I tried to see if the ratio of the square grid would be 4 and found that it would be. If the radius was 3, the diameter would be 6 and the circumference would be 24. 24:6 is equal to 4. The grid with the triangles has a ratio of 3. This is because the circumference is 12 and the diameter is 4 (12:4 is 3). After reading Trevor's blog I became confused. A triangle has a ratio of 3. A square has a ratio of 4. But other than that finding a pattern would be ridiculously hard.
The reason that only triangles, squares, and hexagons can tessalate has to do with the interior angles. The interior angles are respectively 60, 90, and 120. All three of these are factors of 360. If you look at a pentagon, the interior angle is 108 degrees which does not divide into 360. An octagon has an interior angle of 135 degrees which also does not divide into 360.
I think that the Polyhedra tessellations are very similar to regular tessellations. You could glue pyramids, cubes, and hexagonal prisms together but not a pentagonal prism.
http://solmentes.blogspot.com/2010/01/italian-floor-tiles-and-euclid.html
I did find this weird thing called a rhombic dodecahedron. Based on the wikipedia page it looks like it can tessellate. Based on this link i got the impression that it could tessellate (but i'm not sure).
http://en.wikipedia.org/wiki/Rhombic_dodecahedron
