Approximate Self-Assembly of the Sierpinski Triangle

TL;DR

This paper explores the limits of strict self-assembly of structures approximating the Sierpinski triangle within the Tile Assembly Model, establishing bounds based on fractal dimension and demonstrating tightness with new constructions.

Contribution

It introduces bounds on the fractal dimension of strictly self-assembling sets approximating the Sierpinski triangle and presents a tight construction with communication fibers.

Findings

Sets that strictly self-assemble differ from the Sierpinski triangle by a fractal dimension of at least 1.585.

No subset of the Sierpinski triangle with fractal dimension > 1 can strictly self-assemble.

A tight construction with added communication fibers achieves strict self-assembly without disturbing the fractal structure.

Abstract

The Tile Assembly Model is a Turing universal model that Winfree introduced in order to study the nanoscale self-assembly of complex (typically aperiodic) DNA crystals. Winfree exhibited a self-assembly that tiles the first quadrant of the Cartesian plane with specially labeled tiles appearing at exactly the positions of points in the Sierpinski triangle. More recently, Lathrop, Lutz, and Summers proved that the Sierpinski triangle cannot self-assemble in the "strict" sense in which tiles are not allowed to appear at positions outside the target structure. Here we investigate the strict self-assembly of sets that approximate the Sierpinski triangle. We show that every set that does strictly self-assemble disagrees with the Sierpinski triangle on a set with fractal dimension at least that of the Sierpinski triangle (roughly 1.585), and that no subset of the Sierpinski triangle with fractal dimension greater than 1 strictly self-assembles. We show that our bounds are tight, even when restricted to supersets of the Sierpinski triangle, by presenting a strict self-assembly that adds communication fibers to the fractal structure without disturbing it. To verify this strict self-assembly we develop a generalization of the local determinism method of Soloveichik and Winfree.