It would be a clever trick (and maybe not legitimate), but I have discovered that you could use the https://en.wikipedia.org/wiki/Banach–Tarski_paradoxand thus conceptually pack twice as many spheres into the same space with a bit of hypothetical cutting and reshuffling.
"The Banach–Tarski paradox is a theorem
in set-theoretic geometry
, which states the following: Given a solid ball
in 3‑dimensional space, there exists
a decomposition of the ball into a finite number of disjoint subsets
, which can then be put back together in a different way to yield two identical copies of the original ball. Indeed, the reassembly process involves only moving the pieces around and rotating them, without changing their shape. However, the pieces themselves are not "solids" in the usual sense, but infinite scatterings of points."
Maybe useful, maybe not.
I do not seek; I find.
- Pablo Picasso