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Rhombic dodecahedron
(Click here for rotating model)
Type	Catalan solid
Face type	rhombus
Faces	12
Edges	24
Vertices	14
Vertices by type	8{3}+6{4}
Face configuration	V3.4.3.4
Symmetry group	Oh
Dihedral angle	120°
Dual	Cuboctahedron
Properties	convex, face-transitive edge-transitive, zonohedron

The rhombic dodecahedron is a convex polyhedron with 12 rhombic faces. It is an Archimedean dual solid, or a Catalan solid. Its dual is the cuboctahedron.

Contents 1 Properties 2 Cartesian coordinates 3 See also 4 References 5 External links

It is the polyhedral dual of the cuboctahedron and a zonohedron. 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 cos⁻¹(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.

The rhombic dodecahedron can be used to tessellate 3-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-centred 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 forms the hull of the vertex-first projection of a tesseract to 3 dimensions. There are exactly two ways of decomposing a rhombic dodecahedron into 4 congruent parallelepipeds, giving 8 possible parallelepipeds. The 8 cells of the tesseract under this projection map precisely to these 8 parallelepipeds.^{[citation needed]}

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

(±1, ±1, ±1)

The six vertices where four faces meet at their acute angles are given by the permutations of

(0, 0, ±2)

• Rhombic triacontahedron
• Truncated rhombic dodecahedron

• Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X.  (Section 3-9)

• Eric W. Weisstein, Rhombic dodecahedron at MathWorld.
• Virtual Reality Polyhedra – The Encyclopedia of Polyhedra
• Rhombic Dodecahedron Calendar – make a rhombic dodecahedron calendar without glue