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.

Special cases

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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.

Form Cube √2 Rhombohedron Golden Rhombohedron
Angle
constraints
 
Ratio of diagonals 1 √2 Golden ratio
Occurrence Regular solid Dissection of the rhombic dodecahedron Dissection of the rhombic triacontahedron

Solid geometry

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For a unit (i.e.: with side length 1) rhombohedron,[4] 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[5] 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

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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]:

 

See also

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Notes

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

References

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  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.
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