Calculating amount of cubes that fit in a sphere [duplicate]

Question

I know that the problem of finding out how many spheres can fit in a cube is a commonly asked and well documented one, but I am struggling to find anything on the inverse of the problem, namely:

How many cubes of a certain Length x Width x Height fit in a sphere of a certain radius?

As in the spheres-in-a cube problem I am sure it depends on the stacking method but I was wondering if there might also be an optimum as given by the Kepler problem.

I would also then like to expand the analogy to the amount of spheres that can fit in a cylinder of certain diameter and length.

Any insight into the problem will be appreciated, thanks!

Answer 1

Each partition in the xz planes of thickness delta y is a separate problem yeilding a particular solution for “whole” cubes. The partial cubes at the perimeter of each partition are eliminated from the sum. Assuming 8 symmetric sections, solve for 1 quadrant of a hemisphere and multiply your answer by 8. For instance: How many rooms 10 feet by 10 feet by 10 feet will fit into a sphere 1 mile in diameter if the floor and ceiling thickness is also 10 feet and wall thicknesses are 0.5 feet with 4 foot hallways in between every room? First slice the sphere into 264 layers each 20 feet thick. Then translate the partitions up 10 feet to obtain half partitions at the top and bottom for the roof and floor. Then solve for each floor. This is a simple engineering problem that involves geometrical sums obtained from applying parametric limits to 3 dimensional geometric shapes and then tiling the plane with your floor plan.