Hiroshima Mathematical Journal

Paperfolding sequences, paperfolding curves and local isomorphism

Francis Oger

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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

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

First available in Project Euclid: 30 March 2012

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05B45: Tessellation and tiling problems [See also 52C20, 52C22]
Secondary: 52C20: Tilings in $2$ dimensions [See also 05B45, 51M20] 52C23: Quasicrystals, aperiodic tilings

Paperfolding sequence paperfolding curve tiling local isomorphism aperiodic


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

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