Tilings of a Riemann surface and cubic Pisot numbers

Using the reducible algebraic polynomial x x 1 1⁄4 ðx xþ 1Þ ðx x 1Þ, we study two types of tiling substitutions t and s : t generates a tiling of a plane based on five prototiles of polygons, and s 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 t -tiling of P equals a re-tiling of s -tiling of R through the canonical projection of the Riemann surface to the plane.