Search of fractal space-filling curves with minimal dilation
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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 γ[0,1]→[0,1]^d with minimal dilation sup_t_1<t_2γ(t_2)-γ(t_1)^d/t_2-t_1. We report new results of that search in the case of Euclidean norm. One of the results is a new curve that we call "YE", a self-similar (monofractal) plain curve of genus 5× 5 with dilation 543/73=5.5890…, which is best-known among plain monofractals. Moreover, the YE-curve is the unique minimal curve among monofractals of genus ≤ 6× 6. We give a proof of minimality, which relies both on computations and theoretical results. We notice that the classes of facet-gated multifractals are rigid enough to allow an efficient search, and contain many curves with small dilation. In dimension 3 we have found facet-gated bifractals (that we call "Spring") of genus 2×2× 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.
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