How do I calculate the volume of a "solid" sphere with a non-integer dimension, and is it possible to stack such objects so as to fill a non-integer-dimensional space?

Answer 1

You would first have to define what you mean by volume. In a normed space (of any dimension) we usually take a sphere of radius r centered at c to be the set of all points such that norm(x-c)=r for all x. But volume has no such universally accepted definition as far as I know.

Answer 2

V = (4/3)πr³

r has not restrictions over the set of reals

Assume the radius is 0.5 units.

The diameter is 1 unit.

Four spheres could fill a space of 2 x 2 x 1

Assume the radius is 0.7 units

The diameter is 1.4 unit

Four spheres could fill a space of 2.8 x 2.8 x 1