Some SMSG Definitions for Euclidean Geometry ............ 405
F. The ASA Theorem ......................................... 409
References .................................................... 413
Index ......................................................... 427
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