Self-assembly of the discrete Sierpinski carpet and related fractals

TL;DR

This paper introduces a broad class of discrete self-similar fractals, including the Sierpinski carpet, and proves they can all self-assemble using specifically constructed tilesets within the tile assembly model.

Contribution

It generalizes the self-assembly framework to a new class of fractals defined by modular residues, providing explicit tilesets for their assembly.

Findings

Discrete Sierpinski carpet self-assembles with 30 tiles.

All fractals in the class can be assembled with a uniform tileset.

The approach extends tile assembly to new fractal structures.

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 easily 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 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 using a uniformly constructed tileset. As a special case we show that the discrete Sierpinski carpet self-assembles using a set of 30 tiles.