Tilings of the plane and Thurston semi-norm
  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. 
J.-R. Chazottes, J.-M. Gambaudo, F. Gautero
math.MG math.GT