Rhombohedron

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 $\theta$. There are two general forms of the rhombohedron: oblate (flattened) and prolate (stretched).

File:Rhombohedron-oblate.svg	File:Prolate rhombohedron.svg
Oblate rhombohedron, note there is a mistake in the labelling of angles here. All angles labeled theta should be on the acute angles. Here, two are on the obtuse and one is on the acute.	Prolate rhombohedron

In the oblate case $\theta >90^{\circ }$ and in the prolate case $\theta <90^{\circ }$. For $\theta =90^{\circ }$ 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	$\theta =90^{\circ }$
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 $\theta$, with one vertex at the origin (0, 0, 0), and with one edge lying along the x-axis, the three generating vectors are

e₁ : $(1,0,0),$

e₂ : $(\cos \theta ,\sin \theta ,0),$

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

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 $V$ of a 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 a rhombohedron in terms of its side length $a$ and its rhombic acute angle $\theta$ is given by

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

Note:

$h=az$₃ , where $z$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:

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See also

• Lists of shapes

External links

• Weisstein, Eric W.. "Rhombohedron". http://mathworld.wolfram.com/Rhombohedron.html. 
• Volume Calculator https://rechneronline.de/pi/rhombohedron.php