2007/02/13 by Samuel Lelièvre, Lelièvre, Samuel, Robert Silhol +1
Materials Science · Mathematics · #14H15 #30F10 #30F30 #30F45 #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Quasicrystal Structures and Properties #math.GT #msc:14H15 #msc:30F10 #msc:30F30 #msc:30F45
paper · pdf · doi:10.48550/arxiv.math/0702374
35 pages, TeX, 18 figures
arxiv created 2007/02/13 · openalex publication_date 2007/02/13 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Identifying parallel sides of a collection of Euclidean polygons yields a flat surface with cone points of angles multiples of 2 pi, naturally a compact Riemann surface but also an algebraic curve, and a hyperbolic surface. In general two different metrics on a surface have no geodesic arcs in common, but in special cases the surface is decomposed into polygons geodesic for both the flat and the hyperbolic metric. This is the case for certain surfaces which are translation and half-turn tiled by an Euclidean rectangle. We explore them in this paper. Their multi-geodesic tessellation provides a mechanical way to reconstruct a Fuchsian group for them; allows to describe their Teichmueller disk in terms of Fenchel-Nielsen coordinates; allows for an interpretation in terms of fractional Dehn twists of the natural PSL2(Z) action on the PSL2(R)-orbit of such surfaces. In many cases the tiling by rectangles allows to recover an equation for the corresponding algebraic curve, providing a bridge between the algebraic equation and the hyperbolic structure deduced from the multi-geodesic tessellation; in other words solving the uniformization problem for such curves. In fact it also gives a scheme to do uniformization for infinitely many families of curves. We also discuss some number theoretic aspects.