Rhombohedron

Polyhedron with six rhombi as faces
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] or, inaccurately, a rhomboid) is a three-dimensional figure with six faces which are rhombi. It is a special case of a parallelepiped where all edges are the same length. It can be used to define the rhombohedral lattice system, a honeycomb with rhombohedral cells. A cube is a special case of a rhombohedron with all sides square.

In general a rhombohedron can have up to three types of rhombic faces in congruent opposite pairs, Ci symmetry, order 2.

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.[2]

Rhombohedral lattice system

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

Special cases by symmetry

Special cases of the rhombohedron
Form Cube Trigonal trapezohedron Right rhombic prism Oblique rhombic prism
Angle
constraints 		α = β = γ = 90 ∘ {\displaystyle \alpha =\beta =\gamma =90^{\circ }} α = β = γ {\displaystyle \alpha =\beta =\gamma } α = β = 90 ∘ {\displaystyle \alpha =\beta =90^{\circ }} α = β {\displaystyle \alpha =\beta }
Symmetry Oh
order 48 		D3d
order 12 		D2h
order 8 		C2h
order 4
Faces 6 squares 6 congruent rhombi 2 rhombi, 4 squares 6 rhombi
  • Cube: with Oh symmetry, order 48. All faces are squares.
  • Trigonal trapezohedron (also called isohedral rhombohedron):[3] with D3d symmetry, order 12. All non-obtuse internal angles of the faces are equal (all faces are congruent rhombi). This can be seen by stretching a cube on its body-diagonal axis. For example, a regular octahedron with two regular tetrahedra attached on opposite faces constructs a 60 degree trigonal trapezohedron.
  • Right rhombic prism: with D2h symmetry, order 8. It is constructed by two rhombi and four squares. This can be seen by stretching a cube on its face-diagonal axis. For example, two right prisms with regular triangular bases attached together makes a 60 degree right rhombic prism.
  • Oblique rhombic prism: with C2h symmetry, order 4. It has only one plane of symmetry, through four vertices, and six rhombic faces.

Solid geometry

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

The volume V {\displaystyle V} of an isohedral 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 an isohedral 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.

See also

References

  1. ^ "David Mitchell's Origami Heaven - Rhombic Polyhedra".
  2. ^ Court, N. A. (October 1934), "Notes on the orthocentric tetrahedron", American Mathematical Monthly, 41 (8): 499–502, doi:10.2307/2300415, JSTOR 2300415.
  3. ^ a b Lines, L (1965). Solid geometry: with chapters on space-lattices, sphere-packs and crystals. Dover Publications.
  4. ^ "Vector Addition". Wolfram. 17 May 2016. Retrieved 17 May 2016.

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Convex polyhedra
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Degenerate polyhedra are in italics.