Compound of five cubes

Compound of five cubes
Type	Regular compound
Coxeter symbol	2{5,3}[5{4,3}]
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)

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 (I_h).

Geometry

The compound is a faceting of a dodecahedron. Each cube represents one orientation of 8 of 12 vertices within a dodecahedron convex hull.

Five cubes in a dodecahedron

If the shape is considered as a union of five cubes yielding a simple concave 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.

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

File:Stellation of rhombic triacontahedron 5 cubes.png

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:

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References

• . p 360
• .
• .
• 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

• MathWorld: Cube 5-Compound
  • MathWorld: Rhombic Triacontahedron Stellations
• George Hart: Compounds of Cubes
• Steven Dutch: Uniform Polyhedra and Their Duals
• VRML model: [1]
• Richard Klitzing, 3D compound, [2]