Compound of five cubes

The compound is a faceting of a dodecahedron (where pentagrams can be seen correlating to the pentagonal faces). Each cube represents a selection of 8 of the 20 vertices of the dodecahedron.

		Views from 2-fold, 5-fold and 3-fold symmetry axis

If the shape is considered as a union of five cubes yielding a simple nonconvex solid without self-intersecting surfaces, then it has 360 faces (all triangles), 182 vertices (60 with degree 3, 30 with degree 4, 12 with degree 5, 60 with degree 8, and 20 with degree 12), and 540 edges, yielding an Euler characteristic of 182 − 540 + 360 = 2.

Its convex hull is a regular dodecahedron. It additionally shares its edge arrangement with the small ditrigonal icosidodecahedron, the great ditrigonal icosidodecahedron, and the ditrigonal dodecadodecahedron. With these, it can form polyhedral compounds that can also be considered as degenerate uniform star polyhedra; the small complex rhombicosidodecahedron, great complex rhombicosidodecahedron and complex rhombidodecadodecahedron.

Small ditrigonal icosidodecahedron	Great ditrigonal icosidodecahedron	Ditrigonal dodecadodecahedron
Dodecahedron (convex hull)	Compound of five cubes	As a spherical tiling

The compound of ten tetrahedra can be formed by taking each of these five cubes and replacing them with the two tetrahedra of the stella octangula (which share the same vertex arrangement of a cube).

Stellation facets
The yellow area corresponds to one cube face.

This compound can be formed as a stellation of the rhombic triacontahedron. The 30 rhombic faces exist in the planes of the 5 cubes.

• Cromwell, Peter R. (1997), Polyhedra, Cambridge. p 360
• Harman, Michael G. (c. 1974), Polyhedral Compounds, unpublished manuscript.
• Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the Cambridge Philosophical Society, 79: 447–457, doi:10.1017/S0305004100052440, MR 0397554.
• Cundy, H. and Rollett, A. "Five Cubes in a Dodecahedron." §3.10.6 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 135–136, 1989.
• H.S.M. Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, 3.6 The five regular compounds, pp.47-50, 6.2 Stellating the Platonic solids, pp.96-104

• MathWorld: Cube 5-Compound
  • MathWorld: Rhombic Triacontahedron Stellations
• George Hart: Compounds of Cubes
• Steven Dutch: Uniform Polyhedra and Their Duals
• VRML model:
• Klitzing, Richard. "3D compound".