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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\n                    <jats:italic>quine<\/jats:italic>\n                    , a program which prints its own output, to the language of tile-assembly naturally induces a fractal structure. The other construction introduces\n                    <jats:italic>self-describing circuits<\/jats:italic>\n                    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\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\mathbb {Z}^2$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    will generate an aTAM producible DSSF. To this end, we also introduce the\n                    <jats:italic>Tree Pump Theorem<\/jats:italic>\n                    , a result analogous to the important\n                    <jats:italic>Window Movie Lemma<\/jats:italic>\n                    , but with requirements on the set of productions rather than on the self-assembling system itself. This paper is an extension of a version that appeared in the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA\u201925).\n                  <\/jats:p>","DOI":"10.1007\/s00453-025-01340-w","type":"journal-article","created":{"date-parts":[[2026,1,10]],"date-time":"2026-01-10T05:32:00Z","timestamp":1768023120000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Strict Self-Assembly of Discrete Self-Similar Fractals in the Abstract Tile Assembly Model"],"prefix":"10.1007","volume":"88","author":[{"given":"Florent","family":"Becker","sequence":"first","affiliation":[]},{"given":"Daniel","family":"Hader","sequence":"additional","affiliation":[]},{"given":"Matthew","family":"Patitz","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2026,1,10]]},"reference":[{"issue":"7748","key":"1340_CR1","doi-asserted-by":"publisher","first-page":"366","DOI":"10.1038\/s41586-019-1014-9","volume":"567","author":"D Woods","year":"2019","unstructured":"Woods, D., Doty, D., Myhrvold, C., Hui, J., Zhou, F., Yin, P., Winfree, E.: Diverse and robust molecular algorithms using reprogrammable dna self-assembly. 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