Hierarchical Growth is Necessary and (Sometimes) Sufficient to Self-Assemble Discrete Self-Similar Fractals

TL;DR

This paper demonstrates that hierarchical self-assembly models can assemble certain fractals that are impossible to assemble in simpler, non-hierarchical models, highlighting the importance of hierarchical growth in self-assembly.

Contribution

It proves the impossibility of self-assembling specific fractals in the aTAM and shows their assembly in the 2HAM, establishing the significance of hierarchical growth models.

Findings

No aTAM systems can assemble the 'H' and 'U' fractals.

The 2HAM can assemble the 'U' fractal.

Hierarchical growth is sometimes sufficient for self-assembly.

Abstract

In this paper, we prove that in the abstract Tile Assembly Model (aTAM), an accretion-based model which only allows for a single tile to attach to a growing assembly at each step, there are no tile assembly systems capable of self-assembling the discrete self-similar fractals known as the "H" and "U" fractals. We then show that in a related model which allows for hierarchical self-assembly, the 2-Handed Assembly Model (2HAM), there does exist a tile assembly systems which self-assembles the "U" fractal and conjecture that the same holds for the "H" fractal. This is the first example of discrete self similar fractals which self-assemble in the 2HAM but not in the aTAM, providing a direct comparison of the models and greater understanding of the power of hierarchical assembly.