Friday, September 18, 2015

Two Classic Clifford Tori Animations

After much rummaging around my hard drive, I finally found some Clifford tori animations from my old site that clearly give a much better sense of how the (stereographically projected) tori change as the angle $\varphi$ changes from $0$ to $\pi/2$ (in the notation of the last 2 posts on this subject). Here, we've used the Clifford circles to see its effect on them as well.

It starts off with the first degenerate case of one single unit circle, and expands from there. We see that it eventually comes very close to the other degenerate case, that of the straight line, the $z$-axis.

Our next video requires more explanation. This time, we take one particular torus, namely the one of identical radii $\frac{1}{\sqrt{2}}$ (in the video, these two particular circles are highlighted red and blue). Now, a $3$-sphere, like any sphere, can be rotated (by a matrix, or a whole path of matrices, in $SO(4)$). Thus, of course, such a rotation can always be realized as a rigid motion of the ambient $4$-space containing this $3$-sphere. It is possible to continuously rotate it so that the torus within has beginning and ending configurations that look the same, except that the red and blue circles have been swapped. If we restrict ourselves to $3$-space, such a rigid motion is impossible, but if we allow ourselves to let the torus pass through itself, then it, too, can be done. However, visualizing the $3$-sphere version in stereographic projection, with a $4$-space rotation, we effectively allow ourselves to distort distances (actually the $4$-space distance is not distorted; the distortion we see is an artifact of the stereographic projection), and add a "point at infinity," so a continuous rotation is allowed to take things through that point. The rotation of the ambient $3$-sphere does not preserve our usual set of nested tori, as can be seen by letting a matrix in $SO(4)$ act directly on the coordinates of our parametrization: it jumbles up all the components. So, of course, the torus undergoes a completely different kind of motion than in our previous "expander" video.

What happens is we inflate our inner tube, so a part of it gets puffed up to infinity, and wraps back around, turning the torus inside-out. In fact, after wrapping back around, we're "inflating" the outside of the torus. Or equivalently, getting back to donuts with frosting, the dough gets bigger and bigger, and when wrapping back around, almost all of space (plus a point at infinity) is dough, and the frosting bounds an inner-tube-shaped pocket of air.

Anyway, the full turning inside-out (which also swaps the red and blue circles, as promised) occurs exactly halfway through the movie (the rotation continues to restore the torus to its original state in the second half). Notice how the stripes on the torus which started out horizontal now are vertical, and what used to be the "apple core" shape which surrounds the donut hole now has become a "donut segment." Plus it just looks totally awesome!

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