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From Wikipedia, the free encyclopedia
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Rhombic dodecahedron
Rhombicdodecahedron.jpg
(Click here for rotating model)
Type Catalan solid
Coxeter diagram CDel node f1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node f1.png
CDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.png
Conway notation jC
Face type V3.4.3.4
DU07 facets.png

rhombus
Faces 12
Edges 24
Vertices 14
Vertices by type 8{3}+6{4}
Symmetry group Oh, BC3, [4,3], (*432)
Rotation group O, [4,3]+, (432)
Dihedral angle 120°
Properties convex, face-transitive edge-transitive, parallelohedron
Cuboctahedron.png
Cuboctahedron
(dual polyhedron)
Rhombic dodecahedron Net
Net

In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of two types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron.

Properties[edit]

The rhombic dodecahedron is a zonohedron. Its polyhedral dual is the cuboctahedron. The long diagonal of each face is exactly √2 times the length of the short diagonal, so that the acute angles on each face measure arccos(1/3), or approximately 70.53°.

Being the dual of an Archimedean polyhedron, the rhombic dodecahedron is face-transitive, meaning the symmetry group of the solid acts transitively on the set of faces. In elementary terms, this means that for any two faces A and B there is a rotation or reflection of the solid that leaves it occupying the same region of space while moving face A to face B.

The rhombic dodecahedron is one of the nine edge-transitive convex polyhedra, the others being the five Platonic solids, the cuboctahedron, the icosidodecahedron and the rhombic triacontahedron.

A garnet crystal

The rhombic dodecahedron can be used to tessellate three-dimensional space. It can be stacked to fill a space much like hexagons fill a plane.

This tessellation can be seen as the Voronoi tessellation of the face-centered cubic lattice. Some minerals such as garnet form a rhombic dodecahedral crystal habit. Honeybees use the geometry of rhombic dodecahedra to form honeycomb from a tessellation of cells each of which is a hexagonal prism capped with half a rhombic dodecahedron. The rhombic dodecahedron also appears in the unit cells of diamond and diamondoids. In these cases, four vertices (alternate threefold ones) are absent, but the chemical bonds lie on the remaining edges.[1]

The graph of the rhombic dodecahedron is nonhamiltonian.

Dimensions[edit]

If the edge length of a rhombic dodecahedron is a, the radius of an inscribed sphere (tangent to each of the rhombic dodecahedron's faces) is

r_i = \frac{\sqrt{6}}{3}a \approx 0.8164965809a, OEISA157697

the radius of the midsphere is

r_m = \frac{2\sqrt{2}}{3}a \approx 0.94280904158a, OEISA179587.

and the radius of the circumscribed sphere is

r_o = \frac{2\sqrt{3}}{3}a \approx 1.154700538a, OEISA020832.

Area and volume[edit]

The area A and the volume V of the rhombic dodecahedron of edge length a are:

A = 8\sqrt{2}a^2 \approx 11.3137085a^2
V = \frac{16}{9} \sqrt{3}a^3 \approx 3.07920144a^3

Orthogonal projections[edit]

The rhombic dodecahedron has four special orthogonal projections along its axes of symmetry, centered on a face, an edge, and the two types of vertex, threefold and fourfold. The last two correspond to the B2 and A2 Coxeter planes.

Orthogonal projections
Projective
symmetry
[4] [6] [2] [2]
Rhombic
dodecahedron
Dual cube t1 B2.png Dual cube t1.png Dual cube t1 e.png Dual cube t1 v.png
Cuboctahedron
(dual)
3-cube t1 B2.svg 3-cube t1.svg Cube t1 e.png Cube t1 v.png

Cartesian coordinates[edit]

Pyritohedron variations between a cube and rhombic dodecahedron
Expansion of a rhombic dodecahedron

The eight vertices where three faces meet at their obtuse angles have Cartesian coordinates:

(±1, ±1, ±1)

The coordinates of the six vertices where four faces meet at their acute angles are the permutations of:

(±2, 0, 0)

The rhombic dodecahedron can be seen as a degenerate limiting case of a pyritohedron, with permutation of coordinates (±1, ±1, ±1) and (0, 1+h, 1−h2) with parameter h=1.

Variations[edit]

The rhombic dodecahedron is a parallelohedron, a space-filling polyhedron. Other symmetry constructions of the rhombic dodecahedron are also space-filling.

For example, with 4 square faces, and 60-degree rhombic faces.

