Lancon Billard Tiles

Substitution rules c) or d) applied to supertiles with a (\frac{1}{2}, -\frac{1}{2}) edge sequence produce conventional tilings, in the sense that there are no negative regions in the tiling and none of the prototiles in the tiling are cut up, although negative tiles play a role in the construction. For n=5 the even s tiles are identical to the Lancon Billard (LB) supertiles. Our approach shows that the LB tiling can be generalized to all odd n>1. In this generalized LB substitution tilings no negative tiles are involved. LB-like tilings can also be constructed using the set of odd n and odd s tiles, as well as the full set of tiles for n even.

4a supertilesLancon Billard vs our tiles