# Tilings of a Riemann surface and cubic Pisot numbers

## 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.

## Journal

• Hiroshima mathematical journal

Hiroshima mathematical journal 37(2), 181-210, 2007-07

Hiroshima University

## Codes

• NII Article ID (NAID)
110006311484
• NII NACSIS-CAT ID (NCID)
AA00664323
• Text Lang
ENG
• ISSN
00182079
• Data Source
NII-ELS

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