Andreini tessellation

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

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.

This is not correct. There are 28 such tessellations. See B. Grünbaum, Uniform tilings of 3-space. Geombinatorics 4(1994), 49 - 56.

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Andreini tessellation