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Search of Fractal Space-Filling Curves with Minimal Dilation

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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 \(\gamma :[0,1]\rightarrow [0,1]^d\) with minimal dilation

$$\begin{aligned} \sup _{t_1<t_2}\frac{\Vert \gamma (t_2)-\gamma (t_1)\Vert ^d}{t_2-t_1}. \end{aligned}$$

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 \(5+{43}/{73}=5.5890\ldots \)  In dimension 3 we have found facet-gated bifractals (which 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 proven.

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Data availability

The datasets generated during and/or analysed during the current study are available in the GitHub repository at https://github.com/malykhin-yuri/peano.

Notes

  1. https://github.com/malykhin-yuri/peano

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Acknowledgements

The authors express their gratitude to the anonymous referee for his careful work and valuable suggestions.

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  1. Steklov Mathematical Institute, Moscow, Russia

    Yuri Malykhin & Evgeny Shchepin

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  1. Yuri Malykhin
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  2. Evgeny Shchepin
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Correspondence to Yuri Malykhin.

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Malykhin, Y., Shchepin, E. Search of Fractal Space-Filling Curves with Minimal Dilation. Discrete Comput Geom 70, 189–213 (2023). https://doi.org/10.1007/s00454-022-00444-2

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