Dan, It should be easy to prove that for a) you only get two tilings (Delone cells of, respectively, Z^3 and A_3 lattices) by enumerating valid combinations of dihedral and solid angles surrounding any vertex. How many cases of b) have you found? Veit On Oct 15, 2010, at 7:31 AM, Dan Asimov wrote:
There are of course 3 regular tilings of the plane by copies of one regular polygon.
There are 8 more if several regular polygons are used, with the regularity condition being that the group of tile-preserving isometries of the plane is transitive on the set of all vertices: the tiling is "vertex-regular". (9 more if you distinguish between mirror images, but it's more natural not to.)
* * *
The natural generalization to 3-space would ask for all vertex-regular tilings of space by:
a) any collection of regular polyhedra
OR, what is perhaps more interesting,
b) any collection of regular and/or Archimedean polyhedra. (There are 13 Archimedean polyhedra if we don't distinguish between mirror images.)
Has anyone seen a classification of either case a), or better, case b) ???
--Dan
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