Convex uniform honeycomb

The Andreini tessellations are tilings of three-dimensional space using Platonic and Archimedean solids such that all vertices are identical. There are precisely five such tilings:

• The tiling of cubes

• The tiling of octahedra and cuboctahedra

• The tiling of truncated octahedra

• The tiling of octahedra and tetrahedra

• The tiling of tetrahedra and truncated tetrahedra

These and their deformations comprise, together with prisms of 2-D tessellations, all the vertex-uniform tessellations. The tiling of octahedra and tetrahedra is of special importance since its vertices form a cubic close-packing of spheres. All of these are found in crystal arrangements.