Thread by @keenanisalive

Here's another fun question: given two loops around an (infinite) pole, can you remove one loop without breaking it?

Amazingly enough... yes!

This is a surprising example of what's called an "ambient isotopy": a continuous deformation of space taking one shape to another. 1/n

People have made some great drawings of this transformation over the years (sometimes using a loop rather than an infinite pole—which is equivalent), but it can still be hard to interpolate between individual drawings in your head.

(...does a movie make it any clearer?!) 2/n

What's also fun about the motion in the movie above is that it was created without* human input: instead, the computer tries to nudge the shape around so that every point is as far as possible from itself. You can read all about it in this thread:

https://twitter.com/keenanisalive/status/1468011750608056324

3/n

*I'll admit to one small piece of artistic license: the pole itself is also "repulsive," and tries to push the pink surface away. Near the end I increase the strength of repulsion, to make the final holes more even in size.

(Can you identify the moment where this happens?) 4/n

For a high-quality looping video, check out this link:

There's also a longer talk about the method behind the magic here:

And a paper about the algorithms here: cs.cmu.edu/~kmcrane/Proje…

Enjoy! n/n

(Big thanks by the way to @SaulSchleimer for bringing my attention to this example!)