Self-avoiding and plane-filling properties for terdragons and other triangular folding curves

TL;DR

This paper studies n-folding triangular curves, including the well-known terdragon, showing they can form unique plane coverings with self-avoiding, aperiodic properties, and classifies these coverings based on folding sequences.

Contribution

It introduces the concept of complete folding t-curves and proves their unique extension into plane coverings, classifying these coverings by local isomorphism and folding sequences.

Findings

Each complete folding t-curve can be extended to a unique plane covering.

Coverings are classified by their associated folding sequences.

The number of isomorphism classes varies with the number of curves in the covering.

Abstract

We consider $n$-folding triangular curves, or $n$-folding t-curves, obtained by folding $n$ times a strip of paper in $3$, each time possibly left then right or right then left, and unfolding it with $\pi /3$ angles. An example is the well known terdragon curve. They are self-avoiding like $n$-folding curves obtained by folding $n$ times a strip of paper in two, each time possibly left or right, and unfolding it with $\pi /2$ angles. We also consider complete folding t-curves, which are the curves without endpoint obtained as inductive limits of $n$-folding t-curves. We show that each of them can be extended into a unique covering of the plane by disjoint such curves, and this covering satisfies the local isomorphism property introduced to investigate aperiodic tiling systems. Two coverings are locally isomorphic if and only if they are associated to the same sequence of foldings. Each class of locally isomorphic coverings contains exactly $ 2^{\omega }$ (resp. $2^{\omega }$, $2$ or $5$, $0$) isomorphism classes of coverings by $1$ (resp. $2$, $3$, $\geq 4$) curves. These properties are partly similar to those of complete folding curves.