Math 23 Final Project

Fall 2001

Every student in the course will do a final project which will count for 25% of their final grade. There are several choices for how the students can turn in their project:

• as a paper, or
• as a web site, or
• as a Mathematica notebook.

If you decide to create a web site or Mathemtica notebook for your final project, your grade will be based on two things:

• design and visual impact (30%), and
• content (70%).

The project can involve:

• History of non-Euclidean geometry. References can be found in the "Projects" section at the end of Chapters 5 and 6 of Marvin Greenberg's wonderful book Euclidean and Non-Euclidean Geometry.
• Philosophical implications of non-Euclidean geometry. References can be found in the "Some Topics for Essays" section at the end of Chapter 8 of Greenberg.
• Isometry groups of Euclidean and non-Euclidean geometry. This is for people who know some group theory (Math 26). The idea is to carefully describe the isometry groups of Euclidean and non-Euclidean geometry and prove that they are not isomorphic. The Euclidean isometry group is described in the appendix to Chapter 6 of McCleary (he does n-dim space, which is easy to adapt to the plane) and the non-Euclidean case is discussed in Theorem 15.9 of McCleary. See also Chapter 9 of Greenberg.
• Projective Geometry. One geometry we haven't mentioned is projective geometry. For example, the projective plane is obtained by identifying antipodal points on the sphere. This gives a geometry where every pair of lines intesect at a unique point. Some classical theorems of geometry, such as Desargues' Theorem are actually theorems of projective geometry. This is discussed in Chapter 2 of the book by Greenberg. See also the book Projective Geometry by Robin Hartshorne.
• Isometries in neutral geometry. One of my favorite theorems of neutral geometry asserts that every isometry is the composition of at most three reflections. A good project would be to read about this in the book Introduction to Hyperbolic Geometry by Ramsey and Richtmyer.
• Lobacheskii's The Theory of Parallels and Bolyai's The Science of Absolute Space. A nice project is to read some of the original books on non-Euclidean geometry. English translations of these appear in Non-Euclidean Geometry by Roberto Bonola.
• Classic papers in non-Euclidean geometry. The book Sources of Hyperbolic Geometry by John Stillwell contains papers by Beltrami, Klein and Poincar&eacute. These papers develop the models of non-Euclidean geometry studied in Chapter 15 of McCleary.
• Escher and non-Euclidean geometry. Many of the drawings of M. C. Escher embody concepts of non-Euclidean geometry, as expressed in the Poincarémodel. Escher corresponded with the emminent geometer H. S. M. Coxeter about non-Euclidean geometry. There are many books and web-sites related to this topic.
• A Mathemtica notebook on non-Euclidean geometry. For people who know (or want to learn) Mathemtica, a nice project would be to make a Mathemtica notebook which draws asymptotic parallels, horocycles, equidistant curves, non-Euclidean circles, etc.
• Map projections and differential geometry. In class, we will discuss the Mercator projection, which distorts distances and areas but preserves angles (and hence is useful for navigation). But there are many other ways to make a world map. For example, Lambert discovered a projection which messes up distances and angles but preserves areas. This topic has a nice relation to differential geometry. A basic reference is Chapter 8^bis of McCleary.
• Special relativity. There is a section in the book Linear Algebra by Insel, Friedberg and Spence dealing with this subject. There are also relations to hyperbolic geometry (the Weiestrass coordinates discussed on page 418 of Greenberg involve a 3-dimensional analog of special relativity). Also, Chapter 10 of the book "Introduction to Hyperbolic Geometry" by Ramsey and Richtmyer discusses some of this material.
• General relativity. The book The Geometry of Spacetime by Jame Callahan discusses general relativity at the undergraduate level, though it is not light reading. The book hasn't yet been published, but I have a copy of the manuscript that you could borrow.
• Chapter 6 of McCleary has some nice material on evolutes and involutes. The cycloid is an especially interesting example. For instance, you can ride a bicycle with square wheels along a cycloid and get a completely smooth ride.
• Another nice topic is curves in three-dimensional space. Here, in addition to curvature, one can also define the torsion of a curve. This is a measure of how the curve twists in space. One can show that the curvature and torsion characterize a curve uniquely up to rigid motion.
• The rotating disk discussed in class has an interesting metric. One can study geodesics for this metric. Also, it is not easy to understand what time means on the rotating disk, and there are still controversies about what Maxwell's equations mean in this context.

Projects need to be chosen by Friday, November 30, 2001. This is the Friday of the week after Thanksgiving. The project is due on the last day of finals period, which is Wednesday, December 19, 2001.

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