Flippable tilings of constant curvature surfaces

TL;DR

This paper introduces and analyzes flippable tilings of constant curvature surfaces, exploring their properties, existence, and parameterizations using polyhedral surface geometry in spherical and anti-de Sitter spaces.

Contribution

It defines flippable tilings, distinguishes symmetric ones, and provides methods to parameterize and construct these tilings via geometric and metric approaches.

Findings

Existence of flippable tilings established.

Parameterizations of tilings via white face gluings.

Unique determination of symmetric tilings in hyperbolic and spherical cases.

Abstract

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 side and backward on the right 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.