The program-size complexity of self-assembled squares (extended abstract)
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Combinatorial optimization problems in self-assembly
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
On the Decidability of Self-Assembly of Infinite Ribbons
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
Molecular Assembly and Computation: From Theory to Experimental Demonstrations
ICALP '02 Proceedings of the 29th International Colloquium on Automata, Languages and Programming
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
MFCS'05 Proceedings of the 30th international conference on Mathematical Foundations of Computer Science
Random Number Selection in Self-assembly
UC '09 Proceedings of the 8th International Conference on Unconventional Computation
Self-assembly of discrete self-similar fractals
Natural Computing: an international journal
Approximate self-assembly of the Sierpinski triangle
CiE'10 Proceedings of the Programs, proofs, process and 6th international conference on Computability in Europe
Self-assembly of infinite structures: A survey
Theoretical Computer Science
Limitations of self-assembly at temperature 1
Theoretical Computer Science
Synthesizing minimal tile sets for patterned DNA self-assembly
DNA'10 Proceedings of the 16th international conference on DNA computing and molecular programming
Self-assembly of decidable sets
Natural Computing: an international journal
Self-assembling rulers for approximating generalized sierpinski carpets
COCOON'11 Proceedings of the 17th annual international conference on Computing and combinatorics
Exact shapes and turing universality at temperature 1 with a single negative glue
DNA'11 Proceedings of the 17th international conference on DNA computing and molecular programming
Randomized Self-Assembly for Exact Shapes
SIAM Journal on Computing
The power of nondeterminism in self-assembly
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
An introduction to tile-based self-assembly
UCNC'12 Proceedings of the 11th international conference on Unconventional Computation and Natural Computation
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Winfree (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 Rothemund, Papadakis, and Winfree (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 Soloveichik and Winfree (2007).