Rhombic Dodecahedron

The rhombic dodecahedron is a convex polyhedron with twelve identical rhombic faces, fourteen vertices and twenty-four edges, each face a rhombus whose two diagonals are in the ratio of one to the square root of two. It is one of the thirteen Catalan solids, the polyhedra dual to the thirteen Archimedean solids, and it is specifically the dual of the cuboctahedron, its faces corresponding to the cuboctahedron's vertices and its vertices to the cuboctahedron's faces, a duality relationship first worked out in the nineteenth century by the Belgian mathematician Eugene Catalan, after whom the whole class of solids is named. Unlike most Catalan solids, the rhombic dodecahedron is also a space filling polyhedron, meaning identical copies of it can be stacked to fill three dimensional space with no gaps or overlaps, a property closely related to the structure of the face centered cubic lattice used to describe close packed crystals. The same shape occurs in nature as the crystal habit of garnet and as a good geometric approximation of the cell shape bees construct in the interior of a honeycomb.