Search of fractal space-filling curves with minimal dilation

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 $\gamma \text{\hspace{0.17em}⁣}:[0,1]\to [0,1{]}^d$ with minimal dilation $$\underset{t_1<t_2}{\sup }\frac{\mathrm{\parallel }\gamma (t_2)-\gamma (t_1){\mathrm{\parallel }}^d}{t_2-t_1}\mathrm{.}$$ 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 $5\frac{43}{73}=5.5890\dots$. In dimension $3$ we have found facet-gated bifractals (that we call “Spring”) of genus $2\times 2\times 2$ 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.