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Flippable tilings of constant curvature surfaces. (English) Zbl 1296.52011

The authors introduce very special tilings of surfaces which they call flippable tilings. Consider tilings of a constant curvature surface by ‘black’ and ‘white’ faces which are convex polygons. Any oriented edge \(e\) of the tiling is a segment adjacent to a black face and a white face on its right side and similarily on its left side. The lengths of the intersections with \(e\) of the two black faces (respectively the two white faces) are equal. In a right flippable tiling the black face is forward on the right-hand side of \(e\) and backward on the left-hand side. In a left flippable tiling, for each edge \(e\), the black face is forward on the left-hand side of \(e\) and backward on the right-hand side.
For a right flippable tiling the operation called a ‘flip’ consists in pushing all black faces forward on the left-hand side and backward on the right-hand side and yields a left flippable tiling. Similarily a flip of a left flippable tiling is defined. Thus pairs of tilings of a surface, where one tiling is obtained from the other by a flip, are objects under study.
The authors prove some existence theorems for the sphere and hyperbolic surfaces. The space of flippable tilings is studied globally both in the spherical and hyperbolic cases. Also more specific results are obtained on the parametrization of the space of flippable tilings for which the areas of black faces are fixed.
Proofs of the results are based on the geometry of polyhedral surfaces in 3-dimensional spaces modelled either on the sphere or on the anti-de Sitter space.
One of the authors’ motivations for this work is the investigation of a polyhedral version of earthquakes.

MSC:

52B70 Polyhedral manifolds
52A15 Convex sets in \(3\) dimensions (including convex surfaces)
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
86A17 Global dynamics, earthquake problems (MSC2010)
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References:

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