### Paperfolding sequences, paperfolding curves and local isomorphism

Francis Oger
Source: Hiroshima Math. J. Volume 42, Number 1 (2012), 37-75.

#### 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.

First Page:
Primary Subjects: 05B45
Secondary Subjects: 52C20, 52C23
Full-text: Open access