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More Mathematics . . .

This section contains a hodge-podge collection of mathematical entertainments. Some of the links connect to activities that I put together for various courses I've taught, others connect to interesting things that I simply wanted to share. In particular, you can find here streaming feeds (through You-Tube) of various short animated films that I've put together which illustrate different mathematical objects.

I think that every college student should read the following wonderful essay by Prof. T.W. Körner (Cambridge University), which should perhaps be titled:

How to listen to a mathematics lecture

A brief description accompanies each link below. Enjoy!

Geodesics between conjugate points in the 3-sphere

Stereographic Projection

  • This film (approx. 5 mins long) illustrates streographic projection for the circle and the 2-sphere keeping in mind the goal of training your mind to visualize objects in the 3-sphere using stereographic projection. Hopefully, the animations will help you to understand this conformal map.

Cyclic Subgroups of the Loop Group of SU(2)

  • I don't really know what to say about these, other than that I think they are really neat.

Hopf Fibration Experience

  • Alan Hatcher's book on algebraic topology has a very nice description of how to view the Hopf fibration of the 3-sphere by circles. I took up the task of creating an animation which illustrated this geometrically. I literally computed an infinitesimal generator of the circle action in stereographic coordinates and used a Runge-Kutta scheme to evolve the orbits. Low and behold Hatcher's picture emerged! The way I see it, if you're going to make a mathematics movie, why not do it with some style!

Limits with von Koch's Curve, Sierpinski's Gasket and the Chaos Game

  • This was a class activity I put together for my Brief Calculus students. It uses limits to compute the "perimeter" of von Koch's snowflake curve, and the area of Sierpinski's gasket. It finishes with a link to a nice java applet hosted on a German site which plays the chaos game rapidly.

Fun with Möbius strips

  • I put together this activity a number of years ago. It has some accompanying worksheets and is a hands-on exploration of some properties of Möbius strips. Great for junior mathematicians!

Fascinating Property of Circles

  • This is another very amusing hands-on activity which illustrates an interesting fact from classical algebraic geometry. The worksheet shows how to use paper folding to construct an interesting family of lines associated to a given circle. The family lines has an envelope which one can see directly in the folds in the paper. That's all that I am going to say. I don't want to give away what happens.