Rhombic Dodecahedron

$\begin{figure}\BoxedEPSF{RhombicDodecahedron_net.epsf scaled 700}\end{figure}$

The Dual Polyhedron of the Cuboctahedron, also sometimes called the Rhomboidal Dodecahedron (Cotton 1990). Its 14 vertices are joined by 12 Rhombuses, and one possible way to construct it is known as the Bauspiel. The rhombic dodecahedron is a Zonohedron and a Space-Filling Polyhedron. The vertices are given by ( $\pm 1$ , $\pm 1$ , $\pm 1$ ), ( $\pm 2$ , 0, 0), (0, $\pm 2$ , 0), (0, 0, $\pm 2$ ).

$\begin{figure}\begin{center}\BoxedEPSF{CubeOctahedronPoints.epsf scaled 700}\quad\BoxedEPSF{CubeOctahedronRhombs.epsf scaled 700}\end{center}\end{figure}$

The edges of the Cube-Octahedron Compound intersecting in the points plotted above are the diagonals of Rhombuses, and the 12 Rhombuses form a rhombic dodecahedron (Ball and Coxeter 1987).

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 137, 1987.

Cotton, F. A. Chemical Applications of Group Theory, 3rd ed. New York: Wiley, p. 62, 1990.