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What is the maximum number of spheres that can be placed in 3D such that all inter-touch?

One can of course place four unit spheres tetrahedrally and then add a smaller sphere in the middle, so this number must be at least 5.

[By the way, I was trying to extend the "five points in 2D cannot be inter-connected without a crossing" limitation to 3D with a simple statement, but this was sadly the best I could do. If anyone knows a better simple extension, please comment.]

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See Ian Agol's answer to the earlier MO question "Extensions of the Koebe–Andreev–Thurston theorem to sphere packing?" He shows that $K_6$ is not realizable in $\mathbb{R}^3$, using the same argument that Anton provides for $\mathbb{R}^n$:… –  Joseph O'Rourke Sep 1 '12 at 18:35

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up vote 11 down vote accepted

In $\mathbb R^n$, the answer is $n+2$.

You can apply an inversion which sends two of the spheres in to two parallel hyperplanes. The rest of the spheres will have the same radii and their centers lie in a hyperplane. Hence everything follows.

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By "inverse" I imagine you mean "inversion"? –  Lee Mosher Sep 1 '12 at 18:28
@Lee, sure, thanks. –  Anton Petrunin Sep 1 '12 at 20:23
Since the two spheres touch each other at only one point (before the inversion), the two parallel hyperplanes would similarly touch for a contradiction, right? I'm probably mistaken, but it seems you can only send two of the spheres in to "a hyperplane plus a touching sphere". –  bobuhito Sep 1 '12 at 23:05
@bobuhito, the touch-point goes to infinity. –  Anton Petrunin Sep 2 '12 at 1:35
Thanks, very slick! I now wonder about other shapes, so I started another question: –  bobuhito Sep 2 '12 at 7:35

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