{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/search-of-fractal-space-filling-curves-with","title":"Search of fractal space-filling curves with minimal dilation","arxiv_id":"2103.07344","date":"2021-03-12","proceeding":null,"authors":["Yuri Malykhin","Evgeny Shchepin"],"abstract":"We introduce an algorithm for a search of extremal fractal curves in large curve classes. It heavily uses SAT-solvers~ -- heuristic algorithms that find models for CNF boolean formulas. Our algorithm was implemented and applied to the search of fractal surjective curves $\\gamma\\colon[0,1]\\to[0,1]^d$ with minimal dilation $$ \\sup_{t_1<t_2}\\frac{\\|\\gamma(t_2)-\\gamma(t_1)\\|^d}{t_2-t_1}. $$ We report new results of that search in the case of Euclidean norm. We have found a new curve that we call \"YE\", a self-similar (monofractal) plane curve of genus $5\\times 5$ with dilation $5\\frac{43}{73}=5.5890\\ldots$. In dimension $3$ we have found facet-gated bifractals (that we call \"Spring\") of genus $2\\times2\\times 2$ with dilation $<17$. In dimension $4$ we obtained that there is a curve with dilation $<62$. Some lower bounds on the dilation for wider classes of cubically decomposable curves are proved.","url_abs":"https://arxiv.org/abs/2103.07344v2","url_pdf":"https://arxiv.org/pdf/2103.07344v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"search-of-fractal-space-filling-curves-with","repo_url":"https://github.com/malykhin-yuri/peano","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}