Rhombohedron

 ( Prismatoid Polyhedra ) 

prism

6 Rhombus

12

8

C i , 2+,2+, (×), order 2

convex set, 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 rhombus.Coxeter, HSM. Regular Polytopes. Third Edition. Dover. p.26. 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.

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

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 $\backslash theta\; >\; 90^\backslash circ$ and in the prolate case . For $\backslash theta\; =\; 90^\backslash circ$ 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.

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

e₁ : $\backslash biggl(1,\; 0,\; 0\backslash biggr),$

e₂ : $\backslash biggl(\backslash cos\backslash theta,\; \backslash sin\backslash theta,\; 0\backslash biggr),$

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

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

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

We can express the volume $V$ another way :

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

As the area of the (rhombic) base is given by $a^2\backslash sin\backslash theta~$, 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 $\backslash theta$ is given by

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

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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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..

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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

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Notes

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External links

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