Rhombohedron


In geometry, a rhombohedron is a three-dimensional figure like a cuboid, except that its faces are not rectangles but rhombi. It is a special case of a parallelepiped where all edges are the same length. It can be used to define the rhombohedral lattice system, a honeycomb with rhombohedral cells.
In general a rhombohedron can have up to three types of rhombic faces in congruent opposite pairs, Ci symmetry, order 2.
Four points forming non-adjacent vertices of a rhombohedron necessarily form the four vertices of an orthocentric tetrahedron, and all orthocentric tetrahedra can be formed in this way.

Rhombohedral lattice system

The rhombohedral lattice system has rhombohedral cells, with 3 pairs of unique rhombic faces:

Special cases by symmetry

FormCubeTrigonal trapezohedronRight rhombic prismOblique rhombic prism
Angle
constraints		α=β=γ=90°α=β=γα=β=90°α=β
SymmetryOh
order 48		D3d
order 12		D2h
order 8		C2h
order 4
Faces6 squares6 congruent rhombi2 rhombi, 4 squares6 rhombi

For a unit isohedral rhombohedron, with rhombic acute angle, with one vertex at the origin, and with one edge lying along the x-axis, the three generating vectors are
The other coordinates can be obtained from vector addition of the 3 direction vectors: e1 + e2, e1 + e3, e2 + e3, and e1 + e2 + e3.
The volume of an isohedral rhombohedron, in terms of its side length and its rhombic acute angle, is a simplification of the volume of a parallelepiped, and is given by
As the area of the base is given by, and as the height of a rhombohedron is given by its volume divided by the area of its base, the height of an isohedral rhombohedron in terms of its side length and its rhombic acute angle is given by
Note:
The body diagonal between the acute-angled vertices is the longest. By rotational symmetry about that diagonal, the other three body diagonals, between the three pairs of opposite obtuse-angled vertices, are all the same length.