Symmetric colorings of polypolyhedra

Abstract

Polypolyhedra (after R. Lang) are compounds of edge-transitive 1-skeleta. There are 54 topologically different polypolyhedra, and each has icosidodecahedral, cuboctahedral, or tetrahedral symmetry, all are realizable as modular origami models with one module per skeleton edge. Consider a coloring in which each edge of a given component receives a different color, and where the coloring (up to global color permutation) is invariant under the polypolyhedron's symmetry group. On the Five Intersecting Tetrahedra, the edges of each color form visual bands on the model, and correspond to matchings on the dodecahedron graph. We count the number of such colorings and give three proofs. For each of the non-polygon-component polypolyhedra, there is a corresponding matching coloring, and we count the number of these matching colorings. For some of the non-polygon-component polypolyhedra, there is a corresponding visual-band coloring, and we count the number of these band colorings.