Rhombihedron

Not to be confused with Rhombohedron.

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

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.

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".