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

I have done differential topology and geometry for a time in University. There is a lot of visualisation going on in every lecture every day, even if it is not computer visualization. Everything in difftop is drawn at some point, just so you can get an understanding of what's happening. In fact, I would say that topology and subfields are formalisations of these beautiful and silly visual ideas.

One problem, however is that visualizations are by necessity simplifications, even in pure 2d cases like this one. That's not a sphere, that's a tessellation of one. Everything on a screen is differential and "smooth" (or actually, everything is discrete and thus not differentiable), but some things are symbols for "not differentiable here". Even these break down in pathological cases, singularities and such. This doesn't make the visualisation any less useful, but IMHO it can in most cases not be "the" reasoning, only a guide to reasoning.

Besides,as the article points out, there are many non-constructive proofs that don't have any visuals.

Have a look at how vsauce explains Banach-Tarsky [1] or any of 3blue1brown's excellent videos to see the edges of what's possible with visualisations.

[1] https://youtube.com/watch?v=s86-Z-CbaHA [2] https://youtube.com/watch?v=zjMuIxRvygQ

"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/

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.

This is a book I keep recommending as well. It is very fun (and also includes a sphere eversion).

I really like the mathematical impressions from Fomenko

http://chronologia.org/en/math_impressions/images.html

For programmable graphics, I like Penrose https://www.cs.cmu.edu/~kmcrane/Projects/Penrose/Penrose_SIG...

And finally the same author as Penrose (Keenan Crane) has a challenge for creating beautiful visualisation of abstract and difficult math concepts, but at the moment I'm unable to find the link

And the first visualization of the sphere eversion was found by a blind mathematician Bernard Morin.

That's exactly it! I'm sure there are extremely talented 'visual' thinkers who just can't grasp standard mathematical symbology. But is their mathematical ability in any way diminished by this?

Bees make hexagonal combs. Bower birds weave elaborate, decorative structures to attract mates. And spiders and their webs!

Now that we have the tools, it's time to attract those visual/tactile thinkers who don't even realize that they're natural mathematicians...

> Bees make hexagonal combs.

I can't speak to the birds or spiders, but hexagon packing in combs can be explained without endowing bees with mathematical reasoning. It's a geometrically efficient structure that they've likely happened upon as a result of collective behavior (large numbers of bees constructing adjacent cells simultaneously). Check out Philip Ball's Shapes for a detailed account of the "accidental" emergence of many complex natural structures.

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...

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.

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

https://youtu.be/sKqt6e7EcCs

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.

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?

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.

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

Obviously it has to pass through itself. The interesting thing is that it is possible to evert it without pinching or creasing.

Yes, that’s why you can’t do it in 3 dimensions (without self-intersection) - if you have an extra dimension to hand you can invert the sphere using the extra dimension to avoid self-intersection I believe.

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...

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.)

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.

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?

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

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

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.

I'm not sure what sense you're using "the same" in. Eg a cylinder can be transformed into a mobius strip (in 3d space) by cutting and gluing, but if we allow only self-intersection, this can't be done. To me this shows that they're not "the same".

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

This is beautiful

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.

The vid says the surfaces are allowed to pass through each other. I don't see how that corresponds to anything in the real world.

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

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

If you just look at the set of points at each instant in time, the self-intersections would produce discontinuities. So instead of deforming a raw set of points in 3D space, topologists deform a function which takes a point on the sphere and returns a point in 3D space: https://en.wikipedia.org/wiki/Homotopy

Mechanically, you get this deformation by adding another parameter to the function between spaces. In Go-ish pseudo-code, say at each instant of time you have a function

    // lon ranges from -180 to 180
    // lat ranges from -90 to 90
    func eversion_t(lon float, lat float) (x float, y float, z float) {
      // return the xyz point corresponding to the lon/lat
    }

which maps lon/lat points on the sphere to 3D space. Then the homotopy is a single function, parameterized by a t parameter

    // t ranges from 0 to 1 inclusive
    // lon ranges from -180 to 180
    // lat ranges from -90 to 90
    func eversion(t float, lon float, lat float) (x float, y float, z float) {
      // return the xyz point corresponding to the lat/lon at time t
      // see https://arxiv.org/pdf/1711.10466.pdf for the implementation of this function
    }

where this combined function is required to be continuous.

By the way, the "push the ends of the sphere through each other" function is a perfectly valid homotopy. There's no topological way to talk about "creasing" -- you need derivatives for that. In particular, the eversion function is required to be an immersion (https://en.wikipedia.org/wiki/Immersion_(mathematics)) at each point in time, which is an additional constraint beyond just being a homotopy.

I don't even think you need to go this far. Since it's differential topology, everything is defined locally anyway, so there must just be environments of nonzero size around every point that don't intersect themselves.

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

> 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.

It’s not the same. The article goes to that in the second visualisation, and shows that the loop/curl strategy produces a singularity of sorts, making the process non-continuous (two points for the circle, for the sphere it’s a whole circle). Once that is shown it goes on to show it going continuously, even though matter passes through other matter.

Probably because topology isn't concerned with absolute coordinates. It's all about local connectivity.

B-E condensates? Or as someone else said a fluid could model it is you had control over it.

I was wondering if there's some real process that uses the "twist" trick, even if it's not inverting a sphere.

Packing a pop-up tent maybe? Always seems difficult enough..!

The stowable shades people use to cover car windshields and windows use a twist to fold them up without having to crimp/crease the stiff wire on the exterior that gives it its support.

In the real world you usually can't pass a material through itself.

I wonder if that couldn't be possible with colored gaz or liquid (magnetic "liquids" that hold together). That would be quite fascinating to see IRL.

From the article:

> Note that unlike physical objects, self-intersection is permitted.

Spectacular presentation