Compound of five cubes
|Compound of five cubes|
|Stellation core||rhombic triacontahedron|
|Faces||30 squares (visible as 360 triangles)|
|Dual||Compound of five octahedra|
|Symmetry group||icosahedral (Ih)|
|Subgroup restricting to one constituent||pyritohedral (Th)|
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.
It has a density of higher than 1.
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
Dodecahedron (convex hull)
Compound of five cubes
As a spherical tiling
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.
The stellation facets for construction are:
Other 5-cube compounds
A second 5-cube compound, with octahedral symmetry, also exists. It can be generated by adding a fifth cube to the standard 4-cube compound.
- Regular polytopes, pp.49-50, p.98
- 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- Harman, Michael G. (c. 1974), Polyhedral Compounds, unpublished manuscript.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
- George Hart: Compounds of Cubes
- Steven Dutch: Uniform Polyhedra and Their Duals
- VRML model:
- Klitzing, Richard. "3D compound".