Rauzy Tiles

Elsewhere the existence of self-similar tiles associated with Perron numbers is discussed. There the existence of tiles which are the attractors of graph-directed IFSs and are involved in self-similar tilings, but which are not in themselves self-similar was noted. These are known to the mathematical community as Rauzy tiles (in distinction to the Rauzy tile, which is a particular tribonacci tile).

These are also substitution tilings. The mathematical community may have a procedure for generating graph-directed IFSs from substitution rules. However I have conceived a perhaps more easily understood, if perhaps less reliable, technique for finding Rauzy tiles.

Using IFSs of the form

{ T_i: p → ±c^n_i p + P_i(c) } or { T_i: p → c^n_i e^im_iπ p + P_i(c) }

as before, sometimes we get an attractor in which there is an overlap with an "area" which is |c|²ⁿ of that of the whole figure. We can get a figure with a uniform weighted measure by dissecting the overlapping elements, and throwing away all but one of each set of resulting coincident elements. There are two ways this can be used to find new tiles.

Firstly, to find new Perron tiles add a replacement elements of the size of the ones thrown away at positions determined by inspection or by a survey.

Secondly, to find Rauzy tiles. Here we note the dissection equation becomes D(|c|²) = 1 + E(|c|²),where E(|c|²) represents the overlaps. We can solve D*(|c|²) = D(|c|²) - E(|c|²) = 1, and then proceed as for Perron tiles.

Note that I have not actually applied either of those techniques, so I don't have empirical evidence of their viability. I have however successfully applied a third technique related to the first - of removing an element and adding a replacement element at another position.