Symmetry and Pattern in Projective Geometry

Cover......Page 1
Symmetry and Pattern in Projective Geometry......Page 4
Preface......Page 7
Contents......Page 9
1.1 Euclid......Page 12
1.2 Axioms of Projective Geometry......Page 13
1.3 The Art of Perspective......Page 14
1.4 Desargues\' Theorem......Page 16
1.5 The Complete Quadrilateral......Page 18
1.6 Affine Geometry......Page 21
1.7 The Theorem of Pappus......Page 22
1.8 Affine Coordinates......Page 25
1.9 Configurations......Page 28
1.10 Axioms for N-Dimensional Projective Space......Page 35
1.11 Duality......Page 36
1.12 Algebra versus Axioms......Page 37
2.1 Homogeneous Coordinates......Page 38
2.2 More than Two Dimensions......Page 41
2.3 Collineations......Page 42
2.4 A Proof of Pappus\'s Theorem......Page 44
2.5 Proofs of Desargues\' Theorem......Page 45
2.7 Subspaces of a Vector Space......Page 47
2.8 Plücker Coordinates......Page 49
2.9 Grassmann Coordinates......Page 50
3.1 The Projective Line......Page 53
3.2 Cross-Ratio......Page 55
3.3 Involutions......Page 56
3.4 Cross-Ratio in Affine Geometry......Page 57
3.5 The Complex Projective Line......Page 58
3.7 Four Points in a Plane......Page 60
3.8 Configurations in More than Two Dimensions......Page 63
3.9 Five Points in 3-Space......Page 64
3.10 Six Planes in 3-Space......Page 65
3.11 Six Points in 4-Space......Page 68
3.12 Sylvester\'s Duads and Synthemes......Page 70
3.13 Permutations of Six Things......Page 72
3.14 Another Extension of Desargues\' Theorem......Page 73
3.15 Twenty-Seven Lines......Page 74
3.16 Associated Trihedron Pairs......Page 76
3.18 The Polytope 221......Page 78
3.19 Desmic Systems......Page 79
3.20 Baker\'s Configuration......Page 82
4.1 Conics......Page 88
4.2 Tangents......Page 90
4.3 Canonical Forms......Page 91
4.4 Polarity......Page 93
4.5 Self-Polar Triangles......Page 95
4.6 Mutually Polar Triangles......Page 97
4.7 Metric Planes......Page 98
4.8 Pascal\'s Theorem......Page 101
4.9 The Extended Pascal Figure......Page 104
4.10 Quadrics......Page 108
4.11 Pascal\'s Theorem Again......Page 109
4.12 Tangent planes......Page 110
4.13 Polarity......Page 111
4.14 Affine Classification of Quadrics......Page 112
4.15 Reguli......Page 114
4.16 Metric Spaces......Page 116
4.17 Clifford Parallels......Page 117
4.18 Isometries......Page 119
5.1 Plane Cubic Curves......Page 123
5.3 A Canonical Form for a Plane Cubic Curve......Page 125
5.4 Parametric Form......Page 126
5.5 Inflections......Page 127
5.6 Cubics in Affine Geometry......Page 128
5.7 The Twisted Cubic......Page 129
5.8 Chords, Tangent Lines and Osculating Planes......Page 130
5.9 A Net of Quadrics......Page 132
5.10 Cubic Surfaces......Page 133
5.11 Canonical Forms for a Cubic Surface......Page 136
5.12 Twenty-Seven Lines......Page 137
6.1 Algebraic Geometry......Page 141
6.2 The Hessian of a Cubic Surface......Page 142
6.3 Desmic Surfaces......Page 144
6.4 Kummer\'s Quartic Surface......Page 146
7.1 Finite Geometries......Page 153
7.2 PG(2, 2)......Page 154
7.3 PG(2, 3)......Page 155
7.5 Galois Fields......Page 156
7.6 PG(2, 4)......Page 158
7.7 Structure of PG(N,q)......Page 159
7.8 Collineations of PG(N,q)......Page 162
7.9 Finite Projective Lines......Page 163
7.11 PL(5)......Page 164
7.13 PL(7)......Page 168
7.14 Eight Points in PG(3, 2)......Page 169
7.17 Coxeter\'s Configuration (116)......Page 171
7.18 Twelve Points in PG(5, 3)......Page 173
7.19 PL(23)......Page 174
7.20 Twenty-Four Points in PG(11, 2)......Page 175
7.21 Octastigms and Dodecastigms......Page 177
Appendix: Notes and References......Page 180
Bibliography......Page 185
Index......Page 187