Paperfolding sequences, paperfolding curves and local isomorphism

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 R 2 by disjoint such curves, which satisfies the local isomorphism property introduced to investigate aperiodic tiling systems. This covering contains at most six curves.2000 Mathematics Subject Classification. Primary 05B45; Secondary 52C20, 52C23.

The number of paperfolding curves in a covering of the plane

“…Some results similar to those of [4] are also true for folding sequences and curves in Dekking's sense:…”

“…We use the definitions, the notations and the results of [4]. In order to simplify some notations, we identify R 2 with C and Z 2 with the set Z + iZ of Gaussian integers.…”

“…Many examples of ∞-folding sequences are actually constructed in that way, including the positive folding sequence associated to the dragon curve (see [4,Example 3.13]) and the alternating folding sequence (see [4,Example 3.14]). Some of them are used in the present paper.…”

“…The definition of the sets E n (C) for n ∈ N ∪ {∞} and F n (C) for n ∈ N is given in [4]. Their existence follows from [4,Prop. 3.3].…”

“…Remark. According to [4,Theorem 3.10], the covering D given by Theorem 4 is essentially unique: two such coverings only differ by a translation or/and a change in the connections at the E ∞ point.…”