The Changing Shape of Geometry: Celebrating a Century of Geometry and Geometry Teaching

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More About This Textbook

Overview

Celebrating a century of geometry and geometry teaching, this volume includes popular articles on Pythagoras, the golden ratio and recreational geometry. Thirty "Desert Island Theorems" from distinguished mathematicians and educators disclose surprising results. (Contributors include a Nobel Laureate and a Pulitzer Prize winner.) Co-published with The Mathematical Association of America.

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Product Details

  • ISBN-13: 9780521531627
  • Publisher: Cambridge University Press
  • Publication date: 3/1/2003
  • Edition description: New Edition
  • Pages: 560
  • Product dimensions: 6.85 (w) x 9.72 (h) x 1.14 (d)

Table of Contents

Foreword
Acknowledgements
General Introduction: Simplicity, Economy, Elegance 1
Pt. I The Nature of Geometry 11
1.1 What is geometry? 13
1.2 What is geometry? 24
Group A Greek Geometry 31
A1 Pythagoras' Theorem 33
A2 Angle at centre of a circle is twice angle at circumference 35
A3 Archimedes' theorem on the area of a parabolic segment 37
A4 An isoperimetric theorem 40
A5 Ptolemy's Theorem 42
Pt. II The History of Geometry 51
2.1 Introductory Essay: A concise and selective history of geometry from Ur to Erlangen 53
2.2 Greek geometry with special reference to infinitesimals 73
2.3 A straight line is the shortest distance between two points 90
2.4 On geometrical constructions by means of the compass 92
2.5 What is a square root? A study of geometrical representation in different mathematical traditions 100
2.6 An old Chinese way of finding the volume of a sphere 115
2.7 Mathematics and Islamic Art 119
2.8 Jamshid al-Kashi, calculating genius 130
2.9 Geometry and Girard Desargues 136
2.10 Henri Brocard and the geometry of the triangle 146
2.11 The development of geometrical methods 146
Group B Elementary Euclidean Geometry 173
B1 Varignon's Theorem 175
B2 Varignon's big sister? 177
B3 Mid-Edges Theorem 179
B4 Van Schooten's Theorem 184
B5 Ceva's Theorem 187
B6 Descartes Circle Theorem 189
B7 Three Squares Theorem 193
B8 Morley's Triangle Theorem 195
Pt. III Pythagoras' Theorem 199
3.1 Introductory Essay: Pythagoras' Theorem, A Measure of Gold 201
3.2 Pythagoras 211
3.3 Perigal's dissection for the Theorem of Pythagoras 222
3.4 Demonstration of Pythagoras' Theorem in three moves 224
3.5 Pythagoras' Theorem 226
3.6 A neglected Pythagorean-like formula 228
3.7 Pythagoras extended: a geometric approach to the cosine rule 232
3.8 Pythagoras in higher dimensions, I 237
3.9 Pythagoras in higher dimensions, II 239
3.10 Pythagoras inside out 240
3.11 Geometry and the cosine rule 242
3.12 Bride's chair revisited 246
3.13 Bride's chair revisited again! 248
Group C Advanced Euclidean Geometry 251
C1 Desargues' Theorem 253
C2 Pascal's Hexagram Theorem 257
C3 Nine-point Circle 261
C4 Napoleon's Theorem and Doug-all's Theorem 265
C5 Miquel's Six Circle Theorem 272
C6 Eyeball Theorems 274
Pt. IV The Golden Ratio 281
4.1 Introduction 283
4.2 Regular pentagons and the Fibonacci Sequence 286
4.3 Equilateral triangles and the golden ratio 292
4.4 Regular pentagon construction 296
4.5 Discovering the golden section 299
4.6 Making a golden rectangle by paper folding 301
4.7 The golden section in mountain photography 304
4.8 Another peek at the golden section 306
4.9 A note on the golden ratio 308
4.10 Balancing and golden rectangles 310
4.11 Golden earrings 312
4.12 The pyramids, the golden section and 2[pi] 314
4.13 A supergolden rectangle 320
Group D Non-Euclidean Geometry & Topology 327
D1 Four-and-a-half Colour Theorem 329
D2 Euler-Descartes Theorem 333
D3 Euler-Poincare Theorem 339
D4 Two Right Tromino theorems 343
D5 Sum of the angles of a spherical triangle 346
Pt. V Recreational Geometry 349
5.1 Introduction 351
5.2 The cube dissected into three yangma 357
5.3 Folded polyhedra 359
5.4 The use of the pentagram in constructing the net for a regular dodecahedron 362
5.5 Paper patterns: solid shapes from metric paper 363
5.6 Replicating figures in the plane 367
5.7 The sphinx task centre problem 371
5.8 Ezt Rakd Ki: A Hungarian tangram 378
5.9 Dissecting a dodecagon 383
5.10 A dissection puzzle 389
5.11 Two squares from one 393
5.12 Half-squares, tessellations and quilting 396
5.13 From tessellations to fractals 401
5.14 Paper patterns with circles 406
5.15 Tessellations with pentagons [with related correspondence] 421
5.16 Universal games 428
Group E Geometrical Physics 443
E1 Euler's Identity 445
E2 Clifford Parallels 448
E3 Tait Conjectures 450
E4 Kelvin's Circulation Theorem 455
E5 Noether's Theorem 456
E6 Kepler's Packing Theorem 459
Pt. VI The Teaching of Geometry 461
6.1 Introductory Essay: A century of school geometry teaching 463
6.2 The teaching of Euclid 486
6.3 The Board of Education circular on the teaching of geometry 489
6.4 The teaching of geometry in schools 496
6.5 Fifty years of change 500
6.6 Milestone or millstone? 505
6.7 The place of geometry in a mathematical education 515
Appendices 527
App. I Report of the M. A. Committee on Geometry (1902) 529
App. II Euclidean Propositions 537
Index 542
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