Hiroshima Mathematical Journal

Tilings of a Riemann surface and cubic Pisot numbers

Fumihiko Enomoto, Hiromi Ei, Maki Furukado, and Shunji Ito

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Abstract

Using the reducible algebraic polynomial \(x^{5} - x^{4}-1 = \left( x^{2}-x + 1 \right) \left( x^{3} - x - 1 \right),\) we study two types of tiling substitutions $\tau^*$ and $\sigma^*$: $\tau^*$ generates a tiling of a plane based on five prototiles of polygons, and $\sigma^*$ generates a tiling of a Riemann surface, which consists of two copies of the plane, based on ten prototiles of parallelograms. Finally we claim that $\tau^*$-tiling of $\mathcal{P}$ equals a re-tiling of $\sigma^*$-tiling of $\mathcal{R}$ through the canonical projection of the Riemann surface to the plane.

Article information

Source
Hiroshima Math. J., Volume 37, Number 2 (2007), 181-210.

Dates
First available in Project Euclid: 24 August 2007

Permanent link to this document
https://projecteuclid.org/euclid.hmj/1187916318

Digital Object Identifier
doi:10.32917/hmj/1187916318

Mathematical Reviews number (MathSciNet)
MR2345367

Zentralblatt MATH identifier
1154.53026

Subjects
Primary: 52C23: Quasicrystals, aperiodic tilings 37A45: Relations with number theory and harmonic analysis [See also 11Kxx]
Secondary: 28A80: Fractals [See also 37Fxx]

Keywords
Quasi-periodic tiling of a Riemann surface tiling substitution generated by a reducible cubic Pisot number

Citation

Enomoto, Fumihiko; Ei, Hiromi; Furukado, Maki; Ito, Shunji. Tilings of a Riemann surface and cubic Pisot numbers. Hiroshima Math. J. 37 (2007), no. 2, 181--210. doi:10.32917/hmj/1187916318. https://projecteuclid.org/euclid.hmj/1187916318


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