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Fixed OP's link to already-bumped question.
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edit approved Nov 15, 2017 at 6:06
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edit approved Nov 15, 2017 at 6:06
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In 3D, the maximum number of spheres which can inter-touch is 5$5$ (mathoverflow.net/questions/106120MO question Inter-Kissing Number for Spheres of Different Sizes). This maximum reduces to 4$4$ for unit spheres.

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

To avoid "corner touching" (e.g., 8$8$ cubes could all touch at one corner), please additionally require that every "touch-point" have only 1one "official connection" (e.g., only 2$2$ of the 8$8$ cubes can be declared as touching at the corner).

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).

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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bobuhito
bobuhito
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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).