Rhombohedron

Last updated February 12, 2024

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^[1] 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.

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):^[3] 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,^[3] 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 :

{\biggl (}1,0,0{\biggr )},

e₂ :

{\biggl (}\cos \theta ,\sin \theta ,0{\biggr )},

e₃ :

{\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^[4] of the 3 direction vectors: e₁ + e₂ , e₁ + e₃ , e₂ + e₃ , and e₁ + e₂ + e₃ .

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 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.

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References

↑ "David Mitchell's Origami Heaven - Rhombic Polyhedra".
↑ Court, N. A. (October 1934), "Notes on the orthocentric tetrahedron", American Mathematical Monthly , 41 (8): 499–502, doi:10.2307/2300415, JSTOR 2300415 .
1 2 Lines, L (1965). Solid geometry: with chapters on space-lattices, sphere-packs and crystals. Dover Publications.
↑ "Vector Addition". Wolfram. 17 May 2016. Retrieved 17 May 2016.

External links

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[1] "David Mitchell's Origami Heaven - Rhombic Polyhedra".

[2] Court, N. A. (October 1934), "Notes on the orthocentric tetrahedron", American Mathematical Monthly , 41 (8): 499–502, doi:10.2307/2300415, JSTOR 2300415 .

[:0-3] 1 2 Lines, L (1965). Solid geometry: with chapters on space-lattices, sphere-packs and crystals. Dover Publications.

[4] "Vector Addition". Wolfram. 17 May 2016. Retrieved 17 May 2016.

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