Mathematicians Discover New Way for Spheres to 'Kiss'

I'd really love to know what the mathematicians are actually doing when they work this stuff out? Is it all on computers now? Can they somehow visualize 24-dimensional-sphere-packings in their minds? Are they maybe rigorously checking results of a 'test function' that tells them they found a correct/optimal packing? I would love to know more about what the day-to-day work involved in this type of research actually would be!

It's strange the article doesn't even mention just trying to simulate the problem computationally.

Surely it's not too difficult to repeatedly place spheres around a central sphere in 17 dimensions, maximizing how many kiss for each new sphere added, until you get a number for how many fit? And add some randomness to the choices to get a range of answers Monte Carlo-style, to then get some idea of the lower bound? [Edit: I meant upper bound, whoops.]

Obviously ideally you want to discover a mathematically regular approach if you can. But surely computation must also play a role here in narrowing down reasonable bounds for the problem?

And computation will of course be essential if the answer turns out to be chaotic for certain numbers of dimensions, if the optimal solution is just a jumble without any kind of symmetry at all.

Sort of a tangent, but here's a 20m video explaining how to invert a sphere without tearing it:

https://youtu.be/wO61D9x6lNY?si=ecBgnOemKAbYZCrP

The interesting ta for me:

> Had she been one of his graduate students, he would have tried harder to convince her to work on something else. “If they work on something hopeless, it’ll be bad for their career,” he said.

> In two dimensions, the answer is clearly six: Put a penny on a table, and you’ll find that when you arrange another six pennies around it, they fit snugly into a daisylike pattern.

Is there an intuitive reason for why 6 fits so perfectly? Like, it could be a small gap somewhere, like in 3d when it's 12, but it isn't. Something to do with tessellation and hexagons, perhaps?

> They look for ways to arrange spheres as symmetrically as possible. But there’s still a possibility that the best arrangements might look a lot weirder.

Like square packing for 11 looks just crazy (not same problem, but similar): https://en.wikipedia.org/wiki/Square_packing

If you are wondering what sort of idiot would dispute a math problem with Newton, it was not a dispute at all; it's not even clear that they had a discussion at all: https://hsm.stackexchange.com/questions/5148/how-did-newton-...

> “There may be structures without any symmetry at all,” said Gabriele Nebe (opens a new tab) of RWTH Aachen University in Germany. “And no good way to find them.”

She taught Lineare Algebra II when I took it! It was one of the toughest lectures I took during university. I remember looking to the person next to me and one of us asked "do you understand anything?" and the other said "no! I haven't understood anything for like 20 minutes" and we burst out laughing and couldn't get it together until we were asked to quiet down. Wadim if you hang out here, send me a mail or something!

I took a class taught by David Huffman (of Huffman coding) called Cybernetics (IIUC it was the UCSC equivalent of a class Wiener taught at MIT.

The very first day, he started out by talking about kissing spheres and concluded the lecture with "and that's why kissing spheres are easy in 7 dimensions" (or something like that).

Every lecture of his was like being placed in front of a window looking upon a wonderful new world, incomprehensible at first, but slowly becoming more and more clear as he explained. Sometimes I wish I could play in the garden of math.

> Mathematicians often visualize this problem in terms of spheres. You can think of each code word as a high-dimensional point at the center of a sphere. If an error-filled message (when represented as a high-dimensional point) lives inside a given sphere, you know that the code word at the sphere’s center was the intended message. You don’t want these spheres to overlap — otherwise, a received message might be interpreted in more than one way. But the spheres shouldn’t be too far apart, either. Packing the spheres tightly means you can communicate more efficiently.

I went to math prep school for 2 years, attended 12 hours of math class in agebra and analysis per week, which I think proves I've done more math than most people in the general population, and this makes no sense to me. It either lacks introduction required to understand the analogy, or I've become really dumb. I want to understand this based on what the article says, but I can't. I can't represent error-filled messages as high-dimensional points. It's easier for me to imagine what the intersection between 4D spheres would look like in geometry.

I found this for anyone interested in understanding 4D spheres without knowing too much math: https://baileysnyder.com/interactive-4d/4d-spheres/

Mathematicians may have discovered it, but it will take an engineer or experimental physicist to capture a trace on the osculoscope.

two best spheres in a room; they might kiss :^)

if we believe the article, Li did all the work

and yet Cohn is first on the author list :(

what kind of kiss is that? certainly not a french kiss.