This is a new edition of a highly accessible presentation on modern geometry. Each chapter includes a list of suggested resources for applications or related topics in areas such as art and history. This second edition contains new material throughout, including a new chapter on chaos theory and fractal geometry. The book now includes pointers to the author's website, which contains guides for dynamic software explorations of many concepts, using both Cabri Geometry and Geometers Sketchpad.

Judith N. Cederberg is an associate professor of mathematics at St. Olaf College in Minnesota.

Preface to the Second Edition

v

Preface to the First Edition

xiii

Axiomatic Systems and Finite Geometries

1

(32)

Gaining Perspective*

1

(1)

Axiomatic Systems

2

(7)

Finite Projective Planes

9

(9)

An Application to Error-Correcting Codes

18

(7)

Desargues' Configurations

25

(5)

Suggestions for Further Reading

30

(3)

Non-Euclidean Geometry

33

(66)

Gaining Perspective

33

(1)

Euclid's Geometry

34

(13)

Non-Euclidean Geometry*

47

(4)

Hyperbolic Geometry---Sensed Parallels

51

(10)

Hyperbolic Geometry---Asymptotic Triangles

61

(7)

Hyperbolic Geometry---Saccheri Quadrilaterals

68

(6)

Hyperbolic Geometry---Area of Triangles

74

(6)

Hyperbolic Geometry---Ultraparallels

80

(4)

Elliptic Geometry

84

(9)

Significance of the Discovery of Non-Euclidean Geometries

93

(1)

Suggestions for Further Reading

93

(6)

Geometric Transformations of the Euclidean Plane

99

(114)

Gaining Perspective

99

(4)

Exploring Line and Point Reflections*

103

(5)

Exploring Rotations and Finite Symmetry Groups*

108

(8)

Exploring Translations and Frieze Pattern Symmetries*

116

(5)

An Analytic Model of the Euclidean Plane

121

(8)

Transformations of the Euclidean Plane

129

(7)

Isometries

136

(8)

Direct Isometries

144

(10)

Indirect Isometries

154

(11)

Frieze and Wallpaper Patterns

165

(8)

Exploring Plane Tilings*

173

(10)

Similarity Transformations

183

(7)

Affine Transformations

190

(8)

Exploring 3-D Isometries

198

(9)

Suggestions for Further Reading

207

(6)

Projective Geometry

213

(102)

Gaining Perspective

213

(1)

The Axiomatic System and Duality

214

(7)

Perspective Triangles*

221

(2)

Harmonic Sets*

223

(6)

Perspectivities and Projectivities*

229

(11)

Conics in the Projective Plane*

240

(10)

An Analytic Model for the Projective Plane

250

(8)

The Analytic Form of Projectivities

258

(6)

Cross Ratios

264

(6)

Collineations

270

(13)

Correlations and Polarities

283

(15)

Subgeometries of Projective Geometry

298

(13)

Suggestions for Further Reading

311

(4)

Chaos to Symmetry: An Introduction to Fractal Geometry

315

(74)

A Chaotic Background*

316

(18)

Need for a New Geometric Language

334

(13)

Fractal Dimension

347

(13)

Iterated Function Systems*

360

(17)

Finally---What Is a Fractal?

377

(3)

Applications of Fractal Geometry

380

(2)

Suggestions for Further Reading

382

(7)

Appendices

389

(24)

A Euclid's Definitions, Postulates, and the First 30 Propositions of Elements, Book I

389

(6)

B Hilbert's Axioms for Plane Geometry

395

(4)

C Birkhoff's Postulates for Euclidean Plane Geometry

399

(2)

D The SMSG Postulates for Euclidean Geometry

401

(4)

E Some SMSG Definitions for Euclidean Geometry

405

(4)

F The ASA Theorem

409

(4)

References

413

(14)

Index

427

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