Paperfolding sequences, paperfolding curves and local isomorphism

TL;DR

This paper explores the mathematical properties of paperfolding sequences and curves, demonstrating how complete folding curves can be extended to cover the plane in a locally isomorphic, aperiodic tiling pattern with at most six disjoint curves.

Contribution

It introduces a method to extend complete folding curves into plane coverings with local isomorphism, advancing understanding of aperiodic tilings and folding sequences.

Findings

Existence of a standard extension method for complete folding curves

Such coverings can contain at most six disjoint curves

The covering satisfies the local isomorphism property

Abstract

For each integer n, an n-folding curve is obtained by folding n times a strip of paper in two, possibly up or down, and unfolding it with right angles. Generalizing the usual notion of infinite folding curve, we define complete folding curves as the curves without endpoint which are unions of increasing sequences of n-folding curves for n integer. We prove that there exists a standard way to extend any complete folding curve into a covering of the plane by disjoint such curves, which satisfies the local isomorphism property introduced to investigate aperiodic tiling systems. This covering contains at most six curves.