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a parallelepiped bounded by six similar faces (either rhombuses or parallelograms)
having threefold symmetry (同)trigonal

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出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2015/10/22 23:46:58」(JST)

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Rhombohedron

Type	Prism
Faces	6 rhombi
Edges	12
Vertices	8
Symmetry group	C_i, [2⁺,2⁺], (×), Order 2
Properties	convex, zonohedron

In geometry, a rhombohedron is a three-dimensional figure like a cube, except that its faces are not squares but 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.

In general the rhombohedron can have 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.^[1]

Rhombohedral lattice system

The rhombohedral lattice system has rhombohedral cells, with 3 pairs of unique rhombic faces:

Special cases

Form	Cube	Trigonal trapezohedron	Right rhombic prism	General rhombic prism	General rhombohedron
Symmetry	O_h, [4,3], order 48	D_3d, [2+,6], order 12	D_2h, [2,2], order 8	C_2h, [2], order 4	C_i, [2+,2+], order 2
Image
Faces	6 squares	6 identical rhombi	Two rhombi and 4 squares	6 rhombic faces	6 rhombic faces

Cube: with O_h symmetry, order 48. All faces are squares.
Trigonal trapezohedron: with D_3d symmetry, order 12. If all of the non-obtuse internal angles of the faces are equal (all faces are same). This can be see by stretching a cube on its body-digonal axis. For example a regular octahedron with two tetrahedra attached on opposite faces constructs a 60 degree trigonal trapezohedron:.
Right rhombic prism: with D_2h symmetry, order 8. It constructed by two rhombi and 4 squares. This can be see by stretching a cube on its face-digonal axis. For example two triangular prisms attached together makes a 60 degree rhombic prism.
A general rhombic prism: With C_2h symmetry, order 4. It has only one plane of symmetry through four vertices, and 6 rhombic faces.

References

^ Court, N. A. (October 1934), "Notes on the orthocentric tetrahedron", American Mathematical Monthly: 499–502, JSTOR 2300415 .

External links

Weisstein, Eric W., "Rhombohedron", MathWorld.
"golden" rhombohedron
"silver" rhombohedron

This polyhedron-related article is a stub. You can help Wikipedia by expanding it.

English Journal

Host Lipid and Temperature as Important Screening Variables for Crystallizing Integral Membrane Proteins in Lipidic Mesophases. Trials with Diacylglycerol Kinase.

Li D1, Shah ST, Caffrey M.
Crystal growth & design.Cryst Growth Des.2013 Jul 3;13(7):2846-2857.
A systematic study of the crystallization of an α-helical, integral membrane enzyme, diacylglycerol kinase, DgkA, using the lipidic cubic mesophase or in meso method is described. These trials have resulted in the production of blocky, rhombohedron-shaped crystals of diffraction quality currently i
PMID 23956688

Density functional calculations of polysynthetic Brazil twinning in α-quartz.

Grimmer H1, Delley B.
Acta crystallographica. Section A, Foundations of crystallography.Acta Crystallogr A.2012 May;68(Pt 3):359-65. doi: 10.1107/S0108767312008756. Epub 2012 Mar 22.
Polysynthetic Brazil twinning in α-quartz, which occurs commonly in amethyst, is interpreted in the literature as having its composition planes parallel to one of the faces of the major rhombohedron r. It is shown that, instead, the composition planes are parallel to one of the faces of the minor r
PMID 22514068

Surface self-diffusion behavior of individual tungsten adatoms on rhombohedral clusters.

Yang J1, Hu W, Tang J.
Journal of physics. Condensed matter : an Institute of Physics journal.J Phys Condens Matter.2011 Oct 5;23(39):395004. doi: 10.1088/0953-8984/23/39/395004. Epub 2011 Sep 14.
The diffusion of single tungsten adatoms on the surfaces of rhombohedral clusters is studied by means of molecular dynamics and the embedded atom method. The energy barriers for the adatom diffusing across and along the step edge between a {110} facet and a neighboring {110} facet are calculated usi
PMID 21918292

Japanese Journal

Synthesis of Polymerized C60 Bulk and Film in Three Dimensions by Irradiating Free-Electron Laser

Iwata Nobuyuki,Ando Shingo,Nokariya Ryo [他],Yamamoto Hiroshi
Jpn J Appl Phys 47(2), 1412-1415, 2008-02-25
… These results indicate that the polymerized structure is a two-dimensional (2D) rhombohedron in the film plane. … We expect that the (0001) surface of the oriented film will be suitable for a 2D polymerized rhombohedron and that migration energy will contribute to polymerization at an elevated temperature. …
NAID 150000050050

無水硫酸アルミニウムの結晶構造

加藤悦朗,大門啓志
窯業協會誌 87(1011), 590-595, 1979-11-01
… The grain was rhombohedron in shape and surrounded by crystal-faces of {0 1 1 2} planes. …
NAID 110002310621