Rhombic dodecahedral honeycomb 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. The "rhombic dodecahedral honeycomb" can be dissected into a trigonal trapezohedral honeycomb with each rhombic dodecahedron dissected into 4 trigonal trapezohedrons. Each rhombic dodecahedra can also be dissected with a center point into 12 rhombic pyramids of the rhombic pyramidal honeycomb. The trapezo-rhombic dodecahedral honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It consists
Honeycomb (geometry) polyhedral cells, and having all vertices the same (i.e., the group of [isometries of 3-space that preserve the tiling] is "transitive on vertices"). There are 28 convex examples in Euclidean 3-space, also called the Archimedean honeycombs. A honeycomb is called regular if the group of isometries preserving the tiling acts transitively on flags, where a flag is a vertex lying on an edge lying on a face lying on a cell. Every regular honeycomb is automatically uniform. However, there is just one regular honeycomb in Euclidean 3-space, the cubic honeycomb. Two are "quasiregular" (made from two types of regular cells):