Compound of five cubes

Not to be confused with Rhombohedron.

Compound of five cubes
(3D model) (OFF file)
Rank	3
Type	Regular compound
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	${\frac {\sqrt {3}}{2}}\approx 0.86603$
Inradius	${\frac {1}{2}}=0.5$
Volume	5
Dihedral angle	90°
Central density	5
Number of external pieces	360
Level of complexity	18
Related polytopes
Army	Doe, edge length ${\frac {{\sqrt {5}}-1}{2}}$
Regiment	Sidtid
Dual	Compound of five octahedra
Conjugate	Compound of five cubes
Convex core	Rhombic triacontahedron
Abstract & topological properties
Flag count	240
Schläfli type	{4,3}
Orientable	Yes
Properties
Symmetry	H3, order 120
Flag orbits	2
Convex	No
Nature	Tame
History
Discovered by	Edmond Hess
First discovered	1876

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.

Gallery

Vertex coordinates

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

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

along with all even permutations of:

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

External links

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

• Klitzing, Richard. "rhom".
• Wikipedia contributors. "Compound of five cubes".