9780131437487

Experiencing Geometry

by ;
  • ISBN13:

    9780131437487

  • ISBN10:

    0131437488

  • Edition: 3rd
  • Format: Paperback
  • Copyright: 7/28/2004
  • Publisher: Pearson

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Summary

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, 3-spheres and hyperbolic 3-spaces and polyhedra.For mathematics educators and other who need to understand the meaning of geometry.

Table of Contents

Preface xv
Changes in this Edition
xvii
Useful Supplements
xviii
Our Background in Geometry
xix
Acknowledgments for the First Edition
xxi
Acknowledgments for the Second Edition
xxii
Acknowledgments for This Edition
xxiii
How to Use This Book xxv
How We Use this Book in a Course
xxvi
But, Do It Your Own Way
xxvii
Chapter Sequences
xxviii
Chapter 0 Historical Strands of Geometry 1(8)
Art/Pattems Strand
1(2)
Navigation/Stargazing Strand
3(1)
Building Structures Strand
4(2)
Motion/Machines Strand
6(3)
Chapter 1 What Is Straight? 9(16)
History: How Can We Draw a Straight Line?
9(4)
PROBLEM 1.1 When Do You Call a Line Straight?
13(4)
The Symmetries of a Line
17(4)
Local (and Infinitesimal) Straightness
21(4)
Chapter 2 Straightness on Spheres 25(12)
Early History of Spherical Geometry
25(3)
PROBLEM 2.1 What Is Straight on a Sphere?
28(4)
Symmetries of Great Circles
32(3)
Every Geodesic Is a Great Circle
35(1)
Intrinsic Curvature
35(2)
Chapter 3 What Is an Angle? 37(6)
PROBLEM 3.1 What Is an Angle?
37(2)
PROBLEM 3.2 Vertical Angle Theorem (VAT)
39(2)
Hints for Three Different Proofs
41(2)
Chapter 4 Straightness on Cylinders and Cones 43(16)
PROBLEM 4.1 Straightness on Cylinders and Cones
44(2)
Cones with Varying Cone Angles
46(3)
Geodesics on Cylinders
49(1)
Geodesics on Cones
50(1)
PROBLEM 4.2 Global Properties of Geodesics
51(1)
n-Sheeted Coverings of a Cylinder
51(2)
n-Sheeted (Branched) Coverings of a Cone
53(2)
Locally Isometric
55(1)
Is "Shortest" Always "Straight"?
56(1)
Relations to Differential Geometry
57(2)
Chapter 5 Straightness on 59(14)
Hyperbolic Planes
A Short History of Hyperbolic Geometry
59(3)
Description of Annular Hyperbolic Planes
62(2)
Hyperbolic Planes of Different Radii (Curvature)
64(2)
PROBLEM 5.1 What Is Straight in a Hyperbolic Plane?
66
PROBLEM 5.2 Coordinate System on Annular Hyperbolic Plane
63(5)
PROBLEM 5.3 The Pseudosphere Is Hyperbolic
68(3)
Intrinsic/Extrinsic, Local/Global
71(1)
PROBLEM 5.4 Rotations and Reflections on Surfaces
71(2)
Chapter 6 Triangles and Congruencies 73(16)
Geodesics Are Locally Unique
73(1)
PROBLEM 6.1 Properties of Geodesics
74(1)
PROBLEM 6.2 Isosceles Triangle Theorem (ITT)
75(1)
Circles
76(2)
Triangle Inequality
78(1)
PROBLEM 6.3 Bisector Constructions
79(1)
PROBLEM 6.4 Side-Angle-Side (SAS)
80(5)
PROBLEM 6.5 Angle-Side-Angle (ASA)
85(4)
Chapter 7 Area and Holonomy 89(20)
PROBLEM 7.1 The Area of a Triangle on a Sphere
90(1)
PROBLEM 7.2 Area of Hyperbolic Triangles
91(4)
PROBLEM 7.3 Sum of the Angles of a Triangle
95(1)
Introducing Parallel Transport
96(2)
Introducing Holonomy
98(2)
PROBLEM 7.4 The Holonomy of a Small Triangle
100(2)
The Gauss-Bonnet Formula for Triangles
102(1)
PROBLEM 7.5 Gauss-Bonnet Formula for Polygons
103(3)
Gauss-Bonnet Formula for Polygons on Surfaces
106(3)
Chapter 8 Parallel Transport 109(8)
PROBLEM 8.1 Euclid's Exterior Angle Theorem (FEAT)
109(2)
PROBLEM 8.2 Symmetries of Parallel Transported Lines
111(3)
PROBLEM 8.3 Transversals through a Midpoint
114(1)
PROBLEM 8.4 What Is "Parallel"?
115(2)
Chapter 9 SSS, ASS, SAA, and AAA 117(8)
