Rhombohedron
# Rhombohedron

Rhombohedron :

| Type | prism |
| - | - |
| Faces | 6 rhombi |
| Edges | 12 |
| Vertices | 8 |
| Symmetry group | $C_i$ , $[2^+,2^+],$ (×), order 2 |
| Properties | convex, equilateral, zonohedron, parallelohedron |

In geometry, a rhombohedron (also called a rhombic hexahedron or, inaccurately, a rhomboid) is a three-dimensional figure with six faces which are 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. A cube is a special case of a rhombohedron with all sides square.

In general a rhombohedron can have up to three types of rhombic faces in congruent opposite pairs, $C_i$ 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 6 congruent rhombic faces forming a trigonal trapezohedron:

## Special cases by symmetry

| Form | Cube | Trigonal trapezohedron | Right rhombic prism | Oblique rhombic prism |
| - | - | - | - | - |
| Angle constraints | $\alpha =\beta =\gamma =90^{\circ }$ | $\alpha =\beta =\gamma$ | $\alpha =\beta =90^{\circ }$ | $\alpha =\beta$ |
| Symmetry | $O_h$ order 48 | $D_{3d}$ order 12 | $D_{2h}$ order 8 | $C_{2h}$ order 4 |
| Faces | 6 squares | 6 congruent rhombi | 2 rhombi, 4 squares | 6 rhombi |

* Cube: with $O_h$ symmetry, order 48. All faces are squares.
* Trigonal trapezohedron (also called isohedral rhombohedron): with $D_{3d}$ symmetry, order 12. All non-obtuse internal angles of the faces are equal (all faces are congruent rhombi). This can be seen by stretching a cube on its body-diagonal axis. For example, a regular octahedron with two regular tetrahedra attached on opposite faces constructs a 60 degree trigonal trapezohedron.
* Right rhombic prism: with $D_{2h}$ symmetry, order 8. It is constructed by two rhombi and four squares. This can be seen by stretching a cube on its face-diagonal axis. For example, two right prisms with regular triangular bases attached together makes a 60 degree right rhombic prism.
* Oblique rhombic prism: with $C_{2h}$ symmetry, order 4. It has only one plane of symmetry, through four vertices, and six rhombic faces.

## Solid geometry

For a unit (i.e.: with side length 1) isohedral rhombohedron, with rhombic acute angle $\theta ~$, with one vertex at the origin (0, 0, 0), and with one edge lying along the x-axis, the three generating vectors are

    $e_1$ : ${\biggl1,0,0{\biggr )},$
    $e_2$ : ${\biggl\cos \theta ,\sin \theta ,0{\biggr )},$
    $e_3$ : ${\biggl\cos \theta ,{\cos \theta -\cos ^{2}\theta \over \sin \theta },{{\sqrt {1-3\cos ^{2}\theta +2\cos ^{3}\theta }} \over \sin \theta }{\biggr )}.$

The other coordinates can be obtained from vector addition of the 3 direction vectors: $e_1$ + $e_2$ , $e_1$ + $e_3$ , $e_2$ + $e_3$ , and $e_1$ + $e_2$ + $e_3$ .

The volume $V$ of an isohedral rhombohedron, in terms of its side length $a$ and its rhombic acute angle $\theta ~$, is a simplification of the volume of a parallelepiped, and is given by

    $V=a^{3}(1-\cos \theta ){\sqrt {1+2\cos \theta }}=a^{3}{\sqrt {(1-\cos \theta )^{2}(1+2\cos \theta )}}=a^{3}{\sqrt {1-3\cos ^{2}\theta +2\cos ^{3}\theta }}~.$

We can express the volume $V$ another way :

    $V=2{\sqrt {3}}~a^{3}\sin ^{2}\left({\frac {\theta }{2}}\right){\sqrt {1-{\frac {4}{3}}\sin ^{2}\left({\frac {\theta }{2}}\right)}}~.$

As the area of the (rhombic) base is given by $a^{2}\sin \theta ~$, and as the height of a rhombohedron is given by its volume divided by the area of its base, the height $h$ of an isohedral rhombohedron in terms of its side length $a$ and its rhombic acute angle $\theta$ is given by

    $h=a~{(1-\cos \theta ){\sqrt {1+2\cos \theta }} \over \sin \theta }=a~{{\sqrt {1-3\cos ^{2}\theta +2\cos ^{3}\theta }} \over \sin \theta }~.$

Note:

    $h=a~z$$_3$ , where $z$$_3$ is the third coordinate of $e_3$ .

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.

