Difference between revisions of "Self-Assembly of Discrete Self-Similar Fractals"
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|abstract=In this paper, we search for theoretical limitations of the Tile Assembly Model ([[Tile Assembly Model|TAM]]), along with | |abstract=In this paper, we search for theoretical limitations of the Tile Assembly Model ([[Tile Assembly Model|TAM]]), along with | ||
techniques to work around such limitations. Specifically, we investigate the self-assembly of fractal shapes | techniques to work around such limitations. Specifically, we investigate the self-assembly of fractal shapes | ||
| − | in the TAM. We prove that no self-similar fractal weakly self-assembles at temperature 1 in a locally | + | in the [[Tile Assembly Model|TAM]]. We prove that no self-similar fractal weakly self-assembles at temperature 1 in a locally |
deterministic tile assembly system, and that certain kinds of discrete self-similar fractals do not strictly | deterministic tile assembly system, and that certain kinds of discrete self-similar fractals do not strictly | ||
self-assemble at any temperature. Additionally, we extend the fiber construction of Lathrop, Lutz and | self-assemble at any temperature. Additionally, we extend the fiber construction of Lathrop, Lutz and | ||
Summers (2007) to show that any discrete self-similar fractal belonging to a particular class of "nice" | Summers (2007) to show that any discrete self-similar fractal belonging to a particular class of "nice" | ||
| − | discrete self-similar fractals has a fibered version that strictly self-assembles in the TAM. | + | discrete self-similar fractals has a fibered version that strictly self-assembles in the [[Tile Assembly Model|TAM]]. |
|authors=Matthew J. Patitz and Scott M. Summers | |authors=Matthew J. Patitz and Scott M. Summers | ||
|date=2008/03/12 | |date=2008/03/12 | ||
|file=[[media:SADSSF journal.pdf]] | |file=[[media:SADSSF journal.pdf]] | ||
}} | }} | ||
Revision as of 15:38, 1 March 2012
Published on: 2008/03/12
Abstract
In this paper, we search for theoretical limitations of the Tile Assembly Model (TAM), along with techniques to work around such limitations. Specifically, we investigate the self-assembly of fractal shapes in the TAM. We prove that no self-similar fractal weakly self-assembles at temperature 1 in a locally deterministic tile assembly system, and that certain kinds of discrete self-similar fractals do not strictly self-assemble at any temperature. Additionally, we extend the fiber construction of Lathrop, Lutz and Summers (2007) to show that any discrete self-similar fractal belonging to a particular class of "nice" discrete self-similar fractals has a fibered version that strictly self-assembles in the TAM.
Authors
Matthew J. Patitz and Scott M. Summers
