Turning a Torus Inside-Out (Punctured Torus Eversion)

Cut a small hole in a donut. Can you turn it inside-out? The answer, as shown in this looping video, is "yes!" Notice how the blue inside becomes the red outside, and back again. Unlike the classic question of "turning the sphere inside out," which considers non-physical motions where the sphere may pass through itself, here the (punctured) torus turns inside-out without any self-contact. In other words, you could really perform this motion if you had a donut made out of a stretchy rubber sheet. Mathematically speaking, this motion is an "isotopy": a continuous transition between states with no self-intersections (whereas the classic sphere eversion is a "regular homotopy": a continuous transition that can have self intersections, but no pinching or creasing).
The motion of the torus was computed using the algorithms developed in the paper
Chris Yu, Caleb Brakensiek, Henrik Schumacher, Keenan Crane
"Repulsive Curves" (2021)
For more information, see www.cs.cmu.edu/...
To compute a sphere eversion, one typically starts with a "mid-surface," i.e., a symmetric surface that exhibits no preferred orientation, applies a tiny perturbation (to break symmetry) and minimizes an energy that encourages smoothness to flow to the round sphere. To get the full "movie", this motion is then reversed in time, the colors are flipped, and the two movies are concatenated together. The punctured torus eversion was computed in the opposite way: starting with a torus, we make a small hole and then minimize an energy that encourages every point on the surface to repel every other point. To keep the surface from simply growing to infinity, we also constrain the surface area and penalize large increases in boundary length and curvature. This "repulsive energy" pushes the surface into the symmetric configuration seen at the halfway point between the red and blue torus. As with the sphere eversion, two movies are then concatenated to get the full motion.

[@netric9084]

0:05 is lookin a little sus

[@michaellellouch3682]

Sussy shapes

[@joshuasdsmalls2157]

That was a sharp corner.

[@Boomboozled]

Amongus

[@columbus8myhw]

I thought the same thing

[@shadow15kryans23]

That thumbnail lookin hella sus XD

[@HypercopeEmia]

amogus

[@Kram1032]

The half way point looks very pleasing, I like this a lot

[@samisiddiqi5411]

Amogus

[@pykapuka]

I did this in real life and it was very confusing how R_new became r_old/2 and r_new became R_old*2. Now I understand why, thanks a lot! (It is still a little confusing because this means eversion doesnt really work because it will always result in a new torus with R

[@oneCtwo]

Cool ! Actually, if the Torus is embedded in the 3-dimensional sphere instead of the 3-dimensonal euclidean space, then you can turn it inside out without making a hole in it !

[@jonwoods6745]

This looks great, thank you for your effort!

[@American_PhilippineA]

how to creat a wormhole:
turn a torus inside out, but only do it half way.
boom

[@yt249137b]

If there is a hole, you cannot say it as torus from topological point of view

[@AdamGenesisArt]

Any possible application to physics or quantum gravity? It might be possible to model sub atomic particles moving in this orientation. Thanks for taking the time to make such radically awesome math videos.

[@antennatrees]

Technically, can't you puncture any surface and turn it inside out if the material is sufficiently stretchy?. Nice animation. Not sure why you had to show it for 6 minutes over and over again though. Thurston's corrugations, now that's a video I could watch for 6 minutes ;)

[@columbus8myhw]

Perhaps a knotted torus would be more difficult to turn inside out.

[@keenancrane]

Not clear. If the torus is knotted, for instance, does pulling it through a small hole give you something in the same knot class? I’m not sure.
Also, the question is one of geometry rather than just topology: sure, you know that you *can* turn a punctured torus inside out. But what’s a nice-looking motion that actually realizes that transformation? People have spent many hours (and days, and weeks…) figuring out such motions by hand, like Thurston’s corrugated eversion that you mention. Here we wanted to push the boundaries of what can be done automatically, from a variational point of view (i.e., energy minimization). In particular, automatically finding motions that avoid self-intersection (i.e., isotopy rather than regular homotopy) is not something that’s been on people’s radar, because to date the computation and optimization has just been way too hard.

[@columbus8myhw]

@@keenancrane Following the Twitter discussion: no, turns out you can't turn a knotted punctured torus inside-out (without self-intersection)