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.