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

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 $π/3$ 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 $π/2$ 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^{ω}$ (resp. $2^{ω}$, $2$ 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.