Rhombohedron

Not to be confused with Rhombihedron.

Rhombohedron
Rank	3
Space	Spherical
Elements
Faces	2 + 2 + 2 rhombi
Edges	2 + 2 + 2 + 2 + 2 + 2 edges
Vertices	2 + 2 + 2 + 2 vertices
Measures (edge length 1)
Central density	1
Abstract & topological properties
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	S2, order 2
Convex	Yes
Nature	Tame

A rhombohedron or rhombic hexahedron is a polytope with 6 rhombic faces. The rhombohedron is a special case of the parallelepiped where all edges are the same length. It is a zonohedron and can tile Euclidean space in a rhombohedral honeycomb.

Special cases

Golden rhombohedra

Two of the five golden isozonohedra are rhombohedra. They are called the acute golden rhombohedron and the obtuse golden rhombohedron. They are constructed as rhombohedra with golden rhombi as faces. The acute golden rhombohedron is constructed with an angle of $\arctan(2)$ at the apex and the obtuse golden rhombohedron is constructed with an angle of $\pi - \arctan(2)$.

The golden rhombohedra with side length 1 have volumes:

• Acute: $\frac{\sqrt{10+2\sqrt{5}}}{5}$
• Obtuse: $\frac{\sqrt{10-2\sqrt{5}}}{5}$

Polyhedra such as the rhombic hexecontahedron and rhombic triacontahedron can be dissected into golden rhombohedra and thus their volumes can be derived from the volumes of the golden rhombohedra.

Other cases

• Triangular antitegum: isotopic.
• Cube: all faces are squares.

External links

• Wikipedia Contributors. "Rhombohedron".
• Weisstein, Eric W. "Rhombohedron" at MathWorld.