Geometry from Euclid to Knots

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ISBN: 9780486474595 | 0486474593
Cover: Paperback
Copyright: 3/18/2010

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Tracing the formal development of Euclidean geometry, this text closely follows Euclid's classic,Elements.In addition to providing a historical perspective on plane geometry, it covers related topics, including non-neutral Euclidean geometry, circles and regular polygons, projective geometry, symmetries, inversions, knots and links, and informal topology. Includes 1,000 practice problems. Solutions available. 2003 edition.

Preface to the Dover Edition	p. xi
Preface	p. xiii
Other Geometries: A Computational Introduction	p. 1
Spherical Geometry	p. 1
Hyperbolic Geometry	p. 9
Other Geometries	p. 21
The Neutral Geometry of the Triangle	p. 29
Introduction	p. 29
Preliminaries	p. 34
Propositions 1 through 28	p. 46
Postulate 5 Revisited	p. 81
Nonneutral Euclidean Geometry	p. 87
Parallelism	p. 87
Area	p. 99
The Theorem of Pythagoras	p. 112
Consequences of the Theorem of Pythagoras	p. 119
Proportion and Similarity	p. 122
Circles and Regular Polygons	p. 133
The Neutral Geometry of the Circle	p. 133
The Nonneutral Euclidean Geometry of the Circle	p. 141
Regular Polygons	p. 150
Circle Circumference and Area	p. 155
Impossible Constructions	p. 165
Toward Projective Geometry	p. 177
Division of Line Segments	p. 177
Collinearity and Concurrence	p. 184
The Projective Plane	p. 191
Planar Symmetries	p. 197
Translations, Rotations, and Fixed Points	p. 197
Reflections	p. 203
Glide Reflections	p. 210
The Main Theorems	p. 216
Symmetries of Polygons	p. 219
Frieze Patterns	p. 223
Wallpaper Designs	p. 228
Inversions	p. 247
Inversions as Transformations	p. 247
Inversions to the Rescue	p. 255
Inversions as Hyperbolic Motions	p. 259
Symmetry in Space	p. 269
Regular and Semiregular Polyhedra	p. 269
Rotational Symmetries of Regular Polyhedra	p. 281
Monstrous Moonshine	p. 288
Informal Topology	p. 295
Graphs	p. 305
Nodes and Arcs	p. 305
Traversability	p. 308
Colorings	p. 314
Planarity	p. 317
Graph Homeomorphisms	p. 326
Surfaces	p. 333
Polygonal Presentations	p. 333
Closed Surfaces	p. 346
Operations on Surfaces	p. 358
Bordered Surfaces	p. 367
Knots and Links	p. 379
Equivalence of Knots and Links	p. 379
Labelings	p. 385
The Jones Polynomial	p. 393
A Brief Introduction to The Geometer's Sketchpad®	p. 405
Summary of Propositions	p. 409
George D. Birkhoff's Axiomatization of Euclidean Geometry	p. 415
The University of Chicago School Mathematics Project's Geometrical Axioms	p. 417
David Hilbert's Axiomatization of Euclidean Geometry	p. 421
Permutations	p. 425
Modular Arithmetic	p. 429
Solutions and Hints to Selected Problems	p. 433
Bibliography	p. 447
Index	p. 451
Table of Contents provided by Ingram. All Rights Reserved.

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Geometry from Euclid to Knots

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Table of Contents

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