Rhombohedron – Wikipedia

Polyhedron with six rhombi as faces

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, C_i 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[edit]

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

Special cases by symmetry[edit]

Form	Cube	Trigonal trapezohedron	Right rhombic prism	Oblique rhombic prism
Angle constraints	${displaystyle alpha =beta =gamma =90^{circ }}$	${displaystyle alpha =beta =gamma }$	${displaystyle alpha =beta =90^{circ }}$	${displaystyle alpha =beta }$
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 O_h symmetry, order 48. All faces are squares.
• Trigonal trapezohedron (also called isohedral rhombohedron):^[3] with D_3d 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 D_2h 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 C_2h symmetry, order 4. It has only one plane of symmetry, through four vertices, and six rhombic faces.

Solid geometry[edit]

For a unit (i.e.: with side length 1) isohedral rhombohedron,^[3] 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 :

${displaystyle {biggl (}cos theta ,{cos theta -cos ^{2}theta over sin theta },{{sqrt {1-3cos ^{2}theta +2cos ^{3}theta }} over sin theta }{biggr )}.}$

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

${displaystyle V}$

of an isohedral 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

${displaystyle V=a^{3}(1-cos theta ){sqrt {1+2cos theta }}=a^{3}{sqrt {(1-cos theta )^{2}(1+2cos theta )}}=a^{3}{sqrt {1-3cos ^{2}theta +2cos ^{3}theta }}~.}$

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 an isohedral rhombohedron in terms of its side length

${displaystyle a}$

and its rhombic acute angle

${displaystyle theta }$

is given by

${displaystyle h=a~{(1-cos theta ){sqrt {1+2cos theta }} over sin theta }=a~{{sqrt {1-3cos ^{2}theta +2cos ^{3}theta }} over sin theta }~.}$

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.

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