Note: Supplemental materials are not guaranteed with Rental or Used book purchases.
 ISBN: 9780131437487  0131437488
 Cover: Paperback
 Copyright: 7/28/2004
The distinctive approach of Henderson and Taimina's volume stimulates readers to develop a broader, deeper, understanding of mathematics through active experienceincluding discovery, discussion, writing fundamental ideas and learning about the history of those ideas. A series of interesting, challenging problems encourage readers to gather and discuss their reasonings and understanding. The volume provides an understanding of the possible shapes of the physical universe.The authors provide extensive information on historical strands of geometry, straightness on cylinders and cones and hyperbolic planes, triangles and congruencies, area and holonomy, parallel transport, SSS, ASS, SAA, and AAA, parallel postulates, isometries and patterns, dissection theory, square roots, pythagoras and similar triangles, projections of a sphere onto a plane, inversions in circles, projections (models) of hyperbolic planes, trigonometry and duality, 3spheres and hyperbolic 3spaces and polyhedra.For mathematics educators and other who need to understand the meaning of geometry.
Preface  xv  

xvii  

xviii  

xix  

xxi  

xxii  

xxiii  
How to Use This Book  xxv  

xxvi  

xxvii  

xxviii  
Chapter 0 Historical Strands of Geometry  1  (8)  

1  (2)  

3  (1)  

4  (2)  

6  (3)  
Chapter 1 What Is Straight?  9  (16)  

9  (4)  

13  (4)  

17  (4)  

21  (4)  
Chapter 2 Straightness on Spheres  25  (12)  

25  (3)  

28  (4)  

32  (3)  

35  (1)  

35  (2)  
Chapter 3 What Is an Angle?  37  (6)  

37  (2)  

39  (2)  

41  (2)  
Chapter 4 Straightness on Cylinders and Cones  43  (16)  

44  (2)  

46  (3)  

49  (1)  

50  (1)  

51  (1)  

51  (2)  

53  (2)  

55  (1)  

56  (1)  

57  (2)  
Chapter 5 Straightness on  59  (14)  



59  (3)  

62  (2)  

64  (2)  

66  

63  (5)  

68  (3)  

71  (1)  

71  (2)  
Chapter 6 Triangles and Congruencies  73  (16)  

73  (1)  

74  (1)  

75  (1)  

76  (2)  

78  (1)  

79  (1)  

80  (5)  

85  (4)  
Chapter 7 Area and Holonomy  89  (20)  

90  (1)  

91  (4)  

95  (1)  

96  (2)  

98  (2)  

100  (2)  

102  (1)  

103  (3)  

106  (3)  
Chapter 8 Parallel Transport  109  (8)  

109  (2)  

111  (3)  

114  (1)  

115  (2)  
Chapter 9 SSS, ASS, SAA, and AAA  117  (8)  

117  (2)  

119  (2)  

121  (2)  

123  (2)  
Chapter 10 Parallel Postulates  125  (18)  

125  (1)  

126  (2)  

128  (2)  

130  (1)  

131  (3)  

134  (2)  

136  (2)  

138  (2)  

140  (3)  
Chapter 11 Isometries and Patterns  143  (22)  

144  (4)  

148  (1)  

149  (4)  

153  (1)  

154  (4)  

158  (1)  

159  (1)  

159  (1)  

160  (1)  

161  (2)  

163  (2)  
Chapter 12 Dissection Theory  165  (12)  

165  (2)  

167  (1)  

168  (1)  

169  (1)  

170  (1)  

171  (1)  

172  (1)  

173  (4)  
Chapter 13 Square Roots, Pythagoras, and Similar Triangles  177  (20)  

178  (1)  

179  (5)  

184  (5)  

189  (1)  

190  (1)  

191  (2)  

193  (1)  

194  (1)  

195  (2)  
Chapter 14 Projections of a Sphere onto a Plane  197  (8)  

198  (1)  

198  (1)  

199  (1)  

200  (2)  

202  (3)  
Chapter 15 Circles  205  (12)  

206  (2)  

208  (4)  

212  (1)  

213  (4)  
Chapter 16 Inversions in Circles  217  (16)  

217  (1)  

218  (3)  

221  (3)  

224  (2)  

226  (4)  

230  (3)  
Chapter 17 Projections (Models) of Hyperbolic Planes  233  (12)  

234  (2)  

236  (1)  

237  (2)  

239  (2)  

241  (1)  

242  (2)  

244  (1)  
Chapter 18 Geometric 2Manifolds  245  (22)  

