Modern Geometries

This comprehensive, best-selling text focuses on the study of many different geometries -- rather than a single geometry -- and is thoroughly modern in its approach. Each chapter is essentially a short course on one aspect of modern geometry, including finite geometries, the geometry of transformations, convexity, advanced Euclidian geometry, inversion, projective geometry, geometric aspects of topology, and non-Euclidean geometries. This edition reflects the recommendations of the COMAP proceedings on Geometry's Future, the NCTM standards, and the Professional Standards for Teaching Mathematics. References to a new companion text, Active Geometry by David A. Thomas encourage students to explore the geometry of motion through the use of computer software. Using Active Geometry at the beginning of various sections allows professors to give students a somewhat more intuitive introduction using current technology before moving on to more abstract concepts and theorems.

1. Sets of Axioms and Finite Geometries

1

(36)

1.1 Introduction to Geometry

1

(6)

1.2 Development of Modern Geometries

7

(5)

1.3 Introduction to Finite Geometries

12

(5)

1.4 Four-Line and Four-Point Geometries

17

(4)

1.5 Finite Geometries of Fano and Young

21

(5)

1.6 Finite Geometries of Pappus and Desargues

26

(5)

1.7 Other Finite Geometries

31

(6)

2. Geometric Transformations

37

(72)

2.1 Introduction to Transformations

37

(7)

2.2 Groups of Transformations

44

(7)

2.3 Euclidean Motions of the Plane

51

(10)

2.4 Sets of Equations for Motions of the Plane

61

(8)

2.5 Applications of Transformations in Computer Graphics

69

(7)

2.6 Properties of the Group of Euclidean Motions

76

(8)

2.7 Motions and Graphics of Three-Space

84

(8)

2.8 Similarity Transformations

92

(6)

2.9 Introduction to the Geometry of Fractals and Fractal Dimension

98

(4)

2.10 Examples and Applications of Fractals

102

(7)

3. Convexity

109

(46)

3.1 Basic Concepts

109

(8)

3.2 Convex Sets and Supporting Lines

117

(7)

3.3 Convex Bodies in Two-Space

124

(7)

3.4 Convex Bodies in Three-Space

131

(5)

3.5 Convex Hulls

136

(5)

3.6 Width of a Set

141

(6)

3.7 Helly's Theorem and Applications

147

(8)

4. Modern Euclidean Geometry, Theory, and Applications

155

(56)

4.1 Fundamental Concepts and Theorem

155

(10)

4.2 Some Theorems Leading to Modern Synthetic Geometry

165

(7)

4.3 The Nine-Point Circle and Early Nineteenth-Century Synthetic Geometry

172

(5)

4.4 Isogonal Conjugates

177

(5)

4.5 Recent Synthetic Geometry of the Triangle

182

(5)

4.6 Golden Ratio, Tessellations, Packing Problems, and Pick's Theorem

187

(8)

4.7 Extremum Problems, Geometric Probability, Fuzzy Sets, and Bezier Curves

195

(16)

5. Constructions

211

(32)

5.1 The Philosophy of Constructions

211

(5)

5.2 Constructible Numbers

216

(4)

5.3 Constructions in Advanced Euclidean Geometry

220

(6)

5.4 Constructions and Impossibility Proofs

226

(8)

5.5 Constructions by Paper Folding and by Use of Computer Software

234

(3)

5.6 Constructions with Only One Instrument

237

(6)

6. The Transformation of Inversion

243

(30)

6.1 Basic Concepts

243

(6)

6.2 Additional Properties and Invariants Under Inversion

249

(6)

6.3 The Analytic Geometry of Inversion

255

(6)

6.4 Some Applications of Inversion

261

(12)

7. Projective Geometry

273

(62)

7.1 Fundamental Concepts

273

(6)

7.2 Postulation Basis for Projective Geometry

279

(3)

7.3 Duality and Some Consequences

282

(6)

7.4 Harmonic Sets

288

(6)

7.5 Projective Transformations

294

(6)

7.6 Homogeneous Coordinates

300

(8)

7.7 Equations for Projective Transformations

308

(9)

7.8 Special Projectivities

317

(5)

7.9 Conics

322

(6)

7.10 Construction of Conics

328

(7)

8. Geometric Introduction to Topological Transformations

335

(30)

8.1 Topological Transformations

335

(5)

8.2 Simple Closed Curves

340

(7)

8.3 Invariant Points and Networks

347

(4)

8.4 Introduction to the Topology of Surfaces

351

(4)

8.5 Euler's Formula and the Map-Coloring Problem

355

(10)

9. Non-Euclidean Geometries

365

(42)

9.1 Foundations of Euclidean and Non-Euclidean Geometries

365

(5)

9.2 Introduction to Hyerbolic Geometry

370

(5)

9.3 Ideal Points and Omega Triangles

375

(4)

9.4 Quadrilaterals and Triangles

379

(5)

9.5 Pairs of Lines and Area of Triangular Regions

384

(5)

9.6 Curves

389

(4)

9.7 Elliptic Geometry

393

(4)

9.8 Consistency; Other Modern Geometries

397

(10)

Appendix 1 Selected Ideas from Logic

407

(2)

Appendix 2 Review of Elementary Euclidean Geometry

409

(3)

Appendix 3 The First Twenty-eight Propositions of Euclid (Revised)

412

(2)

Appendix 4 Hilbert's Axioms

414

(3)

Appendix 5 Birkhoff's Postulates

417

(1)

Appendix 6 Illustrations of Basic Euclidean Constructions

418

(3)

Bibliography

421

(6)

Answers to Selected Exercises

427

(16)

Index

443

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