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.