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Hierarchical self-assembly of fractals with signal-passing tiles

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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 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.

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Notes

  1. In this paper we refer only to “strict” self-assembly, wherein a shape is made by placing tiles only within the domain of the shape, as opposed to “weak” self-assembly where a pattern representing the shape can be formed embedded within a framework of additional tiles.

  2. Please note that by the generalized construction in the proof of Theorems 2 and 3, the same result holds but with junk assemblies of size \(\le 2\). However, we present this result and construction with its full details and all tile types and signals explicitly displayed here and in Hendricks et al. (2016b) because we feel that the simplifications used for this version are easier to follow.

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Acknowledgements

Matthew J. Patitz research was supported in part by National Science Foundation Grant CCF-1422152. Trent A. Rogers research was supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-1450079, and National Science Foundation Grant CCF-1422152.

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Authors and Affiliations

  1. Department of Computer Science and Information Systems, University of Wisconsin - River Falls, River Falls, WI, USA

    Jacob Hendricks

  2. Fayetteville High School, Fayetteville, AR, USA

    Meagan Olsen & Hadley Thomas

  3. Department of Computer Science and Computer Engineering, University of Arkansas, Fayetteville, AR, USA

    Matthew J. Patitz & Trent A. Rogers

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  1. Jacob Hendricks
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  2. Meagan Olsen
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  3. Matthew J. Patitz
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  4. Trent A. Rogers
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  5. Hadley Thomas
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Correspondence to Matthew J. Patitz.

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Hendricks, J., Olsen, M., Patitz, M.J. et al. Hierarchical self-assembly of fractals with signal-passing tiles. Nat Comput 17, 47–65 (2018). https://doi.org/10.1007/s11047-017-9663-9

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