Rhombic dodecahedral honeycomb

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Rhombic dodecahedral honeycomb

Type	convex uniform honeycomb dual
Cell type	Rhombic dodecahedron V3.4.3.4
Face types	Rhombus
Symmetry group	Fm3m
Dual	tetrahedral-octahedral honeycomb
Properties	edge-transitive, face-transitive, cell-transitive

The rhombic dodecahedra honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is the Voronoi diagram of the face-centered cubic sphere-packing, which is believed to be the densest possible packing of equal spheres in ordinary space (see Kepler conjecture).

It consists of copies of a single cell, the rhombic dodecahedron. All faces are rhombs, with diagonals in the ratio 1:√2. Three cells meet at each edge. The honeycomb is thus cell-transitive, face-transitive and edge-transitive; but it is not vertex-transitive, as it has two kinds of vertex. The vertices with the obtuse rhombic face angles have 4 cells. The vertices with the acute rhombic face angles have 6 cells.

The rhombic dodecahedron can be twisted on one of its hexagonal cross-sections to form a trapezo-rhombic dodecahedron, which is the cell of a somewhat similar tessellation, the Voronoi diagram of hexagonal close-packing.