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!
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Geodesics
between conjugate points in the 3-sphere
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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.
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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.
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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!
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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.
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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!
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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.
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