Mathematics TV clips I have just come out of a talk on presenting science to the public via mainstream media. I became jealous of what other sciences, such as astrophysics, have achieved in these media.
This is tremendously borderline for MO, so feel free to close. But I'd like to solicit ideas for 2-3 minute TV clips about mathematics which could successfully compete with "private lives of celebrities" for the public's attention. Maybe TV producers could then glance through someday, and turn one or more of our fantasy TV clips into real TV clips.
Criteria are:
*

* I'd like us to focus on 2-3 minute pieces.

* No "talking heads". People will change channel rather than watch two people talking. Everything has to be visual.

* The mathematical idea has to play lead role in some sense ("oh the thinks you can think!"), and has to be communicated to some degree. It shouldn't be precise, but it shouldn't be wrong either.

* The scene has to be set, and the viewer's interest must be grabbed- not just some sitting at a desk.

* No "green peas"- nobody "should know". It should be popularization as opposed to education.

* The target audience has to be the general public, as opposed to people predisposed to watching Discover Channel and to reading scientific magazines. 

Here's my concept as I was listening to the talk, to start things off:
[A 2-minute presentation on basic knot theory. No words needed- it can (and maybe should) be a music video.]
A poor boy in India sees a street conjurer making various knots by a wrist movement on a rope with a weight at the end [there's an old Indian gentleman in Oakland who can do this]. It fills him with wonder, and he resolves to upgrade the trick by knotting a knot no knotter has knotted- a bodacious knot trick to shock the world.
Sinking into his imagination [effect], you see him throw the rope, which wraps itself in an amazing Bollywood way into something which looks like really complicated (from this MO question). He shows it to the conjurer, who untangles it in a second without touching the ends [maybe cut to a computer animation of this, which would be cleaner]. He tries again and fails again, maybe a few times. Eventually he brings one of Haken's gordian unknots, which is also untangled.
He returns dejected, and then suddenly "inspiration strikes". He throws a superhuman Bollywood knot confidently in front of the conjurer, who, try as he may, cannot untie it (without touching the ends). Overlaying animation, a tricolouring flashes over the knot like a dragon, which moves and changes with the knot as the conjurer struggles with it, never changing. Maybe "Can't get rid of the colours!" flashes on-screen, to complement the visual cue of what is going on. The boy wins.
However, the conjurer unknots it to a trefoil, which is an insufficiently bodacious knot, and the boy tries a final time and produces a better knot, which isn't a trefoil either because it's 5-colorable. Again, this is shown with visual effects, blending animation with live action. With more time one might try more colourings.
The boy laughs, and the conjurer stares at him in wonderment- the finest knotter who ever has knotted! Maybe then it returns to the real world, and the boy walks away.
 A: I would be interested to make a short video to show how we can use geometry to explain some things related to an optical illusion. The "spinning dancer" is a rotating silhouette, which appear at the second sight that you can't tell in which direction rotates. At first sight it seems that everybody choses a particular direction.
The script is something like this


*

*introduce the illusion

*let the viewer decide in which direction she thinks the ballerina rotates

*explain why two people out of three see the dancer spinning clockwise

*explain why the correct answer is that the dancer spins counter-clockwise 


(I explained the last two points here, using simple geometry in space and some elementary notions on perspective)
A: Many aspects of our mythos are not generally known: the uncountability of the reals, the classification of surfaces, the approximation of waves via simple harmonic oscillators, the infinitude of primes, the conformal mapping theorem, etc. etc. In fact, most of what we take to be well-known is unknown to the layman. In the 1960s Time-Life books, and Bell-labs and other companies popularized various topics. 
Now independent producers are doing the same. Vi Hart comes to mind, mathematical artists, and yes youtubers and math rappers :D
So the desideratum of Daniel may be coming about through the new media. 
It might be better to be on Discover or History if only if late at night. It would certainly be better than the ghost and paranormal stuff --- Wait, how about para-compact stuff! 
Instead of thinking how good it would be if we did this, then do it. Two people, a 200 video recorder, and a tripod is all that you need. As you get better, add production value. 
A: I will be impressed if you can do better than Martin Gardner.  He wrote articles meant for consumption within minutes (although not 2 or 3), and provided challenges to help maintain interest.  He did articles outside of Scientific American, but I don't know that Gardner could compete for audiences that read things like People magazine.
However, many magazines publish puzzles.  If the topic presented include a simple easily solved puzzle and one not so easily solved, that may draw as good an audience as anything, especially if little or no "higher reasoning" is involved.  Even so, squeezing a topic into two or three minutes is a challenge.  At three words a second, that is less than 600 words, which fits into a MathOverflow comment.  Consider a ten minute version instead.
Gerhard "Ask Me About System Design" Paseman, 2011.02.23
A: I want for years to do a short video based on an idea I had in 2004:
Polyhedral groups
which goes like this:


*

*introducing the Platonic solids

*showing by animations the corresponding polyhedral groups

*showing by animations what dual polyhedra are, and why their groups are the same.


So far, this was known stuff. Here is the new part:


*

*showing that we can label the vertices and edges with permutations, so that we can use two identical polyhedra to compose permutations. I describe this idea here.

