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Hierarchical Self-Assembly of Fractals with Signal-Passing Tiles

(Extended Abstract)

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Book cover DNA Computing and Molecular Programming (DNA 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9818))

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Abstract

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.

Matthew J. Patitz—This author’s research was supported in part by National Science Foundation Grant CCF-1422152.

Trent A. Rogers—This author’s 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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Notes

  1. 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.

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Author information

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

Authors
  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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Corresponding author

Correspondence to Matthew J. Patitz .

Editor information

Editors and Affiliations

  1. ESPCI, Laboratoire Gulliver UMR 7083, Paris, France

    Yannick Rondelez

  2. Centre de Recherche Inria de Paris, Paris, France

    Damien Woods

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Hendricks, J., Olsen, M., Patitz, M.J., Rogers, T.A., Thomas, H. (2016). Hierarchical Self-Assembly of Fractals with Signal-Passing Tiles. In: Rondelez, Y., Woods, D. (eds) DNA Computing and Molecular Programming. DNA 2016. Lecture Notes in Computer Science(), vol 9818. Springer, Cham. https://doi.org/10.1007/978-3-319-43994-5_6

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  • DOI: https://doi.org/10.1007/978-3-319-43994-5_6

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