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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 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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