Geometry from Euclid to Knots
by Stahl, SaulFormat: Paperback
Pub. Date: 2010-03-18
Publisher(s): Dover Publications
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Summary
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 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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