## Hiroshima Mathematical Journal

- Hiroshima Math. J.
- Volume 42, Number 1 (2012), 37-75.

### 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.

#### Article information

**Source**

Hiroshima Math. J., Volume 42, Number 1 (2012), 37-75.

**Dates**

First available in Project Euclid: 30 March 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.hmj/1333113006

**Digital Object Identifier**

doi:10.32917/hmj/1333113006

**Mathematical Reviews number (MathSciNet)**

MR2952072

**Zentralblatt MATH identifier**

1309.52009

**Subjects**

Primary: 05B45: Tessellation and tiling problems [See also 52C20, 52C22]

Secondary: 52C20: Tilings in $2$ dimensions [See also 05B45, 51M20] 52C23: Quasicrystals, aperiodic tilings

**Keywords**

Paperfolding sequence paperfolding curve tiling local isomorphism aperiodic

#### Citation

Oger, Francis. Paperfolding sequences, paperfolding curves and local isomorphism. Hiroshima Math. J. 42 (2012), no. 1, 37--75. doi:10.32917/hmj/1333113006. https://projecteuclid.org/euclid.hmj/1333113006