Suppose T, Q, O, D, I represent the *vertex sets* of the 5 regular polyhedra (Platonic solids), assumed to be on the unit sphere S^2 centered at the origin.
For any two of these, i.e., one of
{TT, TQ, TO, TD, TI, QQ, QO, QD, QI, OO, OD, OI, DD, DI, II},
I want to know how far one of each pair can be rotated from the other one.
So for the pair PP' (i.e., P and P'), consider any rotation of g of the sphere S^2 and define:
f_g(x,y) = minimum_{x in P, y in P'} dist(x, g(y))
where dist is arclength over S^2, i.e., central angle.
Now set
F(P, P') = maximum_{g in SO(3)} f_g(x,y)
where SO(3) is the set of all rotations of S^2.
Note this is unchanged if we had used dist(g(x), y) instead of dist(x, g(y)).
Note: In many cases (e.g., if P and P' are dual) the points P' can be rotated to (or already are) a subset of the face centers of P — or vice versa — so for these cases the value of F(P, P') is easy to determine.
Namely, the cases PP' = TT, TO, TI, QO, QI, DI. That leaves the following 9 less-obvious cases:
{TQ, TD, QQ, QD, OO, OD, OI, DD, II}.
I hope I didn't misstate something, but if so I hope someone will point it out.
—Dan
