Tilings of the plane and Thurston semi-norm

TL;DR

The paper links plane tiling problems with the existence of zeros of a convex function related to the Thurston semi-norm, providing a new mathematical framework for understanding tilings.

Contribution

It introduces a novel approach connecting plane tilings to convex analysis and branched surfaces, extending Thurston semi-norm concepts to tiling problems.

Findings

Tiling problems reduce to finding zeros of a convex function.

The approach generalizes Thurston semi-norm to plane tilings.

Provides a new mathematical framework for tiling analysis.

Abstract

We show that the problem of tiling the Euclidean plane with a finite set of polygons (up to translation) boils down to prove the existence of zeros of a non-negative convex function defined on a finite-dimensional simplex. This function is a generalisation, in the framework of branched surfaces, of the Thurston semi-norm originally defined for compact $3$-manifolds.