Concrete Mathematics: A Foundation for Computer Science
Concrete Mathematics: A Foundation for Computer Science
Algorithmic self-assembly of dna
Algorithmic self-assembly of dna
Theory and experiments in algorithmic self-assembly
Theory and experiments in algorithmic self-assembly
Complexity of Self-Assembled Shapes
SIAM Journal on Computing
Strict self-assembly of discrete Sierpinski triangles
Theoretical Computer Science
Self-assembly of the Discrete Sierpinski Carpet and Related Fractals
DNA Computing and Molecular Programming
Self-assembly of discrete self-similar fractals
Natural Computing: an international journal
MFCS'05 Proceedings of the 30th international conference on Mathematical Foundations of Computer Science
Self-assembly of infinite structures: A survey
Theoretical Computer Science
Self-assembling rulers for approximating generalized sierpinski carpets
COCOON'11 Proceedings of the 17th annual international conference on Computing and combinatorics
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Winfree introduced the Tile Assembly Model in order to study the nanoscale self-assembly of DNA crystals. Lathrop, Lutz, and Summers proved that the Sierpinski triangle S cannot self-assemble in the "strict" sense in which tiles are not allowed to appear at positions outside the target structure. Here we investigate the strict self-assembly of sets that approximate S. We show that every set that does strictly self-assemble disagrees with S on a set with fractal dimension at least that of S (≈ 1.585), and that no subset of S with fractal dimension greater than 1 strictly self-assembles. We show that our bounds are tight by presenting a strict self-assembly that adds communication fibers to the fractal structure without disturbing it. To verify this strict self-assembly we develop a generalization of the local determinism method of Soloveichik and Winfree to tile assembly systems that use a blocking technique.