Squared rhombic dodecahedron.png Squared rhombic dodecahedron net.png
Net

This construction has D2h symmetry, order 8. It can be seen as a cuboctahedron with square pyramids augmented on the top and bottom. It has coordinates:

(0, 0, ±2)
(±1, ±1, 0)
(±1, 0, ±1)
(0, ±1, ±1)

Related polyhedra[edit]

Spherical rhombic dodecahedron
Uniform octahedral polyhedra
Symmetry: [4,3], (*432) [4,3]+
(432)
[1+,4,3] = [3,3]
(*332)
[3+,4]
(3*2)
{4,3} t{4,3} r{4,3}
r{31,1}
t{3,4}
t{31,1}
{3,4}
{31,1}
rr{4,3}
s2{3,4}
tr{4,3} sr{4,3} h{4,3}
{3,3}
h2{4,3}
t{3,3}
s{3,4}
s{31,1}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png
CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
= CDel nodes 11.pngCDel split2.pngCDel node.png
CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
= CDel nodes 11.pngCDel split2.pngCDel node 1.png
CDel node h0.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
= CDel nodes.pngCDel split2.pngCDel node 1.png
CDel node 1.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png =
CDel nodes 10ru.pngCDel split2.pngCDel node.png or CDel nodes 01rd.pngCDel split2.pngCDel node.png
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png =
CDel nodes 10ru.pngCDel split2.pngCDel node 1.png or CDel nodes 01rd.pngCDel split2.pngCDel node 1.png
CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node h0.png =
CDel node h.pngCDel split1.pngCDel nodes hh.png
Uniform polyhedron-43-t0.svg Uniform polyhedron-43-t01.svg Uniform polyhedron-43-t1.svg
Uniform polyhedron-33-t02.png
Uniform polyhedron-43-t12.svg
Uniform polyhedron-33-t012.png
Uniform polyhedron-43-t2.svg
Uniform polyhedron-33-t1.png
Uniform polyhedron-43-t02.png
Rhombicuboctahedron uniform edge coloring.png
Uniform polyhedron-43-t012.png Uniform polyhedron-43-s012.png Uniform polyhedron-33-t0.pngUniform polyhedron-33-t2.png Uniform polyhedron-33-t01.pngUniform polyhedron-33-t12.png Uniform polyhedron-43-h01.svg
Uniform polyhedron-33-s012.png
Duals to uniform polyhedra
V43 V3.82 V(3.4)2 V4.62 V34 V3.43 V4.6.8 V34.4 V33 V3.62 V35
CDel node f1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 4.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node fh.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png CDel node fh.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 3.pngCDel node fh.pngCDel 4.pngCDel node.png
CDel node f1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 4.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node f1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 3.pngCDel node fh.pngCDel 3.pngCDel node fh.png
Octahedron.svg Triakisoctahedron.jpg Rhombicdodecahedron.jpg Tetrakishexahedron.jpg Hexahedron.svg Deltoidalicositetrahedron.jpg Disdyakisdodecahedron.jpg Pentagonalicositetrahedronccw.jpg Tetrahedron.svg Triakistetrahedron.jpg Dodecahedron.svg

This polyhedron is a part of a sequence of rhombic polyhedra and tilings with [n,3] Coxeter group symmetry. The cube can be seen as a rhombic hexahedron where the rhombi are squares.

Dimensional family of quasiregular spherical polyhedra and tilings: (3.n)2
Sym.
*n32
[n,3]
Spherical Euclid. Compact hyperb. Paraco. Noncompact hyperbolic
*332
[3,3]
Td
*432
[4,3]
Oh
*532
[5,3]
Ih
*632
[6,3]
p6m
*732
[7,3]
 
*832
[8,3]...
 
*∞32
[∞,3]
 
[12i,3] [9i,3] [6i,3] [4i,3]
Figure Uniform tiling 332-t1-1-.png Uniform tiling 432-t1.png Uniform tiling 532-t1.png Uniform tiling 63-t1.png H2 tiling 237-2.png H2 tiling 238-2.png H2 tiling 23i-2.png H2 tiling 23j12-2.png H2 tiling 23j9-2.png H2 tiling 23j6-2.png H2 tiling 23j4-2.png
Config. r{3,3} r{4,3} r{5,3} r{6,3} r{7,3} r{8,3} r{9,3} r{12i,3} r{9i,3} r{6i,3} r{4i,3}
Coxeter CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel ultra.pngCDel node 1.pngCDel 3.pngCDel node.png
Dual uniform figures
Dual
conf.
Uniform tiling 432-t0.png
V(3.3)2
Spherical rhombic dodecahedron.png
V(3.4)2
Spherical rhombic triacontahedron.png
V(3.5)2
Rhombic star tiling.png
V(3.6)2
Order73 qreg rhombic til.png
V(3.7)2
Uniform dual tiling 433-t01-yellow.png
V(3.8)2
Ord3infin qreg rhombic til.png
V(3.∞)2
Coxeter CDel node.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 5.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 6.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 7.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 8.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel infin.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel ultra.pngCDel node f1.pngCDel 3.pngCDel node.png
Dimensional family of quasiregular polyhedra and tilings: 4.n.4.n
Symmetry
*4n2
[n,4]
Spherical Euclidean Compact hyperbolic Paracompact Noncompact
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]...
*∞42
[∞,4]
 
