Geometry Ancient and Modern

This book offers a guided tour of geometry from euclid through to algebraic geometry. It shows how mathematicians use a variety of techniques to tackle problems , and it links geometry to other branches of mathematics. Many problems and examples are included to aid understanding.

John Silvester teachers geometry to first-year mathematics students at King's College London

History and philosophy

1

(11)

Before Euclid

1

(1)

Euclid

2

(1)

Parallels and non-Euclidean geometry

3

(2)

Hilbert and Birkhoff

5

(1)

Coordinate Geometry

6

(1)

Pythagoras: theorem or axiom?

7

(2)

Philosophy of this book

9

(3)

Advice to readers

10

(2)

Drawings and constructions

12

(12)

Equipment

12

(1)

Ruler and compasses

13

(2)

Bisection, parallels, and subdivision

15

(5)

Triangles

20

(4)

Answers to exercises

23

(1)

Plane geometry

24

(28)

Points

24

(1)

Lines

24

(1)

Vector notation

25

(2)

Distances and orthogonality

27

(4)

Isometries

31

(6)

Angles

37

(7)

Preamble

37

(3)

Abstract trigonometry

40

(2)

Angle as area

42

(2)

Polar coordinates and rotations

44

(2)

Dot products

46

(6)

Distance from point to line

48

(1)

Angle bisectors

48

(1)

Answers to exercises

49

(3)

Triangles and triangle formulae

52

(30)

The centroid

53

(1)

The orthocentre

54

(1)

The sine formula

55

(1)

The circumcentre

56

(1)

The incentre and excentres

57

(3)

The cosine formula

60

(5)

Similarities

65

(2)

Barycentric coordinates

67

(5)

Menelaus and Ceva

72

(10)

Answers to exercises

77

(5)

Isometries of R2

82

(34)

Fixed points and lines

82

(2)

Groups of isometries

84

(8)

Complex formulae for plane isometries

92

(13)

Classification of plane isometries

93

(7)

Orientation

100

(5)

Finite groups of plane isometries

105

(11)

Answers to exercises

110

(6)

Isometries of Rn

116

(49)

Matrix forms for isometries of R2

116

(1)

Matrix forms for isometries of Rn

117

(5)

Classification of isometries

122

(7)

Permutation groups

129

(3)

Polyhedra

132

(7)

The Platonic solids

139

(6)

Finite groups of isometries of R3

145

(20)

Answers to exercises

153

(12)

Circles and other conics

165

(30)

Proper and improper conics

167

(4)

Plane sections of a cone

171

(3)

Classification of conics under isometry

174

(4)

Circles: angle theorems

178

(7)

Circles: rectangular properties

185

(5)

Stereographic projection

190

(5)

Answers to exercises

192

(3)

Beyond isometry

195

(27)

Klein's definition of geometry

195

(1)

Euclidean geometry: similarities

195

(9)

Classification of similarities

196

(3)

Classification of conics (again)

199

(1)

Centres of similitude and the nine-point circle

199

(4)

Dilatations

203

(1)

Affine geometry

204

(18)

Affine regular polygons

206

(3)

Classification of conics (yet again)

209

(2)

A selection of affine theorems

211

(4)

Collineations

215

(3)

Answers to exercises

218

(4)

Infinity

222

(45)

Mobius transformations

222

(22)

Lines and circles

224

(2)

Cross-ratios

226

(3)

Angles

229

(5)

Applications

234

(5)

Hyperbolic geometry

239

(5)

Projective geometry

244

(23)

Homogeneous coordinates

247

(2)

Duality

249

(2)

Projective transformations and cross-ratios

251

(4)

Classification of conics (for the last time)

255

(4)

Elliptic geometry

259

(2)

Answers to exercises

261

(6)

Complex geometry

267

(34)

Complex projective geometry

267

(4)

Algebraic geometry

271

(30)

Intersections of curves

272

(4)

Nine associated points

276

(4)

Applications

280

(6)

Singular and non-singular cubics

286

(11)

Answers to exercises

297

(4)

Bibliography

301

(2)

Notation

303

(2)

Index

305

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Geometry Ancient and Modern

Summary

Author Biography

Table of Contents

Supplemental Materials