Convergence of square tilings to the Riemann map

TL;DR

This paper proves that square tilings of planar domains converge to the Riemann map, extending a known circle packing convergence theorem to square tilings, thus advancing geometric function theory.

Contribution

It establishes the convergence of square tilings to the Riemann map, generalizing the circle packing convergence theorem to a new class of tilings.

Findings

Square tilings converge to the Riemann map as mesh size approaches zero.

The convergence result parallels the circle packing case.

Provides a new geometric approach to conformal mapping approximation.

Abstract

A well-known theorem of Rodin \& Sullivan, previously conjectured by Thurston, states that the circle packing of the intersection of a lattice with a simply connected planar domain $\Omega$ into the unit disc $\mathbb{D}$ converges to a Riemann map from $\Omega$ to $\mathbb{D}$ when the mesh size converges to 0. We prove the analogous statement when circle packings are replaced by the square tilings of Brooks et al.