Dance a dance of four-space.

Got two hours to spare? It could open up a whole new dimension in your life.

No, this is not some Amway scam, new-age Woo, or political revival. It's a series of brilliant videos (along with explanatory text) put together by a French mathematician which explore the existence of a fourth spatial dimension. And it is *very* cool. From ScienceNews:

So can any of these techniques help us visualize Schläfli’s 600-sided, four-dimensional shape? Using a computer, Ghys first passes Schläfli’s regular, four-dimensional shapes through three-dimensional space and looks at the three-dimensional “slices” created. This helps a bit, but just as in two dimensions, it’s not easy to assemble an image of the higher-dimensional shape this way.

Next, he draws the three-dimensional “shadows” of the four-dimensional objects. This turns out to be much better: Rotating the objects around to see different facets of them can give a pretty good feeling for their shapes.

Finally, he uses stereographic projection. The idea is the same as projecting from three to two: You blow the four-dimensional shape up into a ball, and then you place a light at the “north pole” and project the image down into three dimensions. That process is all-but-impossible for us to visualize, just as the process of projecting a three-dimensional ball would be impossible for the lizards to imagine. The results, though, are gloriously easy to make sense of.

OK, for this old dog it is still a bit tough - my imagination is not as supple as it once was. But even I could start to get glimpses, on the first viewing. I plan on taking the time to make at least one more pass at the series. For someone such as myself who lacks the mathematics background to really understand what is going on, this is a very helpful tool. Seriously - give it a try. It could open up a whole new dimension for you.

Jim Downey

(Via MeFi. Cross posted to my blog.)