This work is still unfinished, and so I don't have all that much to tell you here.

We're trying to use the theory of tilings to capture the spirit of William Huff's parquet deformations. Basically, it's a question of interpolation -- discovering a plausible and smooth transition between end shapes.

Until I have more to say on this subject, you can find out about parquet deformations by reading Doug Hofstadter's excellent Metamagical Themas, which contains an article about them. You can also browse over to William Huff's article about parquet deformations, or see one that Alvy Ray Smith designed.

Here are some examples constructed with a prototype tool I've developed, offered without commentary.