Is there a way to generate k points on a n-sphere, say, $x_1,\dots,x_k$ such that $\min_{ i \neq j } \| x_i - x_j \| $ is as large as possible? Approximate solutions are also OK, I just need well separated points on a sphere.
well separated points on sphere
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geometry
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1Possible duplicate: http://math.stackexchange.com/questions/9846/which-tessellation-of-the-sphere-yields-a-constant-density-of-vertices/ See my answer there for links. – 2011-04-07
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0Thanks, I have marked this question as a duplicate. [Update] Actually this question appears a little bit different, anyway... – 2011-04-07
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2One of lhf's links leads to [a page of results on spherical packings](http://www2.research.att.com/~njas/packings/), which is precisely to maximize the minimum distance between a specified number of points on a 2-sphere. Little is known about the problem beyond dimension 5 or so. I looked into that about half a year ago for someone in a newsgroup. – 2011-04-07
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0@hardmath: I disagree with "little is known about the problem beyond dimension 5 or so". Conway & Sloane's "Sphere Packings, Lattices and Groups" is quite a voluminous compendium, and quite a bit of it is about sphere packings in higher dimensions. You can see the table of contents here: http://www.gbv.de/dms/goettingen/245890696.pdf – 2011-04-08
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0@joriki: I should have framed it as a comparative. That's Sloane's page linked above. It may be outdated; the link to his book "Sphere Packings, Lattices, and Groups" with Conway (3ed., 1998, same as your PDF) seems to be broken. Coverage of dimensions > 5 there is sparse and unsystematic, amounting to only a couple of dozen packings, and while the he writes the packings for dimensions 3,4,5 listed in part 1 are "putatively optimal", the higher dimensional packings are only said to be "nice", "good", or "interesting". – 2011-04-08
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0@hardmath: I guess you meant "there's little systematic knowledge" -- "Sphere Packings, ..." takes more of a butterfly collecting approach to the problem :-) – 2011-04-08