Sketch of the Techniques in Matching Rules and Substitution Tilings"

The fundamental idea behind the construction is rather simple: in essence, we wish the tiles to organize themselves into larger and larger images of the in ated tiles (these images are \supertiles"). Each of these supertiles is to be associated with a small amount of information| its alleged position with respect to its parent supertile. This information resides on a \skeleton" of edges; these skeletons are designed so that every edge in the tiling belongs to only nitely many skeletons, and the skeletons of one generation of supertile connect to the skeletons of the previous generation. The proof that the construction succeeds is by induction: if every structure is well-formed for one generation, we show every structure must be well-formed for the previous generation as well.