Geometry from Euclid to Knots

Format: Paperback
Pub. Date: 2010-03-18
Publisher(s): Dover Publications
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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.

Table of Contents

Preface to the Dover Editionp. xi
Prefacep. xiii
Other Geometries: A Computational Introductionp. 1
Spherical Geometryp. 1
Hyperbolic Geometryp. 9
Other Geometriesp. 21
The Neutral Geometry of the Trianglep. 29
Introductionp. 29
Preliminariesp. 34
Propositions 1 through 28p. 46
Postulate 5 Revisitedp. 81
Nonneutral Euclidean Geometryp. 87
Parallelismp. 87
Areap. 99
The Theorem of Pythagorasp. 112
Consequences of the Theorem of Pythagorasp. 119
Proportion and Similarityp. 122
Circles and Regular Polygonsp. 133
The Neutral Geometry of the Circlep. 133
The Nonneutral Euclidean Geometry of the Circlep. 141
Regular Polygonsp. 150
Circle Circumference and Areap. 155
Impossible Constructionsp. 165
Toward Projective Geometryp. 177
Division of Line Segmentsp. 177
Collinearity and Concurrencep. 184
The Projective Planep. 191
Planar Symmetriesp. 197
Translations, Rotations, and Fixed Pointsp. 197
Reflectionsp. 203
Glide Reflectionsp. 210
The Main Theoremsp. 216
Symmetries of Polygonsp. 219
Frieze Patternsp. 223
Wallpaper Designsp. 228
Inversionsp. 247
Inversions as Transformationsp. 247
Inversions to the Rescuep. 255
Inversions as Hyperbolic Motionsp. 259
Symmetry in Spacep. 269
Regular and Semiregular Polyhedrap. 269
Rotational Symmetries of Regular Polyhedrap. 281
Monstrous Moonshinep. 288
Informal Topologyp. 295
Graphsp. 305
Nodes and Arcsp. 305
Traversabilityp. 308
Coloringsp. 314
Planarityp. 317
Graph Homeomorphismsp. 326
Surfacesp. 333
Polygonal Presentationsp. 333
Closed Surfacesp. 346
Operations on Surfacesp. 358
Bordered Surfacesp. 367
Knots and Linksp. 379
Equivalence of Knots and Linksp. 379
Labelingsp. 385
The Jones Polynomialp. 393
A Brief Introduction to The Geometer's Sketchpad®p. 405
Summary of Propositionsp. 409
George D. Birkhoff's Axiomatization of Euclidean Geometryp. 415
The University of Chicago School Mathematics Project's Geometrical Axiomsp. 417
David Hilbert's Axiomatization of Euclidean Geometryp. 421
Permutationsp. 425
Modular Arithmeticp. 429
Solutions and Hints to Selected Problemsp. 433
Bibliographyp. 447
Indexp. 451
Table of Contents provided by Ingram. All Rights Reserved.

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