The number of paperfolding curves in a covering of the plane

TL;DR

This paper classifies plane coverings by disjoint folding curves satisfying local isomorphism, showing they are generated by an infinite-folding curve and can consist of 1, 2, 3, 4, or 6 curves, with specific examples provided.

Contribution

It establishes a classification of such coverings, proves their local isomorphism to a unique generating curve, and extends previous results to new classes of paperfolding curves.

Findings

Coverings are generated by an infinite-folding curve.

Possible numbers of curves in the covering are 1, 2, 3, 4, or 6.

Explicit examples are provided for each case.

Abstract

These results complete our paper in Hiroshima Mathematical Journal, vol. 42, pp. 37-75. Let C be a covering of the plane by disjoint complete folding curves which satisfies the local isomorphism property. We show that C is locally isomorphic to an essentially unique covering generated by an $\infty$-folding curve. We prove that 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 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.