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Can anyone animate my trousers? (Patreon)

Published:
2021-07-19 18:42:47
Imported:
2026-06

Content

Ok, here's a fun maths fact that is so ridiculous it's making me angry. If you sew the two leg-holes of a pair of trousers together: the shape you end up with IS THE SAME AS A PAIR OF TROUSERS.

Fact 1: A pair of trousers is the same as a disk with two holes; which is to say a figure of 8. You can imagine cutting the legs shorter and shorter, past short-shorts, as well as cutting down the waist until you have effectively a 'g string' of material: a figure of 8.

Fact 2: A torus with a punctured hole in it is a figure of 8. You can see that in lovely animations like this: https://www.youtube.com/watch?v=j2HxBUaoaPU

Fact 3: Sewing the two leg-holes of a pair of trousers together gives you one continuous tube, other than the hole at the waist. So it's a torus with a hole. IT'S THE SAME SHAPE.

Fact 4: That. Is. Ridiculous.

I'm about to film a video where I talk about this. I'm going to physically cut down a pair of trousers to show they are a figure of 8 as well as showing how a punctured torus is the same shape, but I feel like it is going to miss something. It needs a CGI animation of a pair of trousers (aka pants) having the two legs joined together and then distorting back into the original pair of trousers. This could loop infinitely many times! Constantly being sewn together.

This is beyond my skillset or anyone I normally work with. Is anyone here able to make such a thing?

Comments

Your Average Chris

If we're modeling these as 2d manifolds with boundaries, then they aren't topologically identical (homeomorphic). You can prove this by noting that the boundary of a pair of trousers is three circles and the boundary of a torus with a hole is one circle. They may very well be homotopy equivalent. Still working on convincing myself of this. Will add more later. I feel like this is an important distinction that you could consider in your video. Another common example is that a ball is not homeomorphic to a point, but they are homotopy equivalent. One thing to note is that this means that any animation between the two is going to involve either planes shrinking down to lines or thickening into 3d. Just something to make sure your animators realize. I may try to code an animation mathematically, but I can't guarantee I'll have time. Edit: typo fix