Self-assembly of discrete self-similar fractals

Abstract

In this paper, we search for theoretical 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 weakly self-assembles at temperature 1 in a locally deterministic tile assembly system, and that certain kinds of discrete self-similar fractals do not strictly self-assemble at any temperature. Additionally, we extend the fiber construction of Lathrop et al. (2009) to show that any discrete self-similar fractal belonging to a particular class of “nice” discrete self-similar fractals has a fibered version that strictly self-assembles in the TAM.