Compound of five cubes(Animation, 3D model)TypeRegular compoundCoxeter symbol 2{5,3}[5{4,3}]Stellation corerhombic triacontahedronConvex hullDodecahedronIndexUC9Polyhedra5 cubesFaces30 squares (visible as 360 triangles)Edges60Vertices20DualCompound of five octahedraSymmetry groupicosahedral (Ih)Subgroup restricting to one constituentpyritohedral (Th)
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 (Ih).
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.
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.
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; respectively, the small complex rhombicosidodecahedron, great complex rhombicosidodecahedron and complex rhombidodecadodecahedron.
Small ditrigonal icosidodecahedronGreat ditrigonal icosidodecahedronDitrigonal dodecadodecahedronDodecahedron (convex hull)Compound of five cubesAs 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).
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.
See also Compound of five octahedra Compound of three cubes Compound of four cubes Compound of six cubes Uniform polyhedron compound
Footnotes
References . . . 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. , 3.6 The five regular compounds, pp.47-50, 6.2 Stellating the Platonic solids, pp.96-104
External links MathWorld: Cube 5-Compound MathWorld: Rhombic Triacontahedron Stellations George Hart: Compounds of Cubes Steven Dutch: Uniform Polyhedra and Their Duals