Colliding Dice Probabilities

LetK, L be congruent regular polyhedra in R. Let g denote a rigid motion of R, that is, g(x) = Φx+τ where Φ is a 3×3 rotation matrix and τ is a translation 3-vector. The polyhedraK, g(L) are said to touch ifK∩g(L) 6= ∅ but int(K)∩int(g(L)) = ∅. Alternatively, we may think of ΦL moving toward K in the direction τ , stopping precisely when the two polyhedra collide. Let us sample the space SO3 of matrices Φ according to the uniform distribution (Haar measure, normalized to 1). The space of vectors τ is slightly harder to describe. Let K − ΦL = {y − Φx : y ∈ K and x ∈ L} be the Minkowski sum of K and the reflected image −ΦL of ΦL. Another way to characterize K − ΦL is as the convex hull of all pairwise sums of vertices of K and −ΦL. Clearly