Self-Assembly of Discrete Self-Similar Fractals

TL;DR

This paper explores the limitations of the Tile Assembly Model in self-assembling fractal shapes, proving certain impossibilities and providing methods to achieve strict self-assembly for specific classes of fractals.

Contribution

It establishes fundamental limitations for self-similar fractals in TAM and extends fiber construction techniques to enable strict self-assembly for certain fractals.

Findings

No self-similar fractal fully weakly self-assembles at temperature 1.

Certain self-similar fractals do not strictly self-assemble at any temperature.

A fibered version of some fractals can strictly self-assemble in TAM.

Abstract

In this paper, we search for {\it absolute} limitations of the Tile Assembly Model (TAM), along with techniques to work around such limitations. Specifically, we investigate the self-assembly of fractal shapes in the TAM. We prove that no self-similar fractal fully weakly self-assembles at temperature 1, and that certain kinds of self-similar fractals do not strictly self-assemble at any temperature. Additionally, we extend the fiber construction from Lathrop et. al. (2007) to show that any self-similar fractal belonging to a particular class of "nice" self-similar fractals has a fibered version that strictly self-assembles in the TAM.