Maximizing Spheres: How Many Can Fit Around One?

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In three-dimensional space (R3), the maximum number of identical spheres that can touch a central sphere is 12, known as the kissing number. While the upper bound of 27 spheres is mentioned, this figure refers to a theoretical volume filling rather than actual contact points. The discussion emphasizes the challenge of calculating the exact number of spheres that can surround a central sphere while maintaining contact. The concept of the kissing number is crucial for understanding this arrangement. Further exploration of geometric packing and sphere arrangements could provide additional insights.
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Suppose I have a bunch of spheres, all with the same radius r. Now, take one sphere and set it aside in R3. The question then, is, how many spheres can I place around this sphere such that at least one part of them is touching the central sphere?

I've arrived at a very strict upper bound of 27 spheres, because 27 spheres would fill the entire volume of space between the surface of the central sphere and the outer sufaces of the surrounding spheres. Other than that, I don't really know how to get to an actual number. Thoughts?
 
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