Compound of five cubes

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Compound of five cubes
Compound of five cubes.png
Type Regular compound
Coxeter symbol 2{5,3}[5{4,3}][1]
Stellation core rhombic triacontahedron
Convex hull Dodecahedron
Index UC9
Polyhedra 5 cubes
Faces 30 squares
Edges 60
Vertices 20
Dual Compound of five octahedra
Symmetry group icosahedral (Ih)
Subgroup restricting to one constituent pyritohedral (Th)
As a stellation

This polyhedral compound is a symmetric arrangement of five cubes. 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 is one of the stellations of the rhombic triacontahedron. It has icosahedral symmetry (Ih).

Edge arrangement[edit]

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.

Small ditrigonal icosidodecahedron.png
Small ditrigonal icosidodecahedron
Great ditrigonal icosidodecahedron.png
Great ditrigonal icosidodecahedron
Ditrigonal dodecadodecahedron.png
Ditrigonal dodecadodecahedron
Dodecahedron (convex hull)
Compound of five cubes.png
Compound of five cubes
Spherical compound of five cubes.png
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[edit]

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:

Stellation of rhombic triacontahedron 5 cubes facets.png

As a facetting[edit]

Five cubes in a dodecahedron

It is also a faceting of a dodecahedron, shown at left.


  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  • Harman, Michael G. (c. 1974), Polyhedral Compounds, unpublished manuscript .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

External links[edit]