Self-assembly of the Discrete Sierpinski Carpet and Related Fractals

Abstract

It is well known that the discrete Sierpinski triangle can be defined as the nonzero residues modulo 2 of Pascal’s triangle, and that from this definition one can construct a tileset with which the discrete Sierpinski triangle self-assembles in Winfree’s tile assembly model. In this paper we introduce an infinite class of discrete self-similar fractals (a class that includes both the Sierpinski triangle and the Sierpinski carpet) that are defined by the residues modulo a prime p of the entries in a two-dimensional matrix obtained from a simple recursive equation. We prove that every fractal in this class self-assembles and that there is a uniform procedure that generates the corresponding tilesets. As a special case we show that the discrete Sierpinski carpet self-assembles using a set of 30 tiles.