[21723]

Ball packing (no, it's not some new thing Bay Area VCs are doing--I mean, it probably is also that but it's not what I'm referring to here; rather, it's the n-dimensional analogue of sphere packing) gets stranger in n-dimensions as well.

If you have an n-dimensional integer lattice with unit n-spheres at each even-numbered point, you get inscribed n-spheres. For n = 2, those circles are tiny: about 29% (1-1/sqrt(2)) of the radius of the bigger (r = 1) ones. At n = 3, they're somewhat bigger but still small. At n = 4, they're the same size. At n = 5+, the inscribed n-spheres are actually larger than the unit spheres, which means they extend farther despite being "inscribed". This is probably impossible to visualize, because it's never that way in our 3-dimensional world.

High dimensional spaces also make it difficult to come up with good distance metrics. In 1000+ dimensions, nearly all the volume is on the boundary, and neighborhoods in the classical sense (e.g. visible clusters) don't exist. It's trivial to come up with metrics that "work" but it often requires domain knowledge to come up with ones that are useful.

[dekhn]

See also: https://en.wikipedia.org/wiki/Kissing_number which had big implications in error correcting codes back in the day.