The Amazing Mathematical Flying Circus

Goal

The main deliverable of this project will be a video designed to convey to non-mathematicians some of the beauty, power and intrigue that mathematics holds in the eyes of its dedicated practitioners.

Movie Synopsis, (a la TV Guide)

The concepts of symmetry and abstraction are explored using an engaging mix of discussion, juggling, and computer graphics in this attempt to show people with little mathematics background the amazing mental acrobatics that are part of the mathematician's everyday life.

Why?

See this project's parents, YAMPOD, and DIIMatR, for a discussion of the motivation for this effort.

Detailed Description

The overall goal of this is to introduce people (students, adults, mathematicians who might not have noticed) to the beauty, power and fascinatingly broad applicability of abstract algebra. This presentation has been given in the form of a conference talk or a special classroom activity on numerous occasions to very receptive audiences.

So, in the most simplistic sense, the goal is just to capture the talk on videotape, with enhancements in the form of computer animations. In order to increase its value as an instructional tool, the talk will need further supporting materials. So, other planned components include a low-bandwidth hypertext-plus-graphics version to be put online, integrated with a simple, interactive application to facilitate further exploration. Also included will be supporting lesson plans and activities that could be used in classroom settings.

Here is a general idea of the talk that the movie will be building on.

Intro to symmetry

It starts off with a demonstration of the symmetry group of an equilateral triangle, first introducing the idea of symmetry, then posing the problem of finding all the triangle's symmetries, considering the symmetries as "motions", and noting the fact that doing one motion after another is equivalent to doing some other motion.

Abstraction: the "Anthenya" operator

When time permits, the audience is given the opportunity to work out the "multiplication table" of the "anthenya" operator. ("Anthenya" comes from lazy-ese for "and then you"--e.g. "if you rotate right, anthenya flip about the vertical, what do you get?" (For those familiar with the concept, this is just a user-friendly term for function composition.) We draw the group table on a whiteboard and look at it, noticing patterns (every element appears exactly once in each column and row, it isn't commutative (the first non-commutative group they have seen, and they will often notice this themselves). We talk about inverse and identity and how those relate to the same things in "normal" addition and multiplication.

The power of abstraction--concept and calculation reusability

Then we show how you can apply the same group to juggling three balls, and how you can actually learn something about juggling from looking at the group. On some occasions we've actually introduced the idea of homomorphism, because the homomorphism into Z₂ is just staring at you, and has a natural physical correlation to "things that flip the triangle over" and "things that leave the triangle on the same side".

A really neat thing to do here is to then show the problem of modeling six clubs passed between two jugglers. This gets really interesting because the patterns the jugglers do, when analyzed using permutation notation, turn out to have quite unexpected subpatterns which you can see, but wouldn't generally notice unless someone (like "the math") pointed them out to you.

Visualizing through computer animation

One of the really cool things that I haven't tried with an audience yet is showing the group of "motions" with actual motion, in the form of computer animation. This has some really exciting possibilities. See this for an example.