Many of the most interesting non-periodic tilings on the plane are generated by substitution rules. Some of these, for example the Penrose and Ammann tilings, can also be constructed as the projection of a slice of a higher dimensional lattice.

I will consider the one dimensional case. Here the substitutions, with a simple higher dimensional structure, are precisely the Sturmian, or invertible two letter, substitutions. These leave invariant some Sturmian word. I will then show how the 2-dimensional case relates to 1 dimension, and how this allows an understanding of a set of substitution tilings related to the Ammann tiling.