The number of paperfolding curves in a covering of the plane

Abstract

This paper completes our previous one in the same journal (vol. 42, pp. 37– 75). Let $\mathscr{C}$ be a covering of the plane by disjoint complete folding curves which satisfies the local isomorphism property. We show that $\mathscr{C}$ is locally isomorphic to an essentially unique covering generated by an $\infty$-folding curve. We prove that $\mathscr{C}$ necessarily consists of 1, 2, 3, 4 or 6 curves. We give examples for each case; the last one is realized if and only if $\mathscr{C}$ is generated by the alternating folding curve or one of its successive antiderivatives. We also extend the results of our previous paper to another class of paperfolding curves introduced by M. Dekking.