Strict Self-Assembly of Discrete Self-Similar Fractals in the abstract Tile-Assembly Model
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| Format: | Preprint |
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2024
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| _version_ | 1866917799692075008 |
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| author | Becker, Florent Hader, Daniel Patitz, Matthew J. |
| author_facet | Becker, Florent Hader, Daniel Patitz, Matthew J. |
| contents | This paper answers a long-standing open question in tile-assembly theory, namely that it is possible to strictly assemble discrete self-similar fractals (DSSFs) in the abstract Tile-Assembly Model (aTAM). We prove this in 2 separate ways, each taking advantage of a novel set of tools. One of our constructions shows that specializing the notion of a quine, a program which prints its own output, to the language of tile-assembly naturally induces a fractal structure. The other construction introduces self-describing circuits as a means to abstractly represent the information flow through a tile-assembly construction and shows that such circuits may be constructed for a relative of the Sierpinski carpet, and indeed many other DSSFs, through a process of fixed-point iteration. This later result, or more specifically the machinery used in its construction, further enable us to provide a polynomial time procedure for deciding whether any given subset of $\mathbb{Z}^2$ will generate an aTAM producible DSSF. To this end, we also introduce the Tree Pump Theorem, a result analogous to the important Window Movie Lemma, but with requirements on the set of productions rather than on the self-assembling system itself. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_19595 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Strict Self-Assembly of Discrete Self-Similar Fractals in the abstract Tile-Assembly Model Becker, Florent Hader, Daniel Patitz, Matthew J. Computational Geometry This paper answers a long-standing open question in tile-assembly theory, namely that it is possible to strictly assemble discrete self-similar fractals (DSSFs) in the abstract Tile-Assembly Model (aTAM). We prove this in 2 separate ways, each taking advantage of a novel set of tools. One of our constructions shows that specializing the notion of a quine, a program which prints its own output, to the language of tile-assembly naturally induces a fractal structure. The other construction introduces self-describing circuits as a means to abstractly represent the information flow through a tile-assembly construction and shows that such circuits may be constructed for a relative of the Sierpinski carpet, and indeed many other DSSFs, through a process of fixed-point iteration. This later result, or more specifically the machinery used in its construction, further enable us to provide a polynomial time procedure for deciding whether any given subset of $\mathbb{Z}^2$ will generate an aTAM producible DSSF. To this end, we also introduce the Tree Pump Theorem, a result analogous to the important Window Movie Lemma, but with requirements on the set of productions rather than on the self-assembling system itself. |
| title | Strict Self-Assembly of Discrete Self-Similar Fractals in the abstract Tile-Assembly Model |
| topic | Computational Geometry |
| url | https://arxiv.org/abs/2406.19595 |