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Self-assembly of discrete self-similar fractals

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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.

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Acknowledgments

We thank David Doty, Jim Lathrop, Jack Lutz, and Aaron Sterling for useful discussions. We would especially like to thank an anonymous reviewer whose detailed comments have substantially improved the final version of this paper.

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Authors and Affiliations

  1. Department of Computer Science, Iowa State University, Ames, IA, 50011, USA

    Matthew J. Patitz & Scott M. Summers

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  1. Matthew J. Patitz
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  2. Scott M. Summers
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Correspondence to Matthew J. Patitz.

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Preliminary version appeared in the Proceedings of The Fourteenth International Meeting on DNA Computing (DNA 14), Prague, Czech Republic, June 2–6, 2008. This author’s research was supported in part by NSF-IGERT Training Project in Computational Molecular Biology Grant number DGE-0504304.

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Patitz, M.J., Summers, S.M. Self-assembly of discrete self-similar fractals. Nat Comput 9, 135–172 (2010). https://doi.org/10.1007/s11047-009-9147-7

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  • DOI: https://doi.org/10.1007/s11047-009-9147-7

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