Steven Dutch, Natural and Applied Sciences, University
of Wisconsin  Green Bay
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Crystallographer A.F. Wells published a long series of papers in the late 1960's that described a number of hyperbolic tesselations.
This lovely and intricate tesselation has five squares meeting at each vertex. It is formed from crosssheped modules, each consisting of a cube and four triangular prisms. The modules lie in three orientations as shown. On the left is a single layer of the pattern, on the right are two layers with the rear layer colored gray.
This pattern can also be considered as made of truncated octahedra joined by hexagomal prisms. This illustration shows the tesselation from that perspective. Two of the crossshaped modules are colored red.
Icosahedra can be connected by octahedral tunnels to create a tiling with seven
triangles meeting at each vertex. Shown here is a sixmembered ring repeat unit.

Sixmembered rings can in turn be connected to form a threedimensional structure. The
icosahedra basically form a lattice identical to the geometric structure of diamond. The
rear layer is shown here in darker tones.

Snub cubes, alternately left and righthanded, can be joined on their square faces to form a tesselation with eight triangles at each vertex. 
Here we have octahedra connected by octahedral tunnels. The linkages, like the tetrahedrallylinked icosahedra above, form the diamond crystal lattice. 
Nine triangles meet at a vertex. Rows of octahedra enclose tetrahedral voids. This is a geologically very important pattern: it's the atomic structure of spinel (MgAl2O4). The oxygen atoms are located at the octahedron vertices, which are in a cubic closepacked arrangement. The aluminum atoms are at the centers of the octahedra and the magnesium atoms at the centers of the tetrahedra. (This is the reverse of what we would normally expect based on the relative sizes and ionic charges of Mg and Al atoms.) 
This tiling, ten triangles around a vertex, is built up of units consisting of a
central octahedral void surrounded by a ring of six octahedra. Only a pair of opposite
faces of the central octahedron are visible (shown here in light blue). Six units can link
into a zigzag ring, as shown here, with another unit on top to show how layers join.

This seems to be the densest tiling known: twelve triangles meeting at a point. It can be considered as made up of units of eight octahedra surrounding a central octahedral hole. The eight octahedra form a truncated cube unit. The enclosing cubes are truncated 3/8 of the cube edge from each corner, and the recessed central vertices are 1/8 of the cube edge in from the center. By omitting one opposing pair of octahedra we obtain the basic unit of the preceding 10 x 3 tiling. 
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Created 10 March 1999, Last Update 10 March 1999
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