# Inter-Kissing Number for Non-Spheres

In 3D, the maximum number of spheres which can inter-touch is 5 (mathoverflow.net/questions/106120). 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 1 "official connection" (e.g., only 2 of the 8 cubes can be declared as touching at the corner).

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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. –  zeb Sep 2 '12 at 8:48
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 –  Gerhard Paseman Sep 2 '12 at 16:49
@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. –  bobuhito Sep 2 '12 at 17:51

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. –  Douglas Zare Sep 2 '12 at 19:37