Self-Replicating Tiles and Their Boundary

All self-replicating lattice tilings of the plane can be constructed using special iterated function systems (IFS). Certain self-replicating curves can be constructed using the recurrent set method (RS). A bijection between the IFS parameters and the RS parameters is such that the curve K produced by the RS parameters is the boundary of the tile T produced by the IFS parameters. The correspondence is algorithmic in that K can be drawn from the IFS data using turtle graphics and T can be drawn from the RS data using an IFS iteration.