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Strict self-assembly of discrete Sierpinski triangles. (English) Zbl 1160.68012
Summary: E. Winfree [Algorithmic self-assembly of DNA. Ph.D. Thesis, California Institute of Technology (1998)] showed that discrete Sierpinski triangles can self-assemble in the Tile Assembly Model. A striking molecular realization of this self-assembly, using DNA tiles a few nanometers long and verifying the results by atomic-force microscopy, was achieved by P. W. K. Rothemund, N. Papadakis, and E. Winfree [“Algorithmic self-assembly of DNA Sierpinski triangles”, PLoS Biology 2(12) (2004)].
Precisely speaking, the above self-assemblies tile completely filled-in, two-dimensional regions of the plane, with labeled subsets of these tiles representing discrete Sierpinski triangles. This paper addresses the more challenging problem of the strict self-assembly of discrete Sierpinski triangles, i.e., the task of tiling a discrete Sierpinski triangle and nothing else.
We first prove that the standard discrete Sierpinski triangle cannot strictly self-assemble in the Tile Assembly Model. We then define the fibered Sierpinski triangle, a discrete Sierpinski triangle with the same fractal dimension as the standard one but with thin fibers that can carry data, and show that the fibered Sierpinski triangle strictly self-assembles in the Tile Assembly Model. In contrast with the simple XOR algorithm of the earlier, non-strict self-assemblies, our strict self-assembly algorithm makes extensive, recursive use of optimal counters, coupled with measured delay and corner-turning operations. We verify our strict self-assembly using the local determinism method of D. Soloveichik and E. Winfree [SIAM J. Comput. 36, 1544–1569 (2007; Zbl 1136.68029).

MSC:
68Q05 Models of computation (Turing machines, etc.) (MSC2010)
52C20 Tilings in \(2\) dimensions (aspects of discrete geometry)
52C45 Combinatorial complexity of geometric structures
28A80 Fractals
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