The program-size complexity of self-assembled squares (extended abstract)
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Running time and program size for self-assembled squares
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Concrete Mathematics: A Foundation for Computer Science
Concrete Mathematics: A Foundation for Computer Science
The Lucas property of a number array
Discrete Mathematics
A route to fractal DNA-assembly
Natural Computing: an international journal
Coding and geometrical shapes innanostructures: A fractal DNA-assembly
Natural Computing: an international journal
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
Approximate self-assembly of the Sierpinski triangle
CiE'10 Proceedings of the Programs, proofs, process and 6th international conference on Computability in Europe
MFCS'05 Proceedings of the 30th international conference on Mathematical Foundations of Computer Science
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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Discrete self-similar fractals have been studied as test cases for self-assembly ever since Winfree exhibited a tile assembly system in which the Sierpinski triangle self-assembles. For strict self-assembly, where tiles are not allowed to be placed outside the target structure, it is an open question whether any self-similar fractal can self-assemble. This has motivated the development of techniques to approximate fractals with strict self-assembly. Ideally, such an approximation would produce a structure with the same fractal dimension as the intended fractal and with specially labeled tiles at positions corresponding to points in the fractal. We show that the Sierpinski carpet, along with an infinite class of related fractals, can approximately self-assemble in this manner. Our construction takes a set of parameters specifying a target fractal and creates a tile assembly system in which the fractal approximately selfassembles. This construction introduces rulers and readers to control the self-assembly of a fractal structure without distorting it. To verify the fractal dimension of the resulting assemblies, we prove a result on the dimension of sets embedded into discrete fractals. We also give a conjecture on the limitations of approximating self-similar fractals.