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Kissing Number

The Kissing Number of an n-dimensional sphere is the number of identically-sized spheres which can touch it simultaneously without any of the spheres overlapping. The kissing number of a 3-dimensional sphere is 12. But, there is a fair bit of play involved. In fact, if you had some sticky spheres, you'd almost think there might be some way that you could get a 13th one in there.

In fact, if you cluster nine of the touching spheres together as closely as you can, then you can rotate the cluster of the remaining three. In this way, you can move spheres around the surface of the center sphere. You form a three-cluster, rotate it a bit, separate out one ball into a different three-cluster, rotate it a bit, until you get the one ball into the place you wanted it.

Now, I'm not sure you have to do the moving in clusters of three like that. You may be able to rotate a two-cluster. And, I think you can just worm one ball around independently by backing obstructions out of the way momentarily to let the one pass.

But, for the purposes of this dream, assume that you have to rotate them in clusters of three.

Yes, that's right... this is a dream...Collapse )