Dodecahedron, regular rhombic

First Brillouin Zone of the Body-Centered Cubic Lattice. The Figure is a Regular Rhombic Dodecahedron... 

In practice, the crystals of the alkaline metals have cubic lattice of I-t5 e which corresponds to the reciprocal lattice of F-type according to Table 3.4 and the first derived Brillouin zone is of type of a regular rhombic dodecahedron, see Figure 3.20-left (Putz, 2006). 

Figure 22. (a, b) SEM and (c) TEM images of slightly truncated rhombic dodecahedral BaZr03 microcrystals. The arrow shown in (b) is indication of truncated comers, (d) Simulated morphology evolution process of the three states of regular rhombic dodecahedron, truncated dodecahedron and sphere 188]. 

Fig. 1.—The arrangement of 45 spheres in icosahedral closest packing. At the left there is shown a single sphere, which constitutes the inner core. Next there is shown the layer of 12 spheres, at the corners of a regular icosahedron. The third model shows the core of 13 spheres with 20 added in the outer layer, each in a triangular pocket corresponding to a face of the icosahedron these 20 spheres lie at the corners of a pentagonal dodecahedron. The third layer is completed, as shown in the model at the right, by adding 12 spheres at corners of a large icosahedron the 32 spheres of the third layer lie at the corners of a rhombic triaconta-hedron. The fourth layer (not shown) contains 72 spheres.

Corresponding to the semi-regular polyhedra of Table 3.3 there are sets of reciprocal bodies named after Catalan, who first described them all in 1865. Of these we need note only the rhombic dodecahedron, which is the reciprocal of the cuboctahedron, and the family of bipyramids which are related in a similar way to the prisms. 

This compares to values of 1.0990 for the planar tetrakaidecahedron 1.1053 for the rhombic dodecahedron and 1.0984 for the regular pentagonal dodecahedron. The latter - though often considered as a unit cell in foam modeling - is not really a viable candidate either, as it not only violates Plateau s laws but is also not space filling.)... 

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