Rhombohedron

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.

Solid Geometry

For a unit rhombohedron^[2] (side length = 1) whose rhombic acute angle is θ and has one vertex is at the origin (0, 0, 0) with one edge lying along the x-axis the three vectors are

e₁:

{\biggl (}1,0,0{\biggr )},

e₂:

{\biggl (}\cos \theta ,\sin \theta ,0{\biggr )},

e₃:

{\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^[3] of the 3 direction vectors so that e₁ + e₂, e₁ + e₃, e₂ + e₃, and e₁ + e₂ + e_3.

The volume of the rhombohedron whose side length is 'a' is a simplification of the volume of a parallelepiped and is given by

{\text{Vol}}=a^{3}(1-\cos \theta ){\sqrt {(1+2\cos \theta )}}.

As the area of the base is given by $a^{2}\sin \theta$ and it follows that the height of a rhombohedron is given by the volume divided by the area. The height, 'h', is given by

h={a(1-\cos \theta ){\sqrt {(1+2\cos \theta )}} \over \sin \theta }.

The body diagonal between the acute-angled vertices is the longest. By rotational symmetry about that diagonal, the other three body diagonals between the six obtuse-angled vertices are all the same length.

References

↑ Court, N. A. (October 1934), "Notes on the orthocentric tetrahedron", American Mathematical Monthly: 499–502, doi:10.2307/2300415, JSTOR 2300415 .
↑ Lines, L (1965). Solid geometry: with chapters on space-lattices, sphere-packs and crystals. Dover Publications.
↑ "Vector Addition". Wolfram. 17 May 2016. Retrieved 17 May 2016.

External links

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[1] Court, N. A. (October 1934), "Notes on the orthocentric tetrahedron", American Mathematical Monthly: 499–502, doi:10.2307/2300415, JSTOR 2300415 .

[2] Lines, L (1965). Solid geometry: with chapters on space-lattices, sphere-packs and crystals. Dover Publications.

[3] "Vector Addition". Wolfram. 17 May 2016. Retrieved 17 May 2016.