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.
> 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.
danwills ·19 days ago
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crazygringo ·19 days ago
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.
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gosub100 ·19 days ago
https://youtu.be/wO61D9x6lNY?si=ecBgnOemKAbYZCrP
nejsjsjsbsb ·19 days ago
> 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.
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matsemann ·19 days ago
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
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