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Rhombohedron - Wikipedia

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

In geometry, a rhombohedron (also called a rhombic hexahedron[1][2] or, inaccurately, a rhomboid[a]) is a special case of a parallelepiped in which all six faces are congruent rhombi.[3] 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.

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

In the oblate case {\displaystyle \theta >90^{\circ }} and in the prolate case {\displaystyle \theta <90^{\circ }}. For {\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
Angle
constraints 		{\displaystyle \theta =90^{\circ }}
Ratio of diagonals 1 √2 Golden ratio
Occurrence Regular solid Dissection of the rhombic dodecahedron Dissection of the rhombic triacontahedron

For a unit (i.e.: with side length 1) rhombohedron,[4] with rhombic acute angle {\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 : {\displaystyle {\biggl (}1,0,0{\biggr )},}
e2 : {\displaystyle {\biggl (}\cos \theta ,\sin \theta ,0{\biggr )},}
e3 : {\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 addition[5] of the 3 direction vectors: e1 + e2 , e1 + e3 , e2 + e3 , and e1 + e2 + e3 .

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

{\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 {\displaystyle V} another way :

{\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 {\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 {\displaystyle h} of a rhombohedron in terms of its side length {\displaystyle a} and its rhombic acute angle {\displaystyle \theta } is given by

{\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:

{\displaystyle h=a~z}3 , where {\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

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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.[6]

Rhombohedral lattice

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

  1. ^ More accurately, rhomboid is a two-dimensional figure.
  1. ^ Miller, William A. (January 1989). "Maths Resource: Rhombic Dodecahedra Puzzles". Mathematics in School. 18 (1): 18–24. JSTOR 30214564.
  2. ^ Inchbald, Guy (July 1997). "The Archimedean honeycomb duals". The Mathematical Gazette. 81 (491): 213–219. doi:10.2307/3619198. JSTOR 3619198.
  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. ^ "Vector Addition". Wolfram. 17 May 2016. Retrieved 17 May 2016.
  6. ^ Court, N. A. (October 1934), "Notes on the orthocentric tetrahedron", American Mathematical Monthly, 41 (8): 499–502, doi:10.2307/2300415, JSTOR 2300415.