Compound of five great rhombihexahedra

Compound of five great rhombihexahedra
Type	Uniform compound
Index	UC66
Polyhedra	5 great rhombihexahedra
Faces	60 squares, 30 octagrams
Edges	240
Vertices	120
Symmetry group	icosahedral (Ih)
Subgroup restricting to one constituent	pyritohedral (Th)

This uniform polyhedron compound is a composition of 5 great rhombihexahedra, in the same vertex arrangement as the compound of 5 truncated cubes.

Filling

There is some controversy on how to colour the faces of this polyhedron compound. Although the common way to fill in a polygon is to just colour its whole interior, this can result in some filled regions hanging as membranes over empty space. Hence, the "neo filling" is sometimes used instead as a more accurate filling. In the neo filling, orientable polyhedra are filled traditionally, but non-orientable polyhedra have their faces filled with the modulo-2 method (only odd-density regions are filled in). In addition, overlapping regions of coplanar faces can cancel each other out.

Traditional filling

"Neo filling"

References

• Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the Cambridge Philosophical Society, 79 (3): 447–457, Bibcode:1976MPCPS..79..447S, doi:10.1017/S0305004100052440, MR 0397554, S2CID 123279687.