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.

Compound of five cubes

(Animation)
TypeRegular compound
Coxeter symbol2{5,3}[5{4,3}][1]
Stellation corerhombic triacontahedron
Convex hullDodecahedron
IndexUC9
Polyhedra5 cubes
Faces30 squares (visible as 360 triangles)
Edges60
Vertices20
DualCompound of five octahedra
Symmetry groupicosahedral (Ih)
Subgroup restricting to one constituentpyritohedral (Th)

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 (Ih).

Geometry

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.

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 (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).

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.

References

  1. 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


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