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 Branko 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]).
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)
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[3]
[4]
[5]. Octet trusses are now one of the most common type of truss used in construction.
Some important examples are:
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