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 $Ømega$ into the unit disc $D$
converges to a Riemann map from $Ømega$ to $D$ 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.
%0 Generic
%1 georgakopoulos2019convergence
%A Georgakopoulos, Agelos
%A Panagiotis, Christoforