The Andreini tessellations are tilings of three-dimensional space using Platonic and Archimedean solids such that all vertices are identical. They are special case of uniform tessellation .

There are 28 such tessellations. See B. Grünbaum, Uniform tilings of 3-space. Geombinatorics 4(1994), 49 - 56.

All 28 Andreini tessellations are found in crystal arrangements.

(There are also a few other quasicrystal arrangements that are not Andreini tessellations. [1] (http://www.cmp.caltech.edu/~lifshitz/symmetry.html)).

The tiling of octahedra and tetrahedra is of special importance since its vertices form a cubic close-packing of spheres. The space-filling trusses of packed octahedra and tetrahedra was apparently first discovered by Alexander Graham Bell and independently re-discovered by Buckminster Fuller (who called it the octet truss and patented it in the 1940s) [2] (http://tabletoptelephone.com/~hopspage/Fuller.html) [3] (http://members.cruzio.com/~devarco/energy.htm) [4] (http://www.n55.dk/MANUALS/DISCUSSIONS/OTHER_TEXTS/CM_TEXT.html) [5] (http://www.cjfearnley.com/fuller-faq-2.html). Octet trusses are now one of the most common type of truss used in construction.

Some important examples are:

• 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

External links

• The Uniform Polyhedra (http://www.mathconsult.ch/showroom/unipoly/)
• Virtual Reality Polyhedra (http://www.georgehart.com/virtual-polyhedra/vp.html) The Encyclopedia of Polyhedra