Rhombohedron

RhombohedronTypeprismFaces6 rhombiEdges12Vertices8Symmetry groupCi , [2+,2+], (×), order 2Propertiesconvex, 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 rhombi.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.

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

In the oblate case and in the prolate case . For 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.

FormCube√2 RhombohedronGolden RhombohedronAngleconstraintsRatio of diagonals 1 √2 Golden ratioOccurrence Regular solid Dissection of the rhombic dodecahedron Dissection of the rhombic triacontahedron

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

e1 :

e2 :

e3 :

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 of a rhombohedron, in terms of its side length and its rhombic acute angle , is a simplification of the volume of a parallelepiped, and is given by

We can express the volume another way :

As the area of the (rhombic) base is given by , and as the height of a rhombohedron is given by its volume divided by the area of its base, the height of a rhombohedron in terms of its side length and its rhombic acute angle is given by

Note: 3 , where 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..

Rhombohedral lattice

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

Rhombohedron

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