Search of fractal space-filling curves with minimal dilation

TL;DR

This paper presents an algorithm using SAT-solvers to find extremal fractal space-filling curves with minimal dilation, discovering new curves with lower dilation values in various dimensions.

Contribution

Introduces a novel SAT-based algorithm for searching extremal fractal curves and reports new minimal dilation curves in dimensions 2, 3, and 4.

Findings

Discovered a new self-similar plane curve 'YE' with dilation 5.5890.

Found 'Spring', a bifractal in 3D with dilation less than 17.

Identified a 4D curve with dilation less than 62.

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.