Illinois Journal of Mathematics

Flippable tilings of constant curvature surfaces

François Fillastre and Jean-Marc Schlenker

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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.

Article information

Illinois J. Math., Volume 56, Number 4 (2012), 1213-1256.

First available in Project Euclid: 6 May 2014

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Zentralblatt MATH identifier

Primary: 52B70: Polyhedral manifolds
Secondary: 52A15: Convex sets in 3 dimensions (including convex surfaces) [See also 53A05, 53C45] 53C50: Lorentz manifolds, manifolds with indefinite metrics


Fillastre, François; Schlenker, Jean-Marc. Flippable tilings of constant curvature surfaces. Illinois J. Math. 56 (2012), no. 4, 1213--1256. doi:10.1215/ijm/1399395829.

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