[iπ/λ,4]
Coxeter CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel 7.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel infin.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel ultra.pngCDel node 1.pngCDel 4.pngCDel node.png
Quasiregular
figures
configuration
Uniform tiling 432-t1.png
4.3.4.3
Uniform tiling 44-t1.png
4.4.4.4
H2 tiling 245-2.png
4.5.4.5
H2 tiling 246-2.png
4.6.4.6
H2 tiling 247-2.png
4.7.4.7
H2 tiling 248-2.png
4.8.4.8
H2 tiling 24i-2.png
4.∞.4.∞
4.∞.4.∞
Dual figures
Coxeter CDel node.pngCDel 3.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 4.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 5.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 6.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 7.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 8.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel infin.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel ultra.pngCDel node f1.pngCDel 4.pngCDel node.png
Dual
(rhombic)
figures
configuration
Rhombicdodecahedron.jpg
V4.3.4.3
Uniform tiling 44-t0.png
V4.4.4.4
Order-5-4 quasiregular rhombic tiling.png
V4.5.4.5
Ord64 qreg rhombic til.png
V4.6.4.6
Ord74 qreg rhombic til.png
V4.7.4.7
Ord84 qreg rhombic til.png
V4.8.4.8
Ord4infin qreg rhombic til.png
V4.∞.4.∞
V4.∞.4.∞

Similarly it relates to the infinite series of tilings with the face configurations V3.2n.3.2n, the first in the Euclidean plane, and the rest in the hyperbolic plane.

Rhombicdodecahedron net2.png
V3.4.3.4
(Drawn as a net)
Tile V3636.svg
V3.6.3.6
Euclidean plane tiling
Rhombille tiling
Uniform dual tiling 433-t01.png
V3.8.3.8
Hyperbolic plane tiling
(Drawn in a Poincaré disk model)

Stellations[edit]

Like many convex polyhedra, the rhombic dodecahedron can be stellated by extending the faces or edges until they meet to form a new polyhedron. Several such stellations have been described by Dorman Luke.[2]

The first stellation, often simply called the stellated rhombic dodecahedron, is well known. It can be seen as a rhombic dodecahedron with each face augmented by attaching a rhombic-based pyramid to it, with a pyramid height such that the sides lie in the face planes of the neighbouring faces:

Three flattened octahedra compound.png

Luke describes four more stellations: the second and third stellations (expanding outwards), one formed by removing the second from the third, and another by adding the original rhombic dodecahedron back to the previous one.

Honeycomb[edit]

The rhombic dodecahedron can tessellate space by translational copies of itself. Interestingly, so can the stellated rhombic dodecahedron.

Rhombic dodecahedra.png

Related polytopes[edit]

In a perfect vertex-first projection two of the tesseract's vertices (marked in green) are projected exactly in the center of the rhombic dodecahedron

The rhombic dodecahedron forms the hull of the vertex-first projection of a tesseract to three dimensions. There are exactly two ways of decomposing a rhombic dodecahedron into four congruent parallelepipeds, giving eight possible parallelepipeds. The eight cells of the tesseract under this projection map precisely to these eight parallelepipeds.

The rhombic dodecahedron forms the maximal cross-section of a 24-cell, and also forms the hull of its vertex-first parallel projection into three dimensions. The rhombic dodecahedron can be decomposed into six congruent (but non-regular) square dipyramids meeting at a single vertex in the center; these form the images of six pairs of the 24-cell's octahedral cells. The remaining 12 octahedral cells project onto the faces of the rhombic dodecahedron. The non-regularity of these images are due to projective distortion; the facets of the 24-cell are regular octahedra in 4-space.

This decomposition gives an interesting method for constructing the rhombic dodecahedron: cut a cube into six congruent square pyramids, and attach them to the faces of a second cube. The triangular faces of each pair of adjacent pyramids lie on the same plane, and so merge into rhombuses. The 24-cell may also be constructed in an analogous way using two tesseracts.

See also[edit]

References[edit]

  1. ^ Dodecahedral Crystal Habit. khulsey.com
  2. ^ Luke, D. (1957). "Stellations of the rhombic dodecahedron". The Mathematical Gazette 337: 189–194. 

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