Geometry from Euclid to Knots

Saul Stahl

• Publisher: Dover Publications
• Year: 2012
• ISBN:
  9780486134987
• Topic: Mathematics
• Subtopic: Geometry
• Index: Mathematics

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.

Chapter 1

Other Geometries: A Computational Introduction

In order to provide a better perspective on Euclidean geometry, three alternative geometries are described. These are the geometry of the surface of the sphere, hyperbolic geometry, and taxicab geometry.

1.1 Spherical Geometry

Due to its relationship with geography and astronomy, spherical geometry was studied extensively by the Greeks as early as 300 B.C. Menelaus (ca. 100) wrote the book Spherica on spherical trigonometry, which was greatly extended by Ptolemy (100–178) in his Almagest. Many later mathematicians, including Leonhard Euler (1707–1783) and Carl Friedrich Gauss (1777–1855), made substantial contributions to this topic. Here it is proposed only to compare and contrast this geometry with that of the plane. Because the time to develop spherical geometry in the same manner as will be done with Euclidean geometry is not available, this discussion is necessarily informal and frequent appeals will be made to the readers' visual intuition.

Strictly speaking there are no straight lines on the surface of a sphere. Instead it is both customary and useful to focus on curves that share the “shortest distance” property with the Euclidean straight lines. The following thought experiment will prove instructive for this purpose. Imagine that two pins have been stuck in a smooth sphere in points that are not diametrically opposite and that a (frictionless) rubber band is held by the pins in a stretched state. Rotate this sphere until one of the two pins is directly above the other right in front of your mind’s eye. It is then hard to avoid the conclusion that the rubber band will be stretched out along the sphere in the plane formed by the two pins and the eye—the plane of the book’s page in Figure 1.1. The inherent symmetry of the sphere dictates that this plane should cut the sphere into two identical hemispheres; in other words, that this plane should pass through the center of the sphere. It is also clear that the tension of the stretched rubber band forces it to describe the shortest curve on the surface of the sphere that connects the two pins. The following may therefore be concluded.

Figure 1.1. A geodesic on the sphere

Proposition 1.1.1 (Spherical geodesics) If A and B are two points on a sphere that are not diametrically opposite, then the shortest curve joining A and B on the sphere is an arc of the circle that constitutes the intersection of the sphere with a plane that contains the sphere's center.

Such circles are called great circles and these arcs are called great arcs or geodesic segments. They are the spherical analogs of the Euclidean line segments.

Diametrically opposite points on the sphere present a dilemma. A stretched rubber band joining them will again lie along a great circle, but this circle is no longer uniquely determined since these points can clearly be joined by an infinite number of great semicircles. For example, assuming for the sake of argument that the earth is an exact sphere, each meridian is a great semicircle that joins the North and South Poles. Hence, the aforementioned analogy between the geodesic segments on the sphere and Euclidean line segments is not perfect. It is necessary either to exclude such meridians from the class of geodesic segments or else to ...

Table of contents

• Cover
• Title Page
• Copyright Page
• Dedication
• Contents
• Preface to the Dover Edition
• Preface
• 1 Other Geometries: A Computational Introduction
• 2 The Neutral Geometry of the Triangle
• 3 Nonneutral Euclidean Geometry
• 4 Circles and Regular Polygons
• 5 Toward Projective Geometry
• 6 Planar Symmetries
• 7 Inversions
• 8 Symmetry in Space
• 9 Informal Topology
• 10 Graphs
• 11 Surfaces
• 12 Knots and Links
• 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
• Appendix F: Permutations
• Appendix G: Modular Arithmetic
• Solutions and Hints to Selected Problems
• Bibliography
• Index