Rhombic dodecahedron
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.
Rhombic dodecahedron  

(Click here for rotating model)  
Type  Catalan solid 
Coxeter diagram  
Conway notation  jC 
Face type  V3.4.3.4 rhombus 
Faces  12 
Edges  24 
Vertices  14 
Vertices by type  8{3}+6{4} 
Symmetry group  O_{h}, B_{3}, [4,3], (*432) 
Rotation group  O, [4,3]^{+}, (432) 
Dihedral angle  120° 
Properties  convex, facetransitive isohedral, isotoxal, parallelohedron 
Cuboctahedron (dual polyhedron) 
Net 
Properties
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 facetransitive, 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 can be viewed as the convex hull of the union of the vertices of a cube and an octahedron. The 6 vertices where 4 rhombi meet correspond to the vertices of the octahedron, while the 8 vertices where 3 rhombi meet correspond to the vertices of the cube.
The rhombic dodecahedron is one of the nine edgetransitive convex polyhedra, the others being the five Platonic solids, the cuboctahedron, the icosidodecahedron and the rhombic triacontahedron.
The rhombic dodecahedron can be used to tessellate threedimensional space. It can be stacked to fill a space much like hexagons fill a plane.
This polyhedron in a spacefilling tessellation can be seen as the Voronoi tessellation of the facecentered cubic lattice. It is the Brillouin zone of body centered cubic (bcc) crystals. Some minerals such as garnet form a rhombic dodecahedral crystal habit. Honey bees use the geometry of rhombic dodecahedra to form honeycombs 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.
A rhombic dodecahedron can be dissected with its center into 4 trigonal trapezohedra. These rhombohedra are the cells of a trigonal trapezohedral honeycomb. This is analogous to the dissection of a regular hexagon dissected into rhombi, and tiled in the plane as a rhombille.
Dimensions
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
the radius of the midsphere is
and the radius of the circumscribed sphere is
Area and volume
The area A and the volume V of the rhombic dodecahedron of edge length a are:
Orthogonal projections
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 B_{2} and A_{2} Coxeter planes.
Projective symmetry 
[4]  [6]  [2]  [2] 

Rhombic dodecahedron 

Cuboctahedron (dual) 
Cartesian coordinates
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:
 (±2, 0, 0), (0, ±2, 0) and (0, 0, ±2)
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 − h^{2}) with parameter h = 1.
Topologically equivalent forms
Parallelohedron
The rhombic dodecahedron is a parallelohedron, a spacefilling polyhedron, dodecahedrille, being the dual to the tetroctahedrille or half cubic honeycomb, and described by two Coxeter diagrams:
The rhombic dodecahedron can tessellate space by translational copies of itself. So can the stellated rhombic dodecahedron. 
The rhombic dodecahedron can be constructed with 4 sets of parallel edges. 
Dihedral rhombic dodecahedron
Other symmetry constructions of the rhombic dodecahedron are also spacefilling, and as parallelotopes they are similar to variations of spacefilling truncated octahedra.[2]
For example, with 4 square faces, and 60degree rhombic faces, and D_{4h} dihedral symmetry, order 16. It can be seen as a cuboctahedron with square pyramids augmented on the top and bottom.
Net 
Coordinates

Bilinski dodecahedron
Bilinski dodecahedron with edges and front faces colored by their symmetry positions. 
Bilinski dodecahedron colored by parallel edges 
In 1960 Stanko Bilinski discovered a second rhombic dodecahedron with 12 congruent rhombus faces, the Bilinski dodecahedron. It has the same topology but different geometry. The rhombic faces in this form have the golden ratio. [3][4]
First form  Second form 

√2:1  √5 + 1/2:1 
Deltoidal dodecahedron
Another topologically equivalent variation, sometimes called a deltoidal dodecahedron[5] or trapezoidal dodecahedron[6][7], is isohedral with tetrahedral symmetry order 24, distorting rhombic faces into kites (deltoids). It has 8 vertices adjusted in or out in alternate sets of 4, with the limiting case a tetrahedral envelope. Variations can be parametrized by (a,b), where b is determined from a for planar faces. (1,1) is the rhombic solution. As (a) approaches 1/2, (b) approaches infinity.
 (±2, 0, 0), (0, ±2, 0), (0, 0, ±2)
 (a, a, a), (−a, −a, a), (−a, a, −a), (a, −a, −a)
 (−b, −b, −b), (−b, b, b), (b, −b, b), (b, b, −b)
(1,1)  (7/8,7/6)  (3/4,3/2)  (2/3,2)  (5/8,5/2)  (9/16,9/2) 

Related polyhedra
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{3^{1,1}} 
t{3,4} t{3^{1,1}} 
{3,4} {3^{1,1}} 
rr{4,3} s_{2}{3,4} 
tr{4,3}  sr{4,3}  h{4,3} {3,3} 
h_{2}{4,3} t{3,3} 
s{3,4} s{3^{1,1}} 
= 
= 
= 

Duals to uniform polyhedra  
V4^{3}  V3.8^{2}  V(3.4)^{2}  V4.6^{2}  V3^{4}  V3.4^{3}  V4.6.8  V3^{4}.4  V3^{3}  V3.6^{2}  V3^{5} 
When projected onto a sphere (see right), it can be seen that the edges make up the edges of two tetrahedra arranged in their dual positions (the stella octangula). This trend continues on with the deltoidal icositetrahedron and deltoidal hexecontahedron for the dual pairings of the other regular polyhedra (alongside the triangular bipyramid if improper tilings are to be considered), giving this shape the alternative systematic name of deltoidal dodecahedron.
Symmetry *n32 [n,3] 
Spherical  Euclid.  Compact hyperb.  Paraco.  

