Rhombihedron

Not to be confused with Rhombohedron.

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

Rhombihedron
(3D model) (OFF file)
Rank	3
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Rhom
Elements
Components	5 cubes
Faces	30 squares
Edges	60
Vertices	20
Vertex figure	Golden hexagram, edge length √2
Measures (edge length 1)
Circumradius	$\frac{\sqrt3}{2} \approx 0.86603$
Inradius	$\frac12 = 0.5$
Volume	5
Dihedral angle	90°
Central density	5
Number of external pieces	360
Level of complexity	18
Related polytopes
Army	Doe
Regiment	Sidtid
Dual	Small icosicosahedron
Conjugate	Rhombihedron
Convex core	Rhombic triacontahedron
Abstract & topological properties
Flag count	240
Schläfli type	{4,3}
Orientable	Yes
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame

It has the same edges as the small ditrigonary icosidodecahedron.

This compound is sometimes considered to be 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.

Gallery

Vertex coordinates

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

• $\left(\pm\frac12,\,\pm\frac12,\,\pm\frac12\right),$

along with all even permutations of:

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

External links

• Bowers, Jonathan. "Polyhedron Category C1: Compound Regulars" (#4).

• Klitzing, Richard. "rhom".

• Wikipedia Contributors. "Compound of five cubes".