Geometry: From Euclid to Knots / Edition 1

Geometry: From Euclid to Knots / Edition 1

by Saul Stahl
     
 

ISBN-10: 0130329274

ISBN-13: 9780130329271

Pub. Date: 07/31/2002

Publisher: Pearson

The main purpose of this book is to inform the reader about the formal, or axiomatic, development of Euclidean geometry. It follows Euclid's classic text Elements very closely, with an excellent organization of the subject matter, and over 1,000 practice exercises provide the reader with hands-on experience in solving geometrical problems. Providing a

Overview

The main purpose of this book is to inform the reader about the formal, or axiomatic, development of Euclidean geometry. It follows Euclid's classic text Elements very closely, with an excellent organization of the subject matter, and over 1,000 practice exercises provide the reader with hands-on experience in solving geometrical problems. Providing a historical perspective about the study of plane geometry, this book covers such topics as other geometries, the neutral geometry of the triangle, non-neutral Euclidean geometry, circles and regular polygons, projective geometry, symmetries, inversions, informal topology, graphs, surfaces, and knots and links.

Product Details

ISBN-13:
9780130329271
Publisher:
Pearson
Publication date:
07/31/2002
Edition description:
New Edition
Pages:
480
Product dimensions:
8.58(w) x 8.30(h) x 0.89(d)

Table of Contents

Prefacexi
1Other Geometries: A Computational Introduction1
1.1Spherical Geometry1
1.2Hyperbolic Geometry9
1.3Other Geometries21
2The Neutral Geometry of the Triangle29
2.1Introduction29
2.2Preliminaries34
2.3Propositions 1 through 2846
2.4Postulate 5 Revisited81
3Nonneutral Euclidean Geometry87
3.1Parallelism87
3.2Area99
3.3The Theorem of Pythagoras112
3.4Consequences of the Theorem of Pythagoras119
3.5Proportion and Similarity122
4Circles and Regular Polygons133
4.1The Neutral Geometry of the Circle133
4.2The Nonneutral Euclidean Geometry of the Circle141
4.3Regular Polygons150
4.4Circle Circumference and Area155
4.5Impossible Constructions165
5Toward Projective Geometry177
5.1Division of Line Segments177
5.2Collinearity and Concurrence184
5.3The Projective Plane191
6Planar Symmetries197
6.1Translations, Rotations, and Fixed Points197
6.2Reflections203
6.3Glide Reflections210
6.4The Main Theorems216
6.5Symmetries of Polygons219
6.6Frieze Patterns223
6.7Wallpaper Designs229
7Inversions249
7.1Inversions as Transformations249
7.2Inversions to the Rescue257
7.3Inversions as Hyperbolic Motions261
8Symmetry in Space271
8.1Regular and Semiregular Polyhedra271
8.2Rotational Symmetries of Regular Polyhedra283
8.3Monstrous Moonshine290
9Informal Topology297
10Graphs307
10.1Nodes and Arcs307
10.2Traversability310
10.3Colorings316
10.4Planarity319
10.5Graph Homeomorphisms328
11Surfaces335
11.1Polygonal Presentations335
11.2Closed Surfaces348
11.3Operations on Surfaces360
11.4Bordered Surfaces369
12Knots and Links381
12.1Equivalence of Knots and Links381
12.2Labelings387
12.3The Jones Polynomial395
Appendix AA Brief Introduction to The Geometer's Sketchpad407
Appendix BSummary of Propositions411
Appendix CGeorge D. Birkhoff's Axiomatization of Euclidean Geometry417
Appendix DThe University of Chicago School Mathematics Project's Geometrical Axioms419
Appendix EDavid Hilbert's Axiomatization of Euclidean Geometry423
Appendix FPermutations427
Appendix GModular Arithmetic431
Solutions and Hints to Selected Problems435
Bibliography449
Index451

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