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That the kissing number of a sphere in dimension 3 is 12 is well known. However, it is also known that there is a lot of empty space between the 12 spheres. I deduce (am I wrong?) that there are many different ways (configurations) of 13 spheres such that a central one is in contact with the remaining 12.

Is anyone aware of an algorithm that can be used to generate such configurations? Of course, one could go by trial end error. I was looking for something more efficient, elegant and intelligent than just "try to put them in some way until there are no overlaps".

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  • $\begingroup$ There are a continuously many configuration. You can continuously move the spheres allowing you to exchange any two. $\endgroup$ Commented Feb 15, 2024 at 15:27
  • $\begingroup$ Yes, I also know this. Thanks for the contribution tho. I guess my question could be reformulated by saying "how to samples from this infinte space?" $\endgroup$ Commented Feb 15, 2024 at 15:45
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    $\begingroup$ You may divide your sphere into, $180 \times 360$ countries, with country $(i, j)$ ranging from longitude $i$ to $i+1$, and altitude $j$ to $j+1$. If the center of a kissing sphere is over country $(i, j)$ then the center of another kissing sphere cannot be over country $(i',j')$ for quite a few cases $(i',j')$. You may precompute the list of excluded pairs $(i',j')$ for all latitudes $i$. Then you prepare a bitmap for all countries; and after sampling a new kissing sphere you set the bit for all counties excluded by that sphere. You sample the next sphere over remaining countries only. $\endgroup$ Commented Feb 15, 2024 at 17:52
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    $\begingroup$ This is still trial an error, but with a considerably greater chance of success. Of course, there are quite a few details of this idea yet to be worked out. Also, there are better ways to divide the sphere into countries. $\endgroup$ Commented Feb 15, 2024 at 17:57

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