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In 3D, the maximum number of spheres which can inter-touch is $5$ (MO question Inter-Kissing Number for Spheres of Different Sizes). This maximum reduces to $4$ for unit spheres.

Is there a different shape (e.g., an egg, or a pyramid) for which these maximums are not $5$ and $4$? If so, what shape has the highest maximum?

To avoid "corner touching" (e.g., $8$ cubes could all touch at one corner), please additionally require that every "touch-point" have only one "official connection" (e.g., only $2$ of the $8$ cubes can be declared as touching at the corner).

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  • $\begingroup$ I feel like you can achieve arbitrarily large inter-kissing numbers by carefully arranging a large number of interlocked octopen with nearly one-dimensional legs... If so, this problem is probably more interesting if we restrict to convex shapes. $\endgroup$
    – zeb
    Commented Sep 2, 2012 at 8:48
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    $\begingroup$ You can fuse two identical rods to form a cross shape where the main constraint on mutual contact is the initial rod length. To get the idea, line up n rods as the columns of an array and n more as the rows on top, and then fuse pairs of them together to get n mutually touching solids. For convex shapes, you can find more in work of Martin Gardner, among others. Gerhard "Ask Me About System Design" Paseman, 2012.09.02 $\endgroup$
    – Gerhard Paseman
    Commented Sep 2, 2012 at 16:49
  • $\begingroup$ @Gerhard, Good answer; it seems the 3rd D lets you "go around" and touch anything...I'll need to focus on convex shapes as everyone has pointed out. $\endgroup$
    – bobuhito
    Commented Sep 2, 2012 at 17:51

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There is no upper bound. There can be arbitrarily many congruent convex solids which pairwise touch face-to-face. See Erickson, J. Kim, S. "Arbitrarily large neighborly families of congruent symmetric convex 3-polytopes," for many references.

     Voronoi cells from helix (source)

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  • $\begingroup$ Wow, I'm especially amazed that everything is the same size. I must admit that it's a little ugly (i.e., the assembly has no symmetry) and I hope somebody will comment if they know of any simpler shapes which just beat the sphere. $\endgroup$
    – bobuhito
    Commented Sep 2, 2012 at 18:11
  • $\begingroup$ There is helical symmetry, although it is hard to see exactly what is going on from the picture. The helix is outside its osculating spheres, so slightly larger spheres can touch the helix just at two nearby points. This means that if you choose a few nearby points, their Voronoi cells will touch pairwise at the centers of spheres which touch the helix at just those two points. Choosing the points to be evenly spaced along the helix lets these cells be congruent. $\endgroup$
    – Douglas Zare
    Commented Sep 2, 2012 at 19:30
  • $\begingroup$ If you don't require convexity, then there is a simple example, the union of two $n \times 1 \times 1$ rectangular solids along a $1 \times 1$ subset of an $n \times 1$ face, so that the projection is a V. Let the left legs of the Vs cover a roughly parallelogram region \\\\\\\ and the right legs cover a roughly parallelogram region ////// , and the right leg of each V can touch the left leg of each region to the right, while the left leg touches the right leg of each region to the left. $\endgroup$
    – Douglas Zare
    Commented Sep 2, 2012 at 19:37
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    $\begingroup$ What about the translates of one convex body? $\endgroup$
    – Ilya Bogdanov
    Commented Sep 2, 2012 at 20:37
  • $\begingroup$ The nonconvex V construction I gave is essentially the same as Gerhard Paseman's cross construction. $\endgroup$
    – Douglas Zare
    Commented Sep 3, 2012 at 0:19

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