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E-raamat: Symmetry and Pattern in Projective Geometry

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  • Formaat: PDF+DRM
  • Ilmumisaeg: 14-Dec-2012
  • Kirjastus: Springer London Ltd
  • Keel: eng
  • ISBN-13: 9781447146315
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Symmetry and Pattern in Projective GeometrySuurem pilt
  • Formaat: PDF+DRM
  • Ilmumisaeg: 14-Dec-2012
  • Kirjastus: Springer London Ltd
  • Keel: eng
  • ISBN-13: 9781447146315
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Symmetry and Pattern in Projective Geometry is a self-contained study of projective geometry which compares and contrasts the analytic and axiomatic methods. The analytic approach is based on homogeneous coordinates, and brief introductions to Plücker coordinates and Grassmann coordinates are presented.

This book looks carefully at linear, quadratic, cubic and quartic figures in two, three and higher dimensions. It deals at length with the extensions and consequences of basic theorems such as those of Pappus and Desargues. The emphasis throughout is on special configurations that have particularly interesting symmetry properties.

The intricate and novel ideas of Donald Coxeter, who is considered one of the great geometers of the twentieth century, are also discussed throughout the text. The book concludes with a useful analysis of finite geometries and a description of some of the remarkable configurations discovered by Coxeter.

This book will be appreciated by mathematics students and those wishing to learn more about the subject of geometry. It makes accessible subjects and theorems which are often considered quite complicated and presents them in an easy-to-read and enjoyable manner.

Arvustused

From the reviews:

Here, readers meet a slice of geometry, intermediate between elementary Euclidean and modern algebraic, where first come the objects, extraordinary, surprising, and intriguing. By meeting and literally seeing these special objects, specifically linear configurations and various projective varieties of low dimension and low degree, undergraduates can--perhaps for the first time--really experience mathematics as a phenomenological science. Summing Up: Highly recommended. Upper-division undergraduates through researchers/faculty. (D. V. Feldman, Choice, Vol. 51 (2), October, 2013)

The book under review presents an introduction to projective geometry as well as some preliminary results on algebraic geometry and finite geometries. This is a well-illustrated, self-contained textbook on projective geometry. Both the basics and the significant facts are presented in an original manner. The book is intended for undergraduate students in mathematics and computer science. (Georgi Hristov Georgiev, Mathematical Reviews, September, 2013)

The author wanted to convey some of the fascination he feels for the subject. Nice figures, some in color, illustrate perspective. this is a nice introduction for those with an interest in geometry. (Arthur Gittleman, Computing Reviews, July, 2013)

1 Foundations: The Synthetic Approach
1(26)
1.1 Euclid
1(1)
1.2 Axioms of Projective Geometry
2(1)
1.3 The Art of Perspective
3(2)
1.4 Desargues' Theorem
5(2)
1.5 The Complete Quadrilateral
7(3)
1.6 Affine Geometry
10(1)
1.7 The Theorem of Pappus
11(2)
1.8 Affine Coordinates
13(5)
1.9 Configurations
18(6)
1.10 Axioms for N-Dimensional Projective Space
24(1)
1.11 Duality
25(1)
1.12 Algebra versus Axioms
25(2)
2 The Analytic Approach
27(16)
2.1 Homogeneous Coordinates
27(3)
2.2 More than Two Dimensions
30(1)
2.3 Collineations
31(2)
2.4 A Proof of Pappus's Theorem
33(1)
2.5 Proofs of Desargues' Theorem
34(2)
2.6 Affine Coordinates
36(1)
2.7 Subspaces of a Vector Space
36(2)
2.8 Plucker Coordinates
38(1)
2.9 Grassmann Coordinates
39(4)
3 Linear Figures
43(36)
3.1 The Projective Line
43(2)
3.2 Cross-Ratio
45(1)
3.3 Involutions
46(1)
3.4 Cross-Ratio in Affine Geometry
47(1)
3.5 The Complex Projective Line
48(2)
3.6 Equianharmonic Points
50(1)
3.7 Four Points in a Plane
50(3)
3.8 Configurations in More than Two Dimensions
53(1)
3.9 Five Points in 3-Space
54(1)
3.10 Six Planes in 3-Space
55(3)
3.11 Six Points in 4-Space
58(2)
3.12 Sylvester's Duads and Synthemes
60(2)
3.13 Permutations of Six Things
62(1)
3.14 Another Extension of Desargues' Theorem
63(1)
3.15 Twenty-Seven Lines
64(2)
3.16 Associated Trihedron Pairs
66(2)
3.17 Segre's Notation
68(1)
3.18 The Polytope 221
68(1)
3.19 Desmic Systems
69(3)
3.20 Baker's Configuration
72(7)
4 Quadratic Figures
79(36)
4.1 Conics
79(2)
4.2 Tangents
81(1)
4.3 Canonical Forms
82(2)
4.4 Polarity
84(2)
4.5 Self-polar Triangles
86(2)
4.6 Mutually Polar Triangles
88(1)
4.7 Metric Planes
89(3)
4.8 Pascal's Theorem
92(3)
4.9 The Extended Pascal Figure
95(4)
4.10 Quadrics
99(1)
4.11 Pascal's Theorem Again
100(1)
4.12 Tangent planes
101(1)
4.13 Polarity
102(1)
4.14 Affine Classification of Quadrics
103(1)
4.15 Reguli
104(3)
4.16 Metric Spaces
107(1)
4.17 Clifford Parallels
108(2)
4.18 Isometries
110(5)
5 Cubic Figures
115(18)
5.1 Plane Cubic Curves
115(2)
5.2 Nine Points
117(1)
5.3 A Canonical Form for a Plane Cubic Curve
117(1)
5.4 Parametric Form
118(1)
5.5 Inflections
119(1)
5.6 Cubics in Affine Geometry
120(1)
5.7 The Twisted Cubic
121(1)
5.8 Chords, Tangent Lines and Osculating Planes
122(2)
5.9 A Net of Quadrics
124(1)
5.10 Cubic Surfaces
125(3)
5.11 Canonical Forms for a Cubic Surface
128(1)
5.12 Twenty-Seven Lines
129(4)
6 Quartic Figures
133(12)
6.1 Algebraic Geometry
133(1)
6.2 The Hessian of a Cubic Surface
134(2)
6.3 Desmic Surfaces
136(2)
6.4 Kummer's Quartic Surface
138(7)
7 Finite Geometries
145(28)
7.1 Finite Geometries
145(1)
7.2 PG(2,2)
146(1)
7.3 PG(2,3)
147(1)
7.4 PG(3,2)
148(1)
7.5 Galois Fields
148(2)
7.6 PG(2,4)
150(1)
7.7 Structure of PG(N,q)
151(3)
7.8 Collineations of PG(N,q)
154(1)
7.9 Finite Projective Lines
155(1)
7.10 PL(3)
156(1)
7.11 PL(5)
156(3)
7.12 Six points in PG(2,4)
159(1)
7.13 PL(7)
160(1)
7.14 Eight Points in PG(3,2)
161(2)
7.15 Steiner Systems
163(1)
7.16 PL(11)
163(1)
7.17 Coxeter's Configuration (116)
163(2)
7.18 Twelve Points in PG(5,3)
165(1)
7.19 PL(23)
166(1)
7.20 Twenty-Four Points in PG(11,2)
167(2)
7.21 Octastigms and Dodecastigms
169(4)
Appendix Notes and References 173(6)
Bibliography 179(2)
Index 181
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