Geometry and Symmetry

This new book for mathematics and mathematics education majors helps students gain an appreciation of geometry and its importance in the history and development of mathematics. The material is presented in three parts. The first is devoted to a rigorous introduction of Euclidean geometry, the second covers various noneuclidean geometries, and the last part delves into symmetry and polyhedra. Historical contexts accompany each topic. Exercises and activities are interwoven with the text to enable the students to explore geometry. Some of the activities take advantage of geometric software so students - in particular, future teachers - gain a better understanding of its capabilities. Others explore the construction of simple models or use manipulatives allowing students to experience the hands-on, creative side of mathematics. While this text contains a rigorous mathematical presentation, key design features and activities allow it to be used successfully in mathematics for teachers courses as well.

Preface xi

I Euclidean geometry 1

1 A brief history of early geometry 3

1.1 Prehellenistic mathematics 3

1.2 Greek mathematics before Euclid 5

1.3 Euclid 9

1.4 The Elements 11

1.5 Projects 14

2 Book I of Euclid’s The Elements 15

2.1 Preliminaries 15

2.2 Propositions I.5–I.26: Triangles 39

2.3 Propositions I.27–I.32: Parallel lines 54

2.4 Propositions I.33–I.46: Area 59

2.5 The Pythagorean Theorem 64

2.6 Hilbert’s axioms for euclidean geometry 71

2.7 Distance and geometry 75

2.8 Projects 78

3 More euclidean geometry 80

3.1 Circletheorems 80

3.2 Similarity 88

3.3 More triangle theorems 92

3.4 Inversioninacircle 101

3.5 Projects 107

4 Constructions 109

4.1 Straightedge and compass constructions 109

4.2 Trisections 121

4.3 Constructions with compass alone 124

4.4 Theoretical origami 129

4.5 Knots and star polygons 139

4.6 Linkages 144

4.7 Projects 153

II Noneuclidean Geometries 155

5 Neutral geometry 157

5.1 Viewsongeometry 157

5.2 Neutralgeometry 159

5.3 Alternate parallel postulates 169

5.4 Projects 176

6 Hyperbolic geometry 178

6.1 The history of hyperbolic geometry 178

6.2 Strangenewuniverse 181

6.3 Models of the hyperbolic plane 186

6.4 Consistency of geometries 197

6.5 Asymptoticparallels 199

6.6 Biangles 203

6.7 Divergentparallels 208

6.8 Triangles in hyperbolic space 210

6.9 Projects 218

7 Other geometries 220

7.1 Exploring the geometry of a sphere 220

7.2 Ellipticgeometry 226

7.3 Comparative geometry 239

7.4 Areaanddefect 242

7.5 Taxicab geometry 254

7.6 Finite geometries 258

7.7 Projects 264

III Symmetry 265

8 Isometries 267

8.1 Transformationgeometry 267

8.2 Rosette groups 289

8.3 Frieze patterns 294

8.4 Wallpaper patterns 301

8.5 Isometries in hyperbolic geometry 314

8.6 Projects 319

9 Tilings 320

9.1 Tilings on the plane 320

9.2 Tilings by irregular tiles 327

9.3 Tilings of noneuclidean spaces 341

9.4 Penrose tilings 345

9.5 Projects 355

10 Geometry in three dimensions 357

10.1 Euclidean geometry in three dimensions 357

10.2 Polyhedra 369

10.3 Volume 387

10.4 Infinite polyhedra 393

10.5 Isometries in three dimensions 397

10.6 Symmetries of polyhedra 406

10.7 Four-dimensional figures 412

10.8 Projects 417

Appendix A Logic and proofs 419

A 1 Mathematical systems 419

A 2 Logic 420

A 3 Structuringproofs 424

A 4 Inventingproofs 427

A 5 Writingproofs 428

A 6 Geometric diagrams 429

A 7 Using geometric software 431

A 8 Van Hiele levels of geometric thought 431

Appendix B Postulates and theorems 434

B 1 Postulates 434

B 2 Book I of Euclid’s The Elements 436

B 3 Moreeuclideangeometry 439

B 4 Constructions 441

B 5 Neutralgeometry 442

B 6 Hyperbolic geometry 443

B 7 Othergeometries 445

B 8 Isometries 446

B 9 Tilings 448

B.10 Geometry in three dimensions 448

Bibliography 450

Index 455

Dr. L. Christine Kinsey is in the Mathematics and Statistics department at Canisius University.

Teresa E. Moore is the author of Geometry and Symmetry, published by Wiley.

Efstratios Prassidis is the author of Geometry and Symmetry, published by Wiley.