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Special cases

The common angle at the two apices is here given as θ {\displaystyle \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 θ > 90 ∘ {\displaystyle \theta >90^{\circ }} and in the prolate case θ < 90 ∘ {\displaystyle \theta <90^{\circ }} . For θ = 90 ∘ {\displaystyle \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
Angleconstraints	θ = 90 ∘ {\displaystyle \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,⁵ with rhombic acute angle θ   {\displaystyle \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 ) , {\displaystyle {\biggl (}1,0,0{\biggr )},} e2 : ( cos ⁡ θ , sin ⁡ θ , 0 ) , {\displaystyle {\biggl (}\cos \theta ,\sin \theta ,0{\biggr )},} e3 : ( cos ⁡ θ , cos ⁡ θ − cos 2 ⁡ θ sin ⁡ θ , 1 − 3 cos 2 ⁡ θ + 2 cos 3 ⁡ θ sin ⁡ θ ) . {\displaystyle {\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 addition⁶ of the 3 direction vectors: e1 + e2 , e1 + e3 , e2 + e3 , and e1 + e2 + e3 .

The volume V {\displaystyle V} of a rhombohedron, in terms of its side length a {\displaystyle a} and its rhombic acute angle θ   {\displaystyle \theta ~} , is a simplification of the volume of a parallelepiped, and is given by

V = a 3 ( 1 − cos ⁡ θ ) 1 + 2 cos ⁡ θ = a 3 ( 1 − cos ⁡ θ ) 2 ( 1 + 2 cos ⁡ θ ) = a 3 1 − 3 cos 2 ⁡ θ + 2 cos 3 ⁡ θ   . {\displaystyle 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 {\displaystyle V} another way :

V = 2 3   a 3 sin 2 ⁡ ( θ 2 ) 1 − 4 3 sin 2 ⁡ ( θ 2 )   . {\displaystyle 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 ⁡ θ   {\displaystyle 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 {\displaystyle h} of a rhombohedron in terms of its side length a {\displaystyle a} and its rhombic acute angle θ {\displaystyle \theta } is given by

h = a   ( 1 − cos ⁡ θ ) 1 + 2 cos ⁡ θ sin ⁡ θ = a   1 − 3 cos 2 ⁡ θ + 2 cos 3 ⁡ θ sin ⁡ θ   . {\displaystyle 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 {\displaystyle h=a~z} 3 , where z {\displaystyle 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.⁷

Rhombohedral lattice

Main article: Rhombohedral lattice

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

See also

• Lists of shapes

Notes

External links

• Weisstein, Eric W. "Rhombohedron". MathWorld.
• Volume Calculator https://rechneronline.de/pi/rhombohedron.php

References

1. Miller, William A. (January 1989). "Maths Resource: Rhombic Dodecahedra Puzzles". Mathematics in School. 18 (1): 18–24. JSTOR 30214564. /wiki/JSTOR_(identifier) ↩

2. Inchbald, Guy (July 1997). "The Archimedean honeycomb duals". The Mathematical Gazette. 81 (491): 213–219. doi:10.2307/3619198. JSTOR 3619198. /wiki/Doi_(identifier) ↩

3. More accurately, rhomboid is a two-dimensional figure. /wiki/Rhomboid ↩

4. Coxeter, HSM. Regular Polytopes. Third Edition. Dover. p.26. ↩

5. Lines, L (1965). Solid geometry: with chapters on space-lattices, sphere-packs and crystals. Dover Publications. ↩

6. "Vector Addition". Wolfram. 17 May 2016. Retrieved 17 May 2016. http://mathworld.wolfram.com/VectorAddition.html ↩

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