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Rhombohedron

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Polyhedron with six rhombi as faces

Rhombohedron

Type	prism
Faces	6rhombi
Edges	12
Vertices	8
Symmetry group	C_i , [2⁺,2⁺], (×), order 2
Properties	convex,equilateral,zonohedron,parallelohedron

Ingeometry, arhombohedron (also called arhombic hexahedron^[1]^[2] or, inaccurately, arhomboid^[a]) is a special case of aparallelepiped in which all six faces are congruentrhombi.^[3] It can be used to define therhombohedral lattice system, ahoneycomb with rhombohedral cells. A rhombohedron has two oppositeapices at which all face angles are equal; aprolate rhombohedron has this common angle acute, and anoblate rhombohedron has an obtuse angle at these vertices. Acube is a special case of a rhombohedron with all sidessquare.

Special cases

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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).


Oblate rhombohedron	Prolate rhombohedron

In the oblate case $\theta >90^{\circ }$ and in the prolate case $\theta <90^{\circ }$ . 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.

Form	Cube	√2 Rhombohedron	Golden Rhombohedron
Angle constraints	$\theta =90^{\circ }$
Ratio of diagonals	1	√2	Golden ratio
Occurrence	Regular solid	Dissection of therhombic dodecahedron	Dissection of therhombic triacontahedron

Solid geometry

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For a unit (i.e.: with side length 1) rhombohedron,^[4] 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^[5] of the 3 direction vectors:e₁ +e₂ ,e₁ +e₃ ,e₂ +e₃ , ande₁ +e₂ +e₃ .

The volume $V {\displaystyle V}$ of a rhombohedron, in terms of its side length $a {\displaystyle a}$ and its rhombic acute angle $\theta ~$ , is a simplification of the volume of aparallelepiped, 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 $V {\displaystyle V}$ 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 $h {\displaystyle h}$ of a rhombohedron in terms of its side length $a {\displaystyle a}$ 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 {\displaystyle z}

₃ is the third coordinate ofe₃ .

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

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Four points forming non-adjacent vertices of a rhombohedron necessarily form the four vertices of anorthocentric tetrahedron, and all orthocentric tetrahedra can be formed in this way.^[6]

Rhombohedral lattice

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Main article:Rhombohedral lattice

The rhombohedral lattice system has rhombohedral cells, with 6 congruent rhombic faces forming atrigonal trapezohedron^{[citation needed]}:

Notes

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^More accurately,rhomboid is a two-dimensional figure.

References

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^Miller, William A. (January 1989). "Maths Resource: Rhombic Dodecahedra Puzzles".Mathematics in School.18 (1):18–24.JSTOR 30214564.
^Inchbald, Guy (July 1997). "The Archimedean honeycomb duals".The Mathematical Gazette.81 (491):213–219.doi:10.2307/3619198.JSTOR 3619198.
^Coxeter, HSM.Regular Polytopes. Third Edition. Dover. p.26.
^Lines, L (1965).Solid geometry: with chapters on space-lattices, sphere-packs and crystals. Dover Publications.
^"Vector Addition". Wolfram. 17 May 2016. Retrieved17 May 2016.
^Court, N. A. (October 1934), "Notes on the orthocentric tetrahedron",American Mathematical Monthly,41 (8):499–502,doi:10.2307/2300415,JSTOR 2300415.