| bgcolor=#e7dcc3 colspan=2 | Rhombohedron | |
|---|---|---|
| align=center colspan=2 | ||
| Type | prism | |
| Faces | 6 rhombi | |
| Edges | 12 | |
| Vertices | 8 | |
| Symmetry group | Ci, [2<sup>+</sup>,2<sup>+</sup>], (×), order 2 | |
| Properties | convex, equilateral, zonohedron, parallelohedron |
In geometry, a rhombohedron (also called a rhombic hexahedron[1] [2] or, inaccurately, a rhomboid) is a special case of a parallelepiped in which all six faces are congruent rhombi.[3] It can be used to define the rhombohedral lattice system, a honeycomb with rhombohedral cells. A rhombohedron has two opposite apices at which all face angles are equal; a prolate rhombohedron has this common angle acute, and an oblate rhombohedron has an obtuse angle at these vertices. A cube is a special case of a rhombohedron with all sides square.
The common angle at the two apices is here given as
\theta
In the oblate case
\theta>90\circ
\theta<90\circ
\theta=90\circ
Certain proportions of the rhombs give rise to some well-known special cases. These typically occur in both prolate and oblate forms.
| Form | Cube | √2 Rhombohedron | Golden Rhombohedron | |
|---|---|---|---|---|
| Angle constraints | \theta=90\circ | |||
| Ratio of diagonals | 1 | √2 | Golden ratio | |
| Occurrence | Regular solid | Dissection of the rhombic dodecahedron | Dissection of the rhombic triacontahedron |
For a unit (i.e.: with side length 1) rhombohedron,[4] with rhombic acute angle
\theta~
e1 :
l(1,0,0r),
e2 :
l(\cos\theta,\sin\theta,0r),
e3 :
l(\cos\theta,{\cos\theta-\cos2\theta\over\sin\theta},{\sqrt{1-3\cos2\theta+2\cos3\theta}\over\sin\theta}r).
The other coordinates can be obtained from vector addition[5] of the 3 direction vectors: e1 + e2, e1 + e3, e2 + e3, and e1 + e2 + e3 .
The volume
V
a
\theta~
V=a3(1-\cos\theta)\sqrt{1+2\cos\theta}=a3\sqrt{(1-\cos\theta)2(1+2\cos\theta)}=a3\sqrt{1-3\cos2\theta+2\cos3\theta}~.
We can express the volume
V
V=2\sqrt{3}~a3
| ||||
| \sin |
\right)\sqrt{1-
| 4 | |
| 3 |
| ||||
| \sin |
\right)}~.
As the area of the (rhombic) base is given by
a2\sin\theta~
h
a
\theta
h=a~{(1-\cos\theta)\sqrt{1+2\cos\theta}\over\sin\theta}=a~{\sqrt{1-3\cos2\theta+2\cos3\theta}\over\sin\theta}~.
Note:
h=a~z
z
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.
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.[6]
The rhombohedral lattice system has rhombohedral cells, with 6 congruent rhombic faces forming a trigonal trapezohedron: