Rhombohedron

In geometry, a rhombohedron (also called a rhombic hexahedron[1][2] or, inaccurately, a rhomboid[lower-alpha 1]) 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.

## Contents

* 1 Special cases
* 2 Solid geometry
* 2.1 Relation to orthocentric tetrahedra
* 3 Rhombohedral lattice
* 4 See also
* 5 Notes
* 6 References
* 7 External links

## Special cases

The common angle at the two apices is here given as θ. There are two general forms of the rhombohedron: oblate (flattened) and prolate (stretched).

File:Rhombohedron-oblate.svg  | File:Prolate rhombohedron.svg

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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 θ>90∘ and in the prolate case θ<90∘. For θ=90∘ 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

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|
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Angle
constraints  | θ=90∘ |  |
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,[4] with rhombic acute angle θ, 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⁡θ,sin⁡θ,0),

e3 : (cos⁡θ,cos⁡θ−cos2θsin⁡θ,1−3cos2θ+2cos3θsin⁡θ).

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 of a rhombohedron, in terms of its side length a and its rhombic acute angle θ, is a simplification of the volume of a parallelepiped, and is given by

V=a3(1−cos⁡θ)1+2cos⁡θ=a3(1−cos⁡θ)2(1+2cos⁡θ)=a31−3cos2θ+2cos3θ.

We can express the volume V another way :

V=23a3sin2(θ2)1−43sin2(θ2).

As the area of the (rhombic) base is given by a2sin⁡θ, 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 θ is given by

h=a(1−cos⁡θ)1+2cos⁡θsin⁡θ=a1−3cos2θ+2cos3θsin⁡θ.

Note:

h=az3 , where z3 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.[6]

## Rhombohedral lattice

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

120px

## See also

* Lists of shapes

## Notes

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

## References

1. ↑ Miller, William A. (January 1989). "Maths Resource: Rhombic Dodecahedra Puzzles". Mathematics in School 18 (1): 18–24.
2. ↑ Inchbald, Guy (July 1997). "The Archimedean honeycomb duals". The Mathematical Gazette 81 (491): 213–219. doi:10.2307/3619198.
3. ↑ Coxeter, HSM. Regular Polytopes. Third Edition. Dover. p.26.
4. ↑ Lines, L (1965). Solid geometry: with chapters on space-lattices, sphere-packs and crystals. Dover Publications.
5. ↑ "Vector Addition". Wolfram. 17 May 2016. http://mathworld.wolfram.com/VectorAddition.html.
6. ↑ Court, N. A. (October 1934), "Notes on the orthocentric tetrahedron", American Mathematical Monthly 41 (8): 499–502, doi:10.2307/2300415 .

## External links

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

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Convex polyhedra

Platonic solids (regular)|

* tetrahedron
* cube
* octahedron
* dodecahedron
* icosahedron

Archimedean solids
(semiregular or uniform)

* truncated tetrahedron
* cuboctahedron
* truncated cube
* truncated octahedron
* rhombicuboctahedron
* truncated cuboctahedron
* snub cube
* icosidodecahedron
* truncated dodecahedron
* truncated icosahedron
* rhombicosidodecahedron
* truncated icosidodecahedron
* snub dodecahedron

Catalan solids
(duals of Archimedean)

* triakis tetrahedron
* rhombic dodecahedron
* triakis octahedron
* tetrakis hexahedron
* deltoidal icositetrahedron
* disdyakis dodecahedron
* pentagonal icositetrahedron
* rhombic triacontahedron
* triakis icosahedron
* pentakis dodecahedron
* deltoidal hexecontahedron
* disdyakis triacontahedron
* pentagonal hexecontahedron

Dihedral regular|

* dihedron
* hosohedron

Dihedral uniform|

* prisms
* antiprisms

duals:|

* bipyramids
* trapezohedra

Dihedral others|

* pyramids
* truncated trapezohedra
* gyroelongated bipyramid
* cupola
* bicupola
* pyramidal frusta
* bifrustum
* rotunda
* birotunda
* prismatoid
* scutoid

Degenerate polyhedra are in italics.

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