The eleven (11) general Tilings of the plane, and their duals
 
 
Laves
(isohedral tilings)
Duals
(Archimedian tilings)

3-3-3-3-3-3

3-3-3-3-3-3

3-3-3-3-6

3-3-3-3-6

3-3-3-4-4

3-3-3-4-4

3-3-4-3-4

3-3-4-3-4

3-4-6-4

3-4-6-4

3-6-3-6

3-6-3-6

3-12-12

3-12-12

4-4-4-4

4-4-4-4

4-6-12

4-6-12

4-8-8

4-8-8

6-6-6

6-6-6

These images taken from the SIGGRAPH 2000 coursenotes for subdivision. http://mrl.nyu.edu/publications/subdiv-course2000/

I have posted these simply because I find them quite interesting. The numbers of the tilings indicate the count of edges entering a vertex for each vertex around a polygon in the Lave. Note that in the lave every tile is the same shape, however this is not true in the Duals. 

The duals are created by placing a vertex in the center of every tile and connecting them.

Some links that were relevant on Sunday October 14th 2001: