Rhombohedron Explained

bgcolor=#e7dcc3 colspan=2Rhombohedron
align=center colspan=2
Typeprism
Faces6 rhombi
Edges12
Vertices8
Symmetry groupCi, [2<sup>+</sup>,2<sup>+</sup>], (×), order 2
Propertiesconvex, equilateral, zonohedron, parallelohedron

In geometry, a rhombohedron (also called a rhombic hexahedron[1] [2] or, inaccurately, a rhomboid) 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

The common angle at the two apices is here given as

\theta

.There are two general forms of the rhombohedron, oblate (flattened) and prolate (stretched.

In the oblate case

\theta>90\circ

and in the prolate case

\theta<90\circ

. For

\theta=90\circ

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.

FormCube√2 RhombohedronGolden Rhombohedron
Angle
constraints

\theta=90\circ

Ratio of diagonals1√2Golden ratio
OccurrenceRegular solidDissection of the rhombic dodecahedronDissection of the rhombic triacontahedron

Solid geometry

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

\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 :

l(1,0,0r),

e2 :

l(\cos\theta,\sin\theta,0r),

e3 :

l(\cos\theta,{\cos\theta-\cos2\theta\over\sin\theta},{\sqrt{1-3\cos2\theta+2\cos3\theta}\over\sin\theta}r).

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

V

of a rhombohedron, in terms of its side length

a

and its rhombic acute angle

\theta~

, is a simplification of the volume of a parallelepiped, and is given by

V=a3(1-\cos\theta)\sqrt{1+2\cos\theta}=a3\sqrt{(1-\cos\theta)2(1+2\cos\theta)}=a3\sqrt{1-3\cos2\theta+2\cos3\theta}~.

We can express the volume

V

another way :

V=2\sqrt{3}~a3

2\left(\theta
2
\sin

\right)\sqrt{1-

4
3
2\left(\theta
2
\sin

\right)}~.

As the area of the (rhombic) base is given by

a2\sin\theta~

, and as the height of a rhombohedron is given by its volume divided by the area of its base, the height

h

of a rhombohedron in terms of its side length

a

and its rhombic acute angle

\theta

is given by

h=a~{(1-\cos\theta)\sqrt{1+2\cos\theta}\over\sin\theta}=a~{\sqrt{1-3\cos2\theta+2\cos3\theta}\over\sin\theta}~.

Note:

h=a~z

3, where

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.

Relation to orthocentric tetrahedra

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

See main article: Rhombohedral lattice. The rhombohedral lattice system has rhombohedral cells, with 6 congruent rhombic faces forming a trigonal trapezohedron:

See also

External links

Notes and References

  1. Maths Resource: Rhombic Dodecahedra Puzzles. William A.. Miller. Mathematics in School. January 1989. 18. 1. 18–24. 30214564.
  2. Inchbald. Guy. July 1997. 10.2307/3619198. 491. The Mathematical Gazette. 3619198. 213–219. The Archimedean honeycomb duals. 81.
  3. Coxeter, HSM. Regular Polytopes. Third Edition. Dover. p.26.
  4. Book: Lines, L. Solid geometry: with chapters on space-lattices, sphere-packs and crystals. Dover Publications. 1965.
  5. Web site: Vector Addition. 17 May 2016. Wolfram. 17 May 2016.
  6. .