# Compound of five cubes

Compound of five cubes
Rank3
TypeWeakly regular compound
Notation
Bowers style acronymRhom
Elements
Components5 cubes
Faces30 squares
Edges60
Vertices20
Vertex figureGolden hexagram, edge length 2
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {3}}{2}}\approx 0.86603}$
Inradius${\displaystyle {\frac {1}{2}}=0.5}$
Volume5
Dihedral angle90°
Central density5
Number of external pieces360
Level of complexity18
Related polytopes
ArmyDoe, edge length ${\displaystyle {\frac {{\sqrt {5}}-1}{2}}}$
RegimentSidtid
DualCompound of five octahedra
ConjugateCompound of five cubes
Convex coreRhombic triacontahedron
Abstract & topological properties
Flag count240
Schläfli type{4,3}
OrientableYes
Properties
SymmetryH3, order 120
Flag orbits2
ConvexNo
NatureTame
History
Discovered byEdmond Hess
First discovered1876

The rhombihedron, rhom, or compound of five cubes is a weakly-regular polyhedron compound. It consists of 30 squares. The vertices coincide in pairs, leading to 20 vertices where 6 squares join.

It has the same edges as the small ditrigonary icosidodecahedron.

This compound is sometimes called regular, but it is not flag-transitive, despite the fact it is vertex-, edge-, and face-transitive. It is however regular if you consider conjugacies along with its other symmetries.

Its quotient prismatic equivalent is the cubic pentagyroprism, which is seven-dimensional.

## Vertex coordinates

The vertices of a rhombihedron of edge length 1 are given by:

• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right)}$,

along with all even permutations of:

• ${\displaystyle \left(0,\,\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}}\right)}$.