How to calculate the sphere within an octahedron of six spheres? Apologies if you've already spent a lot of time answering this one but I've spent a lot of time looking for the answer here without success unfortunately.
If I bunch six spheres together to form an octahedron then how do I calculate the size of sphere that fits at the centre of those six spheres?
I assume the answer to be some expression of the size of spheres containing it?

If the spheres are all the same size, then this isn't too hard to calculate.
Taking a cross section of a regular octahedron gives a square.  If $R$ and $r$ are the radii of the outer and inner spheres, respectively, then $2R + 2r$ equals the diagonal of this square, $R\sqrt2$.  Solving for $r$ yields
$$r = (\sqrt 2 - 1) R \approx 0.414 R.$$