246  (5)  

251  (2)  

253  (3)  

256  (2)  

258  (5)  

263  (1)  

264  (1)  

265  (2)  
Chapter 19 Geometric Solutions of Quadratic and Cubic Equations  267  (18)  

268  (4)  

272  (4)  

276  (4)  

280  (2)  

282  (3)  
Chapter 20 Trigonometry and Duality  285  (14)  

285  (2)  

287  (3)  

290  (2)  

292  (1)  

293  (1)  

294  (1)  

294  (2)  

296  (1)  

297  (2)  
Chapter 21 Mechanisms  299  (20)  

299  (4)  

303  (4)  

307  (3)  

310  (4)  

314  (3)  

317  (2)  
Chapter 22 3Spheres and Hyperbolic 3Spaces  319  (16)  

320  (2)  

322  (3)  

325  (3)  

328  (2)  

330  (2)  

332  (1)  

333  (2)  
Chapter 23 Polyhedra  335  (8)  

335  (1)  

336  (2)  

338  (1)  

339  (1)  

339  (1)  

340  (3)  
Chapter 24 3Manifolds  The Shape of Space  343  (20)  

344  (3)  

347  (2)  

349  (4)  

353  (2)  

355  (1)  

356  (4)  

360  (1)  

361  (2)  
Appendix A Euclid's Definitions, Postulates, and Common Notions  363  (4)  

363  (3)  

366  (1)  

366  (1)  
Appendix B Constructions of  

367  (1)  

367  (1)  

368  (3)  

371  (1)  

371  (1)  

372  (3)  
Bibliography  375  (10)  
Index  385 
We believe that mathematics is a natural and deep part of human experience and that experiences of meaning in mathematics are accessible to everyone. Much of mathematics is not accessible through formal approaches except to those with specialized learning. However, through the use of nonformal experience and geometric imagery, many levels of meaning in mathematics can be opened up in a way that most humans can experience and find intellectually challenging and stimulating.Formalism contains the power of the meaning but not the meaning. It is necessary to bring the power back to the meaning.A formal proof as we normally conceive of it is not the goal of mathematicsit is a toola means to an end. The goal is understanding. Without understanding we will never be satisfiedwith understanding we want to expand that understanding and to communicate it to others. This book is based on a view of proof as aconvincing communication that answersWhy?Many formal aspects of mathematics have now been mechanized and this mechanization is widely available on personal computers or even handheld calculators, but the experience of meaning in mathematics is still a human enterprise that is necessary for creative work.In this book we invite the reader to explore the basic ideas of geometry from a more mature standpoint. We will suggest some of the deeper meanings, larger contexts, and interrelations of the ideas. We are interested in conveying a different approach to mathematics, stimulating the reader to take a broader and deeper view of mathematics and to experience for herself/himself a sense of mathematizing. Through an active participation with these ideas, including exploring and writing about them, people can gain a broader context and experience. This active participation is vital for anyone who wishes to understand mathematics at a deeper level, or anyone wishing to understand something in their experience through the vehicle of mathematics.This is particularly true for teachers or prospective teachers who are approaching related topics in the school curriculum. All too often we convey to students that mathematics is a closed system, with a single answer or approach to every problem, and often without a larger context. We believe that even where there are strict curricular constraints, there is room to change the meaning and the experience of mathematics in the classroom.This book is based on a junior/seniorlevel course that David started teaching in 1974 at Cornell for mathematics majors, high school teachers, future high school teachers, and others. Most of the chapters start intuitively so that they are accessible to a general reader with no particular mathematics background except imagination and a willingness to struggle with ideas. However, the discussions in the book were written for mathematics majors and mathematics teachers and thus assume of the reader a corresponding level of interest and mathematical sophistication.The course emphasizes learning geometry using reason, intuitive understanding, and insightful personal experiences of meanings in geometry. To accomplish this the students are given a series of inviting and challenging problems and are encouraged to write and speak their reasonings and understandings.Most of the problems are placed in an appropriate history perspective and approached both in the context of the plane and in the context of a sphere or hyperbolic plane (and sometimes a geometric manifold). We find that by exploring the geometry of a sphere and a hyperbolic plane, our students gain a deeper understanding of the geometry of the (Euclidean) plane.We introduce the modern notion of "parallel transport along a geodesic," which is a notion of parallelism that makes sense on the plane but also on a sphere or hyperbolic plane (in fact, on any surface). While exploring parallel transport on a sphere, s