Open Access
Winter 2012 Flippable tilings of constant curvature surfaces
François Fillastre, Jean-Marc Schlenker
Illinois J. Math. 56(4): 1213-1256 (Winter 2012). DOI: 10.1215/ijm/1399395829


We call “flippable tilings” of a constant curvature surface a tiling by “black” and “white” faces, so that each edge is adjacent to two black and two white faces (one of each on each side), the black face is forward on the right side and backward on the left side, and it is possible to “flip” the tiling by pushing all black faces forward on the left-hand side and backward on the right-hand side. Among those tilings, we distinguish the “symmetric” ones, for which the metric on the surface does not change under the flip. We provide some existence statements, and explain how to parameterize the space of those tilings (with a fixed number of black faces) in different ways. For instance, one can glue the white faces only, and obtain a metric with cone singularities which, in the hyperbolic and spherical case, uniquely determines a symmetric tiling. The proofs are based on the geometry of polyhedral surfaces in 3-dimensional spaces modeled either on the sphere or on the anti-de Sitter space.


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François Fillastre. Jean-Marc Schlenker. "Flippable tilings of constant curvature surfaces." Illinois J. Math. 56 (4) 1213 - 1256, Winter 2012.


Published: Winter 2012
First available in Project Euclid: 6 May 2014

zbMATH: 1296.52011
MathSciNet: MR3231480
Digital Object Identifier: 10.1215/ijm/1399395829

Primary: 52B70
Secondary: 52A15 , 53C50

Rights: Copyright © 2012 University of Illinois at Urbana-Champaign

Vol.56 • No. 4 • Winter 2012
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