Comment by crazygringo
2 days ago
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.
Maybe the author thought it was so obvious that you could get some lower bounds that way, that it didn't seem worth mentioning! :) Wikipedia has a list, where I presume the lower bounds are mostly demonstrated constructively. Upper bounds must be demonstrated non-constructively, so I presume computers don't really help there.
https://en.wikipedia.org/wiki/Kissing_number#Some_known_boun...
Even in dimension 5, the kissing number is apparently known only as "42 plus or minus 2."
Here is an example of that sort of thing, using gradient descent as a starting point: https://arxiv.org/abs/math/0611451. It is technically about spherical codes rather than the kissing problem specifically, but they are closely related: https://en.wikipedia.org/wiki/Spherical_code