The boundary of Rauzy fractal and discrete tilings

Abstract

We present two methods of constructing the Rauzy fractal by partitioning all points within it into disjoint sets, refer to as layers. We show how these layered structures of the Rauzy fractal can plot the boundary of the fractal effectively. By generalizing the self-replicating pattern of this structure, we demonstrate a new way of discrete tilings of two-dimensional plane.