Geometry from Euclid to Knots

(Dover Books on Mathematics)

Saul Stahl

Dover Publications | 2010 | 480 páginas | rar - epub | 22,7 Mb

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Designed to inform readers about the formal development of Euclidean geometry and to prepare prospective high school mathematics instructors to teach Euclidean geometry, this text closely follows Euclid's classic, Elements. The text augments Euclid's statements with appropriate historical commentary and many exercises — more than 1,000 practice exercises provide readers with hands-on experience in solving geometrical problems.

In addition to providing a historical perspective on plane geometry, this text covers non-Euclidean geometries, allowing students to cultivate an appreciation of axiomatic systems. Additional topics include circles and regular polygons, projective geometry, symmetries, inversions, knots and links, graphs, surfaces, and informal topology. This republication of a popular text is substantially less expensive than prior editions and offers a new Preface by the author.

Contents
Preface to the Dover Edition
Preface
1 Other Geometries: A Computational Introduction
1.1 Spherical Geometry
1.2 Hyperbolic Geometry
1.3 Other Geometries
2 The Neutral Geometry of the Triangle
2.1 Introduction
2.2 Preliminaries
2.3 Propositions 1 through 28
2.4 Postulate 5 Revisited
3 Nonneutral Euclidean Geometry
3.1 Parallelism
3.2 Area
3.3 The Theorem of Pythagoras
3.4 Consequences of the Theorem of Pythagoras
3.5 Proportion and Similarity
4. Circles and Regular Polygons
4.1 The Neutral Geometry of the Circle
4.2 The Nonneutral Euclidean Geometry of the Circle
4.3 Regular Polygons
4.4 Circle Circumference and Area
4.5 Impossible Constructions
5 Toward Projective Geometry
5.1 Division of Line Segments
5.2 Collinearity and Concurrence
5.3 The Projective Plane
6 Planar Symmetries
6.1 Translations, Rotations, and Fixed Points
6.2 Reflections
6.3 Glide Reflections
6.4 The Main Theorems
6.5 Symmetries of Polygons
6.6 Frieze Patterns
6.7 Wallpaper Designs
7 Inversions
7.1 Inversions as Transformations
7.2 Inversions to the Rescue
7.3 Inversions as Hyperbolic Motions
8 Symmetry in Space
8.1 Regular and Semiregular Polyhedra
8.2 Rotational Symmetries of Regular Polyhedra
8.3 Monstrous Moonshine
9. Informal Topology
10 Graphs
10.1 Nodes and Arcs
10.2 Traversability
10.3 Colorings
10.4 Planarity
10.5 Graph Homeomorphisms
11 Surfaces
11.1 Polygonal Presentations
11.2 Closed Surfaces
11.3 Operations on Surfaces
11.4 Bordered Surfaces
12 Knots and Links
12.1 Equivalence of Knots and Links
12.2 Labelings
12.3 The Jones Polynomial
Appendix A: A Brief Introduction to The Geometer's Sketchpad®
Appendix B: Summary of Propositions
Appendix C: George D. Birkhoff's Axiomatization of Euclidean Geometry
Appendix D: The University of Chicago School Mathematics Project's Geometrical Axioms
Appendix E: David Hilbert's Axiomatization of Euclidean Geometry