by wendy » Mon Sep 24, 2012 9:06 am
Consider first, the grand antiprism. It has 300 tetrahedral faces, none of which touch the pentagonal antiprisms. Colour these all the same coulor (say red).
Now, the remaining tetrahedra lie, 100 on each side of the red ones, in little pockets of two, which we can freely colour black and white. This is important that we can freely rotate them, because we're going to put the tetrahedra over the pentagonal antiprisms.
Now, the ring of 10 pentagonal APs correspond to 150 tetrahedra, or 10 sets of 15. One can pack red, white and black tetrahedra, as long as we don't force a pair of white or black triangles over the edge of the PAP. The tetrahedra in the pods can be freely rotated without disturbing the scheme, so there is a degree of freedom here that is worth the explore.
Still, it does mean that three colours suffice.