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### RE: (tye)Re2: Packaging Algorithm

by extremely (Priest)
 on Nov 08, 2000 at 07:18 UTC ( #40464=note: print w/replies, xml ) Need Help??

in reply to (tye)Re2: Packaging Algorithm

Don't let someone's proof that some problem is impossible to solve prevent you from solving the problem well enough to get your work done!
(smartest thing said so far in this whole discussion.)

Well, again, "the sphere packing problem" is different than that. In fact, there have been some neat breakthru's in the field. We have 9600 baud and up modems thanks to a trellis-code based on packing spheres efficiently in 8 dimensions. Turns out a single sphere can be touched by 1024 spheres in a tightly-packed regular array. =) That result is basically cool in anyone's book.

The original problem was that given a bunch of spheres that are the same size, how many can you get to touch a single sphere at the same time. In 2d, the answer is clearly 6. (try it with pennies.) In 3d, 12 is the answer but if you look at the spherical cone of impact that each outer sphere makes, it would seem that 13 COULD be possible. The deal is that no one has found a function that provably states for each dimension what the number is. Only a special case exists for multiples of 8. High-weirdness, plain and simple.

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(tye)Re3: Packaging Algorithm
by tye (Sage) on Nov 08, 2000 at 07:35 UTC

I suggest you do a search for "sphere packing problem" and see the variety of problems that fall under this category. I ran into several before hitting the "kissing number" problem for idential spheres in different numbers of dimensions that you seem to think is the only one.

From what I read, the original problem was popularized by Kepler when he guessed how tightly you could pack identical spheres in 3 dimensions. This one was recently solved (by proving the "obvious").

- tye (but my friends call me "Tye")
I'm aware of them. And in a way, you are correct, the kissing spheres is a subset of the regular vs. irregular tightest packing series of questions. As I recall (and it has been years) Solving the kissing problem in "n" dimensions leads to solutions to the regular packing density question.

And this has now officially gone completely off-topic and gotten rude so that's it for me on this topic.

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