*232 [2,3] 
*332 [3,3] 
*432 [4,3] 
*532 [5,3] 
*632 [6,3] 
*732 [7,3] 
*832 [8,3]... 
*∞32 [∞,3]  
Figure Config. 
V3.4.2.4 
V3.4.3.4 
V3.4.4.4 
V3.4.5.4 
V3.4.6.4 
V3.4.7.4 
V3.4.8.4 
V3.4.∞.4 
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.
Symmetry mutations of dual quasiregular tilings: V(3.n)^{2}  

*n32  Spherical  Euclidean  Hyperbolic  
*332  *432  *532  *632  *732  *832...  *∞32  
Tiling  
Conf.  V(3.3)^{2}  V(3.4)^{2}  V(3.5)^{2}  V(3.6)^{2}  V(3.7)^{2}  V(3.8)^{2}  V(3.∞)^{2} 
*n42 symmetry mutations of quasiregular dual tilings: V(4.n)^{2}  

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]  
Tiling Conf. 
V4.3.4.3 
V4.4.4.4 
V4.5.4.5 
V4.6.4.6 
V4.7.4.7 
V4.8.4.8 
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.
V3.4.3.4 (Drawn as a net) 
V3.6.3.6 Euclidean plane tiling Rhombille tiling 
V3.8.3.8 Hyperbolic plane tiling (Drawn in a Poincaré disk model) 
Stellations
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.[8]
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 rhombicbased pyramid to it, with a pyramid height such that the sides lie in the face planes of the neighbouring faces:
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.
Great rhombic dodecahedron : 12 concave octagonal faces
Great stellated rhombic dodecahedron : 12 concave dodecagonal faces
Related polytopes
The rhombic dodecahedron forms the hull of the vertexfirst projection of a tesseract to three dimensions. There are exactly two ways of decomposing a rhombic dodecahedron into four congruent rhombohedra, giving eight possible rhombohedra as projections of the tesseracts 8 cubic cells. One set of projective vectors are: u=(1,1,1,1), v=(1,1,1,1), w=(1,1,1,1).
The rhombic dodecahedron forms the maximal crosssection of a 24cell, and also forms the hull of its vertexfirst parallel projection into three dimensions. The rhombic dodecahedron can be decomposed into six congruent (but nonregular) square dipyramids meeting at a single vertex in the center; these form the images of six pairs of the 24cell's octahedral cells. The remaining 12 octahedral cells project onto the faces of the rhombic dodecahedron. The nonregularity of these images are due to projective distortion; the facets of the 24cell are regular octahedra in 4space.
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 24cell may also be constructed in an analogous way using two tesseracts.[9]
See also
 Dodecahedron
 Rhombic triacontahedron
 Truncated rhombic dodecahedron
 24cell – 4D analog of rhombic dodecahedron
 Archimede construction systems
 Fully truncated rhombic dodecahedron
References
 Dodecahedral Crystal Habit Archived 20090412 at the Wayback Machine. khulsey.com
 Order in Space: A design source book, Keith Critchlow, p.56–57
 Branko Grünbaum (2010). "The Bilinski Dodecahedron and Assorted Parallelohedra, Zonohedra, Monohedra, Isozonohedra, and Otherhedra" (PDF). 32 (4): 5–15. Archived from the original (PDF) on 20150402. Cite journal requires
journal=
(help)  H.S.M Coxeter, "Regular polytopes", Dover publications, 1973.
 Economic Mineralogy: A Practical Guide to the Study of Useful Minerals, p.8
 http://mathworld.wolfram.com/Isohedron.html
 http://loki3.com/poly/transforms.html
 Luke, D. (1957). "Stellations of the rhombic dodecahedron". The Mathematical Gazette. 337: 189–194.
 https://www.youtube.com/watch?v=oJ7uOj2LRso
Further reading
 Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 048623729X. (Section 39)
 Wenninger, Magnus (1983). Dual Models. Cambridge University Press. doi:10.1017/CBO9780511569371. ISBN 9780521543255. MR 0730208. (The thirteen semiregular convex polyhedra and their duals, Page 19, Rhombic dodecahedron)
 The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim GoodmanStrass, ISBN 9781568812205 (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, p. 285, Rhombic dodecahedron )
External links
 Eric W. Weisstein, Rhombic dodecahedron (Catalan solid) at MathWorld.
 Virtual Reality Polyhedra – The Encyclopedia of Polyhedra
Computer models
Paper projects
 Rhombic Dodecahedron Calendar – make a rhombic dodecahedron calendar without glue
 Another Rhombic Dodecahedron Calendar – made by plaiting paper strips
Practical applications
 Archimede Institute Examples of actual housing construction projects using this geometry