Hierarchical Self-Assembly of Fractals with Signal-Passing Tiles

TL;DR

This paper introduces hierarchical tile-based self-assembly methods to construct complex fractals, including the Sierpinski triangle, at scale factor 1 and temperature 1, revealing new insights into natural and artificial self-assembling systems.

Contribution

It presents the first constructions for arbitrary discrete self-similar fractals at scale factor 1 and many at temperature 1, using signal-passing tiles with local control.

Findings

Successfully assembled the Sierpinski triangle at temperature 1

Developed hierarchical assembly techniques for complex fractals

Provided insights into natural self-assembly processes

Abstract

In this paper, we present high-level overviews of tile-based self-assembling systems capable of producing complex, infinite, aperiodic structures known as discrete self-similar fractals. Fractals have a variety of interesting mathematical and structural properties, and by utilizing the bottom-up growth paradigm of self-assembly to create them we not only learn important techniques for building such complex structures, we also gain insight into how similar structural complexity arises in natural self-assembling systems. Our results fundamentally leverage hierarchical assembly processes, and use as our building blocks square "tile" components which are capable of activating and deactivating their binding "glues" a constant number of times each, based only on local interactions. We provide the first constructions capable of building arbitrary discrete self-similar fractals at scale factor 1, and many at temperature 1 (i.e. "non-cooperatively"), including the Sierpinski triangle.