Imagine I have some spheres, all transparent and of equal radius $r$. Each sphere, at its center, has a light source of identical intensity $i$ at $r$.
I want to take some number of spheres and aggregate it in a roughly spherical shape, until the light seen by an observer exactly in the middle reaches a certain level. Obviously, the more spheres I add, the more light there is, but also you get diminishing returns. But the rate of the diminishing also decreases, because the outer layers are larger. The question is how many spheres would I need to reach a certain intensity $I$ where $I \gg i$.
I can see how you would approximate it as "shells". So, say 10 spheres are packed around the first one at equal distance of $2r$, and the light from them can be found with the inverse square. The second shell, at $4r$ distance, would have an area that is 4x more, therefore 40 spheres, and so on. This gives a very rough estimate.
Of course it is probably more realistic to assume an dense sphere packing rather than such "shells". But it is hard to calculate how many spheres would be at each given distance if densely packed. I'm certain the relation would not be algebraically complex, and I could simply calculate a few numbers for smaller sphere counts, fit a curve and extrapolate from there.
I thought about trying with cube packing instead of spheres. However, that seems to be even worse, since now the distances are not consistent.
What I'm wondering is, is there a reasonably simple way to derive the equation, so as to give $n$ the number of spheres in terms of $I$, $i$ and $r$? Or is it bound to be a very complex and ugly formula?
For context, I was trying to figure out if it's possible to pack solar systems so close that the light of the galaxy would be as bright as the Sun: https://astronomy.stackexchange.com/a/55005/52397