That was great, the only thing missing was why. I am sure there are some interesting applications of the principles they explored and I would have loved to have heard a few at the end. Feel free to fill me in.
When you avoid tight bends and creases (singularities---places where the derivative vanishes), you have a nice structure on which you can do useful things like take derivatives, measure lengths and angles, find "straight" lines, and such. In short, the structure behaves, as long as you're small enough, just like the space we live in and have studied for centuries.
Now you can extend this notion from a property of one surface to a property of a continuous family of surfaces, and then look at the beginning and the end (imagine it varying with respect to time, just like in the video). The niceness property above, if maintained throughout, gives you more things to say about what happens to substructures. For example, you can say something like "if I start with a triangle inside this surface, and morph it into another surface keeping it nice and smooth the whole time, I'll still get a triangle, and it won't do {big class of bad things}".
Was there a practical reason for the restrictions on tight bends and creases?
sigh You'll understand when you're building biological spaceships: you see, the inversion of n-spheres under those conditions is the simplest workable model for how evolutionary phenomena interact with each other.
The inversion of an n-sphere of dimension 3 is representative of the evolution in just one layer of a system. After you discover the generalization of inversion across n-space in 2072, it will just take one ambitious biologists to apply it to genetic data, and generalize to most of the evolutionary phenomena around you: memes and cultures. By 2103, Human Directive 72 will be achieved, and all children under the age of four will be able to conceive of the basic geometries of infinite-dimensional topological models.
There is a large branch of mathematics which this is tied to called Knot Theory, I'll bet you could find some practical applications associated to it. (http://en.wikipedia.org/wiki/Knot_theory)
So well spent that it didn't even feel like 20 minutes. This is what videos should be like-- not essays delivered more slowly and painfully, but real useful things presented visually in a way that takes full advantage of the medium.
Sorry for the lack of clarity. I meant that the form of presentation should be useful for aiding understanding, not coincidental. I didn't mean to implythat the content itself should be useful (in the sense that it has practical applications). It's OK for a video to just be a little mental puzzle.
Another one of their videos is "Not Knot": http://www.youtube.com/watch?v=AGLPbSMxSUM
But I can't find a link to their latest video, called "The Shape of Space".
This was made in 2004. I wonder how much easier it would be to do this with 16 years of advances in 3d modeling?
I wish it were more common...