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 systems 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.
M. J. Patitz—This author’s research was supported in part by National Science Foundation Grants CCF-1422152 and CAREER-1553166.
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- 1.
Note that we use the standard DSSF definition in which DSSF’s are contained within quadrant I of \(\mathbb {N}^2\). However, 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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Hendricks, J., Opseth, J., Patitz, M.J., Summers, S.M. (2018). Hierarchical Growth Is Necessary and (Sometimes) Sufficient to Self-assemble Discrete Self-similar Fractals. In: Doty, D., Dietz, H. (eds) DNA Computing and Molecular Programming. DNA 2018. Lecture Notes in Computer Science(), vol 11145. Springer, Cham. https://doi.org/10.1007/978-3-030-00030-1_6
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