http://www.amherst.edu/~dacox/math23/

Math 23 Topics in Geometry

Fall 2001
Monday, Wednesday, Friday 1 pm
Seeley Mudd 205

David A. Cox
404 Seeley Mudd
Phone: 413-542-2082
Email: dac@cs.amherst.edu

Overview of the Course

The parallel axiom of Euclid asserts the existence of a unique line through a given line and parallel to a given line. This axiom is the foundation of much of the Euclidean geometry learned in high school, including the Pythagorean Theorem and the fact that the sum of the angles in a triangle sum to 180 degrees.

The "big idea" of Math 23 is to study what happens when we no longer assume this axiom. For Fall 2001, the main topics of Math 23 will be:

• Neutral Geometry, which is the study of what theorems in geometry are valid with no assumptions about parallel lines. We will see, for example, that one can prove that the sum of the angles in a triangle is at most 180 degrees.
• Non-Euclidean Geometry, which is the geometry we get when we assume the existence of more than one parallel line. This wonderful geometry was discovered by Gauss, Bolyai, and Lobachevski. In this geometry, the sum of the angles in a triangle is strictly less than 180 degrees. Furthermore, this sum determines the area of the triangle, so that there are no similar triangles in Non-Euclidean geometry.
• Differential Geometry, which generalizes non-Euclidean geometry. The central notion here is Gaussian curvature. A sphere has constant positive curvature, and we will see that non-Euclidean geometry has constant negative curvature. The latter will enable us to construct some explicit models of non-Euclidean geometry.

One subsidiary topics we will explore is:

• General relativity, which asserts that gravity warps space and makes it non-Euclidean. We will discuss Einstein's 1912 thought-experiment which showed that the geometry of a rotating disk is not Euclidean!
  Note: Our treatment of relativity will not assume any previous knowledge of physics.

The figure shown above is from Models of the Hyperbolic Plane by Evelyn Sander. Click here for the full text of the article. Many interesting geometric images can be found at the graphics archive of the Geometry Center at the University of Minnesota.