Rhombohedron

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

In geometry, a rhombohedron (also called a rhombic hexahedron or, inaccurately, a rhomboid) is a special case of a parallelepiped in which all six faces are congruent rhombi. 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.

Special cases

The common angle at the two apices is here given as ${\displaystyle \theta }$. There are two general forms of the rhombohedron: oblate (flattened) and prolate (stretched).

Oblate rhombohedron	Prolate rhombohedron

In the oblate case ${\displaystyle \theta >90^{\circ }}$ and in the prolate case ${\displaystyle \theta <90^{\circ }}$. For ${\displaystyle \theta =90^{\circ }}$ the figure is a cube.

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	${\displaystyle \theta =90^{\circ }}$
Ratio of diagonals	1	√2	Golden ratio
Occurrence	Regular solid	Dissection of the rhombic dodecahedron	Dissection of the rhombic triacontahedron

Solid geometry

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

e₁ : ${\displaystyle {\biggl (}1,0,0{\biggr )},}$

e₂ : ${\displaystyle {\biggl (}\cos \theta ,\sin \theta ,0{\biggr )},}$

e₃ : ${\displaystyle {\biggl (}\cos \theta ,{\cos \theta -\cos ^{2}\theta \over \sin \theta },{{\sqrt {1-3\cos ^{2}\theta +2\cos ^{3}\theta }} \over \sin \theta }{\biggr )}.}$

The other coordinates can be obtained from vector addition of the 3 direction vectors: e₁ + e₂ , e₁ + e₃ , e₂ + e₃ , and e₁ + e₂ + e₃ .

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

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

We can express the volume ${\displaystyle V}$ another way :

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

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

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

Note:

${\displaystyle h=a~z}$₃ , where ${\displaystyle z}$₃ is the third coordinate of e₃ .

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.

Relation to orthocentric tetrahedra

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

The rhombohedral lattice system has rhombohedral cells, with 6 congruent rhombic faces forming a trigonal trapezohedron^{[citation needed]}:

See also

• Lists of shapes

Notes

1. More accurately, rhomboid is a two-dimensional figure.