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It is well known that the discrete Sierpinski triangle can be defined as thenonzero residues modulo 2 of Pascal's triangle, and that from this definitionone can easily construct a tileset with which the discrete Sierpinski triangleself-assembles in Winfree's tile assembly model. In this paper we introduce aninfinite class of discrete self-similar fractals that are defined by theresidues modulo a prime p of the entries in a two-dimensional matrix obtainedfrom a simple recursive equation. We prove that every fractal in this classself-assembles using a uniformly constructed tileset. As a special case we showthat the discrete Sierpinski carpet self-assembles using a set of 30 tiles.

Self-assembly of the discrete Sierpinski carpet and related fractals