PROBLEM 9.1 Side-Side-Side (SSS)
117(2)
PROBLEM 9.2 Angle-Side-Side (ASS)
119(2)
PROBLEM 9.3 Side-Angle-Angle (SAA)
121(2)
PROBLEM 9.4 Angle-Angle-Angle (AAA)
123(2)
Chapter 10 Parallel Postulates 125(18)
Parallel Lines on the Plane Are Special
125(1)
PROBLEM 10.1 Parallel Transport on the Plane
126(2)
PROBLEM 10.2 Parallel Postulates Not Involving (Non-)Intersecting Lines
128(2)
Equidistant Curves on Spheres and Hyperbolic Planes
130(1)
PROBLEM 10.3 Parallel Postulates Involving (Non-)Intersecting Lines
131(3)
PROBLEM 10.4 EFP and HSP on Sphere and Hyperbolic Plane
134(2)
Comparisons of Plane, Spheres, and Hyperbolic Planes
136(2)
Parallel Postulates within the Building Structures Strand
138(2)
Non-Euclidean Geometries within the Historical Strands
140(3)
Chapter 11 Isometries and Patterns 143(22)
PROBLEM 11.1 Isometries
144(4)
PROBLEM 11.2 Three Points Determine an Isometry
148(1)
PROBLEM 11.3 Classification of Isometries
149(4)
Klein's Erlangen Program
153(1)
Symmetries and Patterns
154(4)
PROBLEM 11.4 Examples of Patterns
158(1)
PROBLEM 11.5 Classification of Discrete Strip Patterns
159(1)
PROBLEM 11.6 Classification of Finite Plane Patterns
159(1)
PROBLEM 11.7 Regular Tilings with Polygons
160(1)
Other Periodic (and Non-Periodic) Patterns
161(2)
Geometric Meaning of Abstract Group Terminology
163(2)
Chapter 12 Dissection Theory 165(12)
What Is Dissection Theory?
165(2)
A Dissection Puzzle from 250 B.C. Solved in 2003
167(1)
History of Dissections in the Theory of Area
168(1)
PROBLEM 12.1 Dissect Plane Triangle and Parallelogram
169(1)
Dissection Theory on Spheres and Hyperbolic Planes
170(1)
PROBLEM 12.2 Khayyam Quadrilaterals
171(1)
PROBLEM 12.3 Dissect Spherical and Hyperbolic Triangles and Khayyam Parallelograms
172(1)
PROBLEM 12.4 Spherical Polygons Dissect to Lunes
173(4)
Chapter 13 Square Roots, Pythagoras, and Similar Triangles 177(20)
Square Roots
178(1)
PROBLEM 13.1 A Rectangle Dissects into a Square
179(5)
Baudhayana's Sulbasutram
184(5)
PROBLEM 13.2 Equivalence of Squares
189(1)
Any Polygon Can Be Dissected into a Square
190(1)
History of Dissections
191(2)
PROBLEM 13.3 More Dissection-Related Problems
193(1)
Three-Dimensional Dissections and Hilbert's Third Problem
194(1)
PROBLEM 13.4 Similar Triangles
195(2)
Chapter 14 Projections of a Sphere onto a Plane 197(8)
PROBLEM 14.1 Charts Must Distort
198(1)
PROBLEM 14.2 Gnomic Projection
198(1)
PROBLEM 14.3 Cylindrical Projection
199(1)
PROBLEM 14.4 Stereographic Projection
200(2)
History of Stereographic Projection and Astrolabe
202(3)
Chapter 15 Circles 205(12)
PROBLEM 15.1 Angles and Power of Points for Circles in the Plane
206(2)
PROBLEM 15.2 Power of Points for Circles on Spheres
208(4)
PROBLEM 15.3 Applications of Power of a Point
212(1)
PROBLEM 15.4 Trisecting Angles and Other Constructions
213(4)
Chapter 16 Inversions in Circles 217(16)
Early History of Inversions
217(1)
PROBLEM 16.1 Inversions in Circles
218(3)
PROBLEM 16.2 Inversions Preserve Angles and Preserve Circles (and Lines)
221(3)
PROBLEM 16.3 Using Inversions to Draw Straight Lines
224(2)
PROBLEM 16.4 Apollonius' Problem
226(4)
Expansions of the Notion of Inversions
230(3)
Chapter 17 Projections (Models) of Hyperbolic Planes 233(12)
Distortion of Coordinate Systems
234(2)
PROBLEM 17.1 A Conformal Coordinate System
236(1)
PROBLEM 17.2 Upper Half-Plane Is Model of Annular Hyperbolic Plane
237(2)
PROBLEM 17.3 Properties of Hyperbolic Geodesics
239(2)
PROBLEM 17.4 Hyperbolic Ideal Triangles
241(1)
PROBLEM 17.5 Poincare Disk Model
242(2)
PROBLEM 17.6 Projective Disk Model
244(1)
Chapter 18 Geometric 2-Manifolds 245(22)
PROBLEM 18.1 Flat Torus and Flat Klein Bottle
246(5)
PROBLEM 18.2 Universal Covering of Flat 2-Manifolds
251(2)
PROBLEM 18.3 Spherical 2-Manifolds
253(3)
Coverings of a Sphere
256(2)
PROBLEM 18.4 Hyperbolic Manifolds
258(5)
PROBLEM 18.5 Area and Euler Number
263(1)
Triangles on Geometric Manifolds
264(1)
PROBLEM 18.6 Can the Bug Tell Which Manifold?
265(2)
Chapter 19 Geometric Solutions of Quadratic and Cubic Equations 267(18)
PROBLEM 19.1 Quadratic Equations
268(4)
PROBLEM 19.2 Conic Sections and Cube Roots
272(4)
PROBLEM 19.3 Solving Cubic Equations Geometrically
276(4)
PROBLEM 19.4 Algebraic Solution of Cubics
280(2)
What Does This All Point To?
282(3)
Chapter 20 Trigonometry and Duality 285(14)
PROBLEM 20.1 Circumference of a Circle
285(2)
PROBLEM 20.2 Law of Cosines
287(3)
PROBLEM 20.3 Law of Sines
290(2)
Duality on a Sphere
292(1)
PROBLEM 20.4 The Dual of a Small Triangle on a Sphere
293(1)
PROBLEM 20.5 Trigonometry with Congruences
294(1)
Duality on the Projective Plane
294(2)
PROBLEM 20.6 Properties on the Projective Plane
296(1)
Perspective Drawing and History
297(2)
Chapter 21 Mechanisms 299(20)
Interactions of Mechanisms with Mathematics
299(4)
PROBLEM 21.1 Four-Bar Linkages
303(4)
PROBLEM 21.2 Universal Joint
307(3)
PROBLEM 21.3 Reuleaux Triangle and Constant Width Curves
310(4)
Involutes
314(3)
Linkages Interact with Mathematics
317(2)
Chapter 22 3-Spheres and Hyperbolic 3-Spaces 319(16)
PROBLEM 22.1 Explain 2-Sphere in 3-Space to a 2-Dimensional Bug
320(2)
What Is 4-Space? Vector Spaces and Bases
322(3)
PROBLEM 22.2 A 3-Sphere in 4-Space
325(3)
PROBLEM 22.3 Hyperbolic 3-Space, Upper Half-Space
328(2)
PROBLEM 22.4 Disjoint Equidistant Great Circles
330(2)
PROBLEM 22.5 Hyperbolic and Spherical Symmetries
332(1)
PROBLEM 22.6 Triangles in 3-Dimensional Spaces
333(2)
Chapter 23 Polyhedra 335(8)
Definitions and Terminology
335(1)
PROBLEM 23.1 Measure of a Solid Angle
336(2)
PROBLEM 23.2 Edges and Face Angles
338(1)
PROBLEM 23.3 Edges and Dihedral Angles
339(1)
PROBLEM 23.4 Other Tetrahedral Congruence Theorems
339(1)
PROBLEM 23.5 The Five Regular Polyhedra
340(3)
Chapter 24 3-Manifolds - The Shape of Space 343(20)
Space as an Oriented Geometric 3-Manifold
344(3)
PROBLEM 24.1 Is Our Universe Non-Euclidean?
347(2)
PROBLEM 24.2 Euclidean 3-Manifolds
349(4)
PROBLEM 24.3 Dodecahedral 3-Manifolds
353(2)
PROBLEM 24.4 Some Other Geometric 3-Manifolds
355(1)
Cosmic Background Radiation
356(4)
PROBLEM 24.5 Circle Patterns May Show the Shape of Space
360(1)
Latest Evidence on the Shape of Space
361(2)
Appendix A Euclid's Definitions, Postulates, and Common Notions 363(4)
Definitions
363(3)
Postulates
366(1)
Common Notions
366(1)
Appendix B Constructions of
Hyperbolic Planes
367(1)
The Hyperbolic Plane from Paper Annuli
367(1)
How to Crochet the Hyperbolic Plane
368(3)
{3, 7} and {7, 3} Polyhedral Constructions
371(1)
Hyperbolic Soccer Ball Construction
371(1)
"{3, 6%2}" Polyhedral Construction
372(3)
Bibliography 375(10)
Index 385

Excerpts

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 mathematics--it is a tool--a means to an end. The goal is understanding. Without understanding we will never be satisfied--with 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 answers--Why?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/senior-level 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

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