Planar Substitutions to Lebesgue type Space-Filling Curves and Relatively Dense Fractal-like Sets in the Plane

TL;DR

This paper generalizes Lebesgue's space-filling curve using planar substitutions, creating new curves and dense fractal-like sets in the plane, expanding understanding of space-filling structures.

Contribution

It introduces a method to generate space-filling curves from any planar substitution satisfying a mild condition, and explores their fractal-like dense sets.

Findings

Generated new space-filling curves from known substitutions.

Identified conditions for dense fractal-like sets in the plane.

Extended Lebesgue's construction to a broader class of curves.

Abstract

Lebesgue curve is a space-filling curve that fills the unit square through linear interpolation. In this study, we generalise Lebesgue's construction to generate space-filling curves from any given planar substitution satisfying a mild condition. The generated space-filling curves for some known substitutions are elucidated. Some of those substitutions further induce relatively dense fractal-like sets in the plane, whenever some additional assumptions are met.