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Analagous to circle packing in a circle, let's consider sphere packing on a spherical surface. The little spheres all have the same radius. I think there could be 3 possibilities of packing: a. every smaller sphere touches the surface of the main sphere from the inside; b. every smaller sphere touches the surface of the main sphere from the outside; c. every smaller sphere keeps their radius on the surface of the main sphere. As the number of little spheres goes to inf, what's the limit density (area of little spheres/area of big spherical surface)?

This is not trivial without proving it because any of the gaps between the little spheres will be multiplied by a number proportional to the number of the little spheres, which is a huge number. It's not clear how the sum of the areas of the small spheres is to be calculated. Shall we account for the full surface area of each little sphere or half of it?

In order for the above limit to approach 1, what area of each little sphere should be calculated? The purpose is to find the packing method (out of the above 3 possibilities) and the right area calculation for this limit to approach 1.

feynman
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  • To clarify, the situation is that we have one main hollow sphere, we are putting as many smaller spheres (of same size) as possible inside the main sphere such that every smaller sphere touches the surface of the main sphere from the inside?

    If so, could you clarify the area density you propose, as the extension to 3D space is not obvious given that surfaces are curved. Or do you want volume density, which is a more appropriate extension?

     – Tony Mathew Aug 07 '23 at 10:40
  • Do the little spheres all have the same radius? Also, as per @TonyMathew 's comment, I don't see how the area is going to be approximated rather than the volume. – Chris Lewis Aug 07 '23 at 12:28
  • thanks for the comments! I reedited. Did that answer the question of the area density? – feynman Aug 07 '23 at 12:55
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    The three scenarios each have some issues. If a little sphere is tangent to the large sphere, it only meets the large sphere at a point (zero area). The scenario where they intersect makes more sense, but a radius (straight line) can't lie on a sphere. Perhaps you want the centres of the small spheres to lie on the surface of the large sphere? Then you can ask how much of the surface of the large sphere lies within the set of little spheres. I suspect, though, that this will reduce to the 2D case from the previous question. – Chris Lewis Aug 07 '23 at 13:27
  • I don't see any problem when a little sphere is tangent to the large sphere and it only meets the large sphere at a point. When the size of the little spheres goes to 0, the little spheres adhere to the spherical surface and can approximate the spherical surface area. All I want is finding out how to approximate the large spherical surface area by these little sphears and taking the limit in the end. – feynman Aug 07 '23 at 20:05

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