Compound of five cubes
The compound of five cubes is one of the five regular polyhedral compounds. It was first described by Edmund Hess in 1876.
Its vertices are those of a regular dodecahedron. Its edges form pentagrams, which are the stellations of the pentagonal faces of the dodecahedron.
It is one of the stellations of the rhombic triacontahedron. Its dual is the compound of five octahedra. It has icosahedral symmetry.
The compound of five cubes can also be known as a rhombihedron.
Geometry
The compound is a faceting of the dodecahedron. Each cube represents a selection of 8 of the 20 vertices of the dodecahedron.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, 182 vertices, and 540 edges, yielding an Euler characteristic of 182 − 540 + 360 = 2.
Edge arrangement
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.| 100px Small ditrigonal icosidodecahedron | 100px Great ditrigonal icosidodecahedron | 100px Ditrigonal dodecadodecahedron |
| 100px Dodecahedron | 100px Compound of five cubes | 100px 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.
As a stellation
This compound can be formed as a stellation of the rhombic triacontahedron.The 30 rhombic faces exist in the planes of the 5 cubes.