Plane-filling curves on all uniform grids

TL;DR

This paper explores plane-filling curves on various uniform grids, identifying all such curves for small orders and demonstrating how to derive edge-traversing curves from point-traversing ones.

Contribution

It systematically finds all small-order plane-filling curves on three grids and shows how to generate edge-traversing curves for all uniform grids.

Findings

All small-order curves on three grids have been identified.

Methods to derive edge-traversing curves from point-traversing ones are provided.

Explicit curves traversing all edges are described for four uniform grids.

Abstract

We describe a search for plane-filling curves traversing all edges of a grid once. The curves are given by Lindenmayer systems with only one non-constant letter. All such curves for small orders on three grids have been found. For all uniform grids we show how curves traversing all points once can be obtained from the curves found. Curves traversing all edges once are described for the four uniform grids where they exist.