Definitions

# Rhombic dodecahedron

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.

## Properties

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.

## Area and volume

The area A and the volume V of the rhombic dodecahedron of edge length a are:
$A = 8sqrt\left\{2\right\}a^2 approx 11.3137085a^2$

$V = frac\left\{16\right\}\left\{9\right\} sqrt\left\{3\right\}a^3 approx 3.07920144a^3$

## Cartesian coordinates

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)

## Related polyhedra

This polyhedron is related to an 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) V3.6.3.6Euclidean plane tilingQuasiregular rhombic tiling V3.8.3.8Hyperbolic plane tiling(Drawn in a Poincaré disk model)

## Related polytopes

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.

The rhombic dodecahedron forms the maximal cross-section of a 24-cell, and also forms the hull of its vertex-first parallel projection into 3 dimensions. The rhombic dodecahedron can be decomposed into 6 congruent (but non-regular) square dipyramids meeting at a single vertex in the center; these form the images of 6 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 6 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.

## References

• 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)