Rhombohedron

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).

In the oblate case \theta 90^\circ and in the prolate case \theta . 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.

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 : \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 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~{(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~z3 , 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.

Rhombohedral lattice

Main article: Rhombohedral lattice

The rhombohedral lattice system has rhombohedral cells, with 6 congruent rhombic faces forming a trigonal trapezohedron: :[[File:Rhombohedral.svg|120px]]

Notes

References

References

1. Miller, William A.. (January 1989). "Maths Resource: Rhombic Dodecahedra Puzzles". Mathematics in School.
2. Inchbald, Guy. (July 1997). "The Archimedean honeycomb duals". The Mathematical Gazette.
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. (17 May 2016). "Vector Addition". Wolfram.
6. Court, N. A.. (October 1934). "Notes on the orthocentric tetrahedron". [[American Mathematical Monthly]].