
Sphere Eversion - colinprince
https://rreusser.github.io/explorations/sphere-eversion/
======
J253
I love the fact that a field like topology, with all its underlying
mathematical complexity, also has a such a beautiful visual aspect to it.

Does anyone know if people try to tackle topology problems from the visual
side? Before computers I imagine it wasn’t really considered. But say one is
curious about a particular geometry, any researchers just whip it up in
software and start contorting things to see what happens?

Beautiful visualization, by the way. Very cool use of Idyll. Watching the
sphere evert reminded me of trying to solve those complex wooden burr puzzles.

[https://wikipedia.org/wiki/Burr_puzzle](https://wikipedia.org/wiki/Burr_puzzle)

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Jarmsy
"A topological picturebook" by George K Francis is full of wonderful hand
drawn visual explorations of topological concepts.
[http://www.probehead.com/log/texts/Francis/](http://www.probehead.com/log/texts/Francis/)

~~~
sevensor
I had the great fortune to get my start programming as an REU for George. We
were visualizing non-Euclidean spaces in VR, in the UIUC CAVE around the turn
of the millenium. He also had an animated (minimax) sphere eversion with
special audio cues that would play when interesting things happened. His
website is still up, and I think some of his more recent REUs have been
working on WebGL versions. Only about 5 percent of the math stuck, since I was
not a math major, but I really came to appreciate affine transformations.

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Smaug123
There's a proof project taking place in the Lean theorem prover right now! I
don't know whether the method is the same as that in the article (I'm pretty
sure it's not).

[https://leanprover-community.github.io/sphere-
eversion/bluep...](https://leanprover-community.github.io/sphere-
eversion/blueprint/dep_graph.html)

~~~
kmill
Yeah, it's not. They're following something like Smale's proof as best as I
understand. It will give some general useful topological tools for mathlib.

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magnio
I watched a great YT video on this. Still kinda hard to believe it's possible.

[https://youtu.be/sKqt6e7EcCs](https://youtu.be/sKqt6e7EcCs)

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Causality1
This is a topic I read about every few years and I always end up having to
just trust that they're right because I can't follow the math and they're
clearly using definitions for "crease" and "smooth" that don't match my own.

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war1025
I think the part that turns me off to it is "Look we can flip a sphere inside
out" followed shortly after by "You can't do this in real life".

So what's the point then? Who cares?

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caconym_
The point is that it's a solution to a difficult mathematical problem, and a
particularly interesting one since it has a different result with one less
dimension and naively there's no reason to expect any different in 3D.

A lot of math is like this, in the sense that you can't set it up in the real
world and prove it's true with physical objects, but it's still interesting
and often practically useful. I guess the difference here is that it's _close_
to being physically possible, causing frustration here among HN readers? But
it's odd to see so much disparagement of the result. This is a famous result
in topology, with good reason.

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heavenlyblue
Doesn’t the sphere intersect itself while this operation is performed?

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cevn
Thanks, this confused me greatly watching the graphics, I'm no
mathematician... OK if we can just push the sphere through itself then yeah I
guess you can invert it...

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caconym_
Can you do the same with a circle?

(In case you didn't read the article carefully—no, you can't, and that's why
this is interesting.)

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roenxi
Piece of advice for the lazy: examine the links closely. The "Outside In"
video, linked in the article, is substantially more interesting than the text.

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bane
This trick reminds me of that game ten year old children play where one of
them shoots the other with a "gun" and the child who was "hit" suddenly
declares a new rule, "I have a forcefield!".

Meaning, you can't pinch or cut. So sphere eversion is obviously
impossible...but the suddenly, "I can pass surfaces through one another!"

So I guess the game here is not in finding interesting solutions within a set
of constraints (no pinching or cutting!) but in just making up arbitrary new
rules when the problem becomes intractable?

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zodiac
If you think the constraints make it trivial, try producing a solution with
the same constraints. (intersection allowed but pinching and cutting not)

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bane
I think the constraints make it impossible. It's obvious that allowing free
passage through a surface is the same as cutting.

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aesthesia
Self-intersection is not the same as cutting. The disconnect you’re
experiencing is between the mathematical description of the problem—-does
there exist a smooth homotopy of immersions between two embedding a of the
2-sphere in 3-dimensional space—and the colloquial description. The
mathematical phenomenon corresponding to cutting would be discontinuity of
that homotopy, which is different from the description of self-intersection,
which amounts to the difference between an embedding and an immersion.

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IshKebab
This is amazing. Must have been a lot of work to make!

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solveit
This is beautiful

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The_rationalist
Why hasn't this been achieved in the real world?I strongly feel they need a
mathematical trick inexistant in the real world, either a tear or a
supplementary dimension.

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om2
Real material objects can't self-intersect. Self-intersection is the only
trick this uses that is nonexistent in the real world.

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pdonis
How is self-intersection allowed in the math? Isn't that inconsistent with
maintaining the topology?

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riffraff
Exactly my question. Also why isn't the same allowed in 2d for the circle at
the beginning?

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caconym_
> why isn't the same allowed in 2d for the circle at the beginning?

It is. The picture illustrating the impending cusps is after the two sides of
the circle have passed through each other.

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ridaj
Spectacular presentation

