The boundary of Rauzy fractal and discrete tilings

The Rauzy fractal is a domain in the two-dimensional plane constructed by the Rauzy substitution, a substitution rule on three letters. The Rauzy fractal has a fractal-like boundary, and the currently known its constructions is not only for its boundary but also for the entire domain. In this paper, we show that all points in the Rauzy fractal have a layered structure. We propose two methods of constructing the Rauzy fractal using layered structures. We show how such layered structures can be used to construct the boundary of the Rauzy fractal with less computation than conventional methods. There is a self-replicating pattern in one of the layered structure in the Rauzy fractal. We introduce a notion of self-replicating word and visualize how some self-replicating words on three letters creates discrete tiling of the two dimensional plane.