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Hierarchical growth is necessary and (sometimes) sufficient to self-assemble discrete self-similar fractals

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Abstract

In this paper, we prove that in the abstract Tile Assembly Model (aTAM), an accretion-based model which only allows for a single tile to attach to a growing assembly at each step, there are no tile assembly systems capable of self-assembling the discrete self-similar fractals known as the “H” and “U” fractals. We then show that in a related model which allows for hierarchical self-assembly, the 2-Handed Assembly Model (2HAM), there does exist a tile assembly system which self-assembles the “U” fractal and conjecture that the same holds for the “H” fractal. This is the first example of discrete self similar fractals which self-assemble in the 2HAM but not in the aTAM, providing a direct comparison of the models and greater understanding of the power of hierarchical assembly.

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Notes

  1. Note that we use the standard DSSF definition, restricted to the first quadrant of \(\mathbb {Z} \times \mathbb {Z}\). Our impossibility result proofs could be trivially modified to hold for alternate definitions which allow for DSSFs to occupy any set of quadrants.

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Authors and Affiliations

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

    Jacob Hendricks & Joseph Opseth

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

    Matthew J. Patitz

  3. Computer Science Department, University of Wisconsin–Oshkosh, Oshkosh, USA

    Scott M. Summers

Authors
  1. Jacob Hendricks
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  2. Joseph Opseth
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  3. Matthew J. Patitz
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  4. Scott M. Summers
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Correspondence to Matthew J. Patitz.

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M. J. Patitz’s research was supported in part by National Science Foundation Grants CCF-1422152 and CAREER-1553166.

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Hendricks, J., Opseth, J., Patitz, M.J. et al. Hierarchical growth is necessary and (sometimes) sufficient to self-assemble discrete self-similar fractals. Nat Comput 19, 357–374 (2020). https://doi.org/10.1007/s11047-019-09777-z

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