[math-fun] variants of the Kotani ant problem

The Kotani ant problem asks us to consider distances between points on the surface of a cube where the distance between two points is the length of the shortest path between them that stays on the surface. (See Dick Hess' article "Kotani's Ant Problem" in the book "Puzzler's Tribute: A Feast for the Mind".) Has anyone considered analogues of the Kotani ant problem for polyhedra other than cubes and rectilinear blocks, e.g., the other Platonic solids? (I'm looking for a research project for some students, and this one came to mind as something that doesn't require any post-high-school background.) Jim Propp