Hierarchical Self-Assembly of Fractals with Signal-Passing Tiles (extended abstract)
Jacob Hendricks, Meagan Olsen, Matthew John Patitz, Trent A. Rogers, Hadley Thomas · arXiv (Cornell University) · 2016
In this extended abstract, 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.