Experiencing Geometry : In Euclidean, Spherical and Hyperbolic Spaces

by Henderson, David W.

• ISBN13: 9780130309532
• ISBN10: 0130309532
• Edition: 2nd
• Format: Hardcover
• Copyright: 2001-01-01
• Publisher: Pearson College Div

Summary

For undergraduate-level courses in Geometry. Henderson invites students to explore the basic ideas of geometry beyond the formulation of proofs. The text conveys a distinctive approach, stimulating students to develop a broader, deeper understanding of mathematics through active participationincluding discovery, discussion, and writing about fundamental ideas. It provides a series of interesting, challenging problems, then encourages students to gather their reasonings and understandings of each problem and discuss their findings in an open forum.

Table of Contents

Preface	xv
Useful Supplements	xvii
My Teaching Background	xviii
Acknowledgments for the First Edition	xviii
Acknowledgments for This Edition	xx
How to Use This Book	xxii
How I Use this Book in a Course	xxiii
But, Do It Your Own Way	xxiv
Chapter Sequences	xxv
Message to the Reader	xxvii
Proof as Convincing Argument That Answers---Why?	xxix
What is Straight?	1	(14)
When Do You Call a Line Straight?	2	(13)
How Do You Construct a Straight Line?	4	(2)
The Symmetries of a Line	6	(5)
Local (and Infinitesimal) Straightness	11	(4)
Straightness on Spheres	15	(10)
What Is Straight on a Sphere?	15	(10)
Symmetries of Great Circles	20	(2)
*Every Geodesic Is a Great Circle	22	(1)
*Intrinsic Curvature	23	(2)
What Is an Angle?	25	(8)
Vertical Angle Theorem (VAT)	25	(1)
What Is an Angle?	26	(5)
Hints for Three Different Proofs	29	(2)
Duality between Points and Lines	31	(2)
Straightness on Cylinders and Cones	33	(12)
Straightness on Cylinders and Cones	34	(11)
Cones with Varying Cone Angles	36	(3)
Geodesics on Cylinders	39	(1)
Geodesics on Cones	40	(1)
Locally Isometric	41	(1)
Is ``Shortest'' Always ``Straight''?	41	(2)
Relations to Differential Geometry	43	(2)
Straightness on Hyperbolic Planes	45	(18)
A Short History of Hyperbolic Geometry	45	(3)
Constructions of Hyperbolic Planes	48	(7)
Hyperbolic Planes of Different Radii (Curvature)	55	(2)
What Is Straight in a Hyperbolic Plane?	57	(1)
The Pseudosphere is Hyperbolic	57	(3)
Rotations and Reflections on Surfaces	60	(3)
Triangles and Congruencies	63	(16)
*Geodesics Are Locally Unique	63	(1)
Properties of Geodesics	64	(1)
Isosceles Triangle Theorem (ITT)	65	(4)
Circles	66	(3)
Bisector Constructions	69	(1)
Side-Angle-Side (SAS)	69	(5)
Angle-Side-Angle (ASA)	74	(5)
Area and Holonomy	79	(20)
The Area of a Triangle on a Sphere	80	(1)
Area of Hyperbolic Triangles	81	(4)
Sum of the Angles of a Triangle	85	(4)
Introducing Parallel Transport and Holonomy	85	(4)
The Holonomy of a Small Triangle	89	(3)
The Gauss-Bonnet Formula for Triangles	91	(1)
Gauss-Bonnet Formula for Polygons	92	(7)
*Gauss-Bonnet Formula for Polygons on Surfaces	95	(4)
Parallel Transport	99	(10)
Euclid's Exterior Angle Theorem (EEAT)	99	(2)
Symmetries of Parallel Transported Lines	101	(3)
Transversals through a Midpoint	104	(1)
What is ``Parallel''?	105	(4)
SSS, ASS, SAA, and AAA	109	(8)
Side-Side-Side (SSS)	109	(2)
Angle-Side-Side (ASS)	111	(2)
Side-Angle-Angle (SAA)	113	(2)
Angle-Angle-Angle (AAA)	115	(2)
Parallel Postulates	117	(16)
Parallel Lines on the Plane Are Special	117	(1)
Parallel Transport on the Plane	118	(2)
Parallel Postulates Not Involving (Non-) Intersecting Lines	120	(3)
Equidistant Curves on Spheres and Hyperbolic Planes	122	(1)
Parallel Postulates Involving (Non-) Intersecting Lines	123	(3)
EFP and PPP on Sphere and Hyperbolic Plane	126	(7)
Comparisons of Plane, Spheres, and Hyperbolic Planes	128	(2)
Some Historical Notes on the Parallel Postulates	130	(3)
Isometries and Patterns	133	(18)
Isometries	133	(8)
Symmetries and Patterns	137	(4)
Examples of Patterns	141	(2)
Isometry Determined by Three Points	143	(1)
Classification of Isometries	143	(3)
Classification of Discrete Strip Patterns	146	(1)
Classification of Finite Plane Patterns	146	(1)
Regular Tilings with Polygons	147	(4)
*Geometric Meaning of Abstract Group Terminology	149	(2)
Dissection Theory	151	(10)
What Is Dissection Theory?	151	(2)
Dissect Plane Triangle and Parallelogram	153	(2)
Dissection Theory on Spheres and Hyperbolic Planes	154	(1)
Khayyam Quadrilaterals	155	(1)
Dissect Spherical and Hyperbolic Triangles and Khayyam Parallelograms	156	(1)
Spherical Polygons Dissect to Lunes	157	(4)
Square Roots, Pythagoras, and Similar Triangles	161	(16)
Square Roots	161	(2)
A Rectangle Dissects into a Square	163	(9)
Baudhayana's Sulbasutram	168	(4)
Equivalence of Squares	172	(2)
Any Polygon Can Be Dissected into a Square	173	(1)
Similar Triangles	174	(3)
Three-Dimensional Dissections and Hilbert's Third Problem	175	(2)
Circles in the Plane	177	(10)
Angles and Power Points of Plane Circles	177	(3)
Inversions in Circles	180	(3)
Applications of Inversions	183	(4)
Projections of a Sphere onto a Plane	187	(6)
Charts Must Distort	188	(1)
Gnomic Projection	188	(1)
Cylindrical Projection	189	(1)
Stereographic Projection	190	(3)
Projections (Models) of Hyperbolic Planes	193	(14)
The Upper Half Plane Model	194	(4)
Upper Half Plane Is Model of Annular Hyperbolic Plane	198	(2)
Properties of Hyperbolic Geodesics	200	(1)
Hyperbolic Ideal Triangles	201	(2)
Poincare Disk Model	203	(2)
Projective Disk Model	205	(2)
Geometric 2-Manifolds and Coverings	207	(26)
Geodesics on Cylinders and Cones	208	(4)
n-Sheeted Coverings of a Cylinder	209	(1)
n-Sheeted (Branched) Coverings of a Cone	210	(2)
Flat Torus and Flat Klein Bottle	212	(5)
Universal Covering of Flat 2-Manifolds	217	(2)
Spherical 2-Manifolds	219	(5)
*Coverings of a Sphere	222	(2)
Hyperbolic Manifolds	224	(4)
Area, Euler Number, and Gauss-Bonnet	228	(3)
*Triangles on Geometric Manifolds	230	(1)
Can the Bug Tell Which Manifold?	231	(2)
Geometric Solutions of Quadratic and Cubic Equations	233	(18)
Quadratic Equations	234	(4)
Conic Sections and Cube Roots	238	(4)
Roots of Cubic Equations	242	(4)
Algebraic Solution of Cubics	246	(5)
So What Does This All Point To?	248	(3)
Trigonometry and Duality	251	(14)
Circumference of a Circle	251	(2)
Law of Cosines	253	(4)
Law of Sines	257	(3)
Duality on a Sphere	258	(2)
The Dual of a Small Triangle	260	(1)
Trigonometry with Congruences	261	(1)
Duality on the Projective Plane	261	(1)
Properties on the Projective Plane	262	(3)
Perspective Drawings and Vision	263	(2)
3-Spheres and Hyperbolic 3-Spaces	265	(14)
Explain 3-Space to 2-D Person	266	(3)
A 3-Sphere in 4-Space	269	(3)
Hyperbolic 3-Space, Upper Half Space	272	(2)
Disjoint Equidistant Great Circles	274	(2)
Hyperbolic and Spherical Symmetries	276	(1)
Triangles in 3-Dimensional Spaces	277	(2)
Polyhedra	279	(8)
Definitions and Terminology	279	(1)
Measure of a Solid Angle	280	(1)
Edges and Face Angles	281	(2)
Edges and Dihedral Angles	283	(1)
Other Tetrahedra Congruence Theorems	283	(1)
The Five Regular Polyhedra	284	(3)
3-Manifolds---the Shape of Space	287	(18)
Space as an Oriented Geometric 3-Manifold	288	(3)
Is Our Universe Non-Euclidean?	291	(2)
Euclidean 3-Manifolds	293	(3)
Dodecahedral 3-Manifolds	296	(3)
Some Other Geometric 3-Manifolds	299	(4)
Cosmic Background Radiation	300	(3)
Circle Patterns Show the Shape of Space	303	(2)
Appendix A Euclid's Definitions, Postulates, and Common Notions	305	(4)
Definitions	305	(3)
Postulates	308	(1)
Common Notions	308	(1)
Appendix B Square Roots in the Sulbasutram	309	(12)
Introduction	309	(2)
Construction of the Savisesa for the Square Root of Two	311	(5)
Fractions in the Sulbasutram	316	(1)
Comparing with the Divide-and-Average (D&A) Method	317	(2)
Conclusions	319	(2)
Annotated Bibliography	321	(26)
AT Ancient Texts	321	(3)
CG Computers and Geometry	324	(1)
DG Differential Geometry	325	(2)
Di Dissections	327	(1)
DS Dimensions and Scale	328	(1)
GC Geometry in Different Cultures	329	(1)
Hi History	330	(3)
MP Models, Polyhedra	333	(1)
Na Nature	334	(1)
NE Non-Euclidean Geometries (Mostly Hyperbolic)	335	(2)
Ph Philosophy	337	(1)
RN Real Numbers	338	(1)
SE Surveys and General Expositions	338	(1)
SG Symmetry and Groups	339	(2)
SP Spherical and Projective Geometry	341	(1)
TG Teaching Geometry	341	(1)
Tp Topology	342	(1)
Tx Geometry Texts	343	(2)
Un The Physical Universe	345	(1)
Z Miscellaneous	346	(1)
Index	347