Compound of five cubes
The compound of five cubes is one of the five regular polyhedral compounds. This compound was first described by Edmund Hess in 1876.
It is one of five regular compounds, and dual to the compound of five octahedra. It can be seen as a faceting of a regular dodecahedron.
It is one of the [|stellations] of the rhombic triacontahedron. It has icosahedral symmetry.
Geometry
The compound is a faceting of a 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.Small ditrigonal icosidodecahedron | Great ditrigonal icosidodecahedron | Ditrigonal dodecadodecahedron |
Dodecahedron | 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.