# Compound of five cubes

Compound of five cubes
Rank3
TypeRegular compound
SpaceSpherical
Notation
Bowers style acronymRhom
Elements
Components5 cubes
Faces30 squares
Edges60
Vertices20
Vertex figureGolden hexagram, edge length 2
Measures (edge length 1)
Circumradius${\displaystyle \frac{\sqrt3}{2} \approx 0.86603}$
Inradius${\displaystyle \frac12 = 0.5}$
Volume5
Dihedral angle90°
Central density5
Number of external pieces360
Level of complexity18
Related polytopes
ArmyDoe, edge length ${\displaystyle \frac{\sqrt5-1}{2}}$
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
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 pentachoroorthowedge, which is seven-dimensional.

## Vertex coordinates

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

• ${\displaystyle \left(\pm\frac12,\,\pm\frac12,\,\pm\frac12\right)}$,

along with all even permutations of:

• ${\displaystyle \left(0,\,\pm\frac{\sqrt5-1}{4},\,\pm\frac{1+\sqrt5}{4}\right)}$.