# Rhombohedron

Rhombohedron Rank3
SpaceSpherical
Elements
Faces2 + 2 + 2 rhombi
Edges2 + 2 + 2 + 2 + 2 + 2 edges
Vertices2 + 2 + 2 + 2 vertices
Measures (edge length 1)
Central density1
Abstract & topological properties
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryS2, order 2
ConvexYes
NatureTame

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.