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Fractal dimension of assemblies in the abstract tile assembly model

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Abstract

In this paper, we investigate the power of systems in the abstract Tile Assembly Model (aTAM) to self-assemble shapes having fractal dimensions between 1 and 2. We introduce the concept of sparsity as a tool for investigating such systems and demonstrate its utility by proving how it relates to fractal dimension. We then prove several results regarding the strict self-assembly of certain classes of fractal shapes in the aTAM including the construction of a universal tileset which, given the correct seed assembly, strictly self-assembles with nearly any desired fractal dimension. Additionally, we discuss a long standing conjecture in tile-assembly, that a class of fractals called discrete self-similar fractals cannot strictly self-assemble in the aTAM, and provide evidence that sparsity, rather than fractal dimension, is a more promising differentiating factor between shapes that can and cannot strictly self-assemble.

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Notes

  1. Here a standard counter gadget refers to commonly used log-width counter gadgets. It is unknown whether or not counter-like gadgets can be implemented in a sparse way

  2. There are universal Turing machines which induce asymptotically smaller runtime blowups, but choosing one with a quadratic blow up makes analysis of the final fractal dimension easier.

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Acknowledgements

This work was supported in part by National Science Foundation grant CAREER-1553166.

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Authors and Affiliations

  1. University of Arkansas, Fayetteville, AR, 72703, USA

    Daniel Hader & Matthew J. Patitz

  2. University of Wisconsin-Oshkosh, Oshkosh, WI, 54901, USA

    Scott M. Summers

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  1. Daniel Hader
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  2. Matthew J. Patitz
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  3. Scott M. Summers
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Correspondence to Matthew J. Patitz.

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Hader, D., Patitz, M.J. & Summers, S.M. Fractal dimension of assemblies in the abstract tile assembly model. Nat Comput (2023). https://doi.org/10.1007/s11047-023-09942-5

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