rhombohedra

A rhombohedron is a convex polyhedron with six faces, each face a rhombus. It is a special kind of parallelepiped in which all edges have the same length and opposite faces are parallel. Because of these properties, a rhombohedron can be viewed as a skewed cube obtained by shearing a cube while keeping edge lengths fixed.

In a rhombohedron there are eight vertices, twelve edges, and three pairs of parallel faces. The six faces come in three sets, each consisting of rhombi whose sides are the edges of the polyhedron. The dihedral angles along the three edge directions can vary independently, so the shape ranges from nearly cubic to highly elongated or flattened forms.

A common way to describe a rhombohedron is by three equal-length edge vectors a, b, and c emanating from a common vertex; these define a parallelepiped whose faces are rhombi. If the angles between the vectors are all right angles, the figure is a cube. If the angles differ from 90 degrees, the resulting rhombohedron is a non-orthogonal deformation of the cube.

In crystallography, the rhombohedron appears as a conventional representation of the rhombohedral (trigonal) lattice’s primitive cell. The term also appears in tiling and geometric modeling as a basic unit that can fill space in rhombohedral honeycombs.

Volume can be computed as V = |a · (b × c)|, and for equal edge lengths with inter-edge angles α, β, γ, this depends on the cosines of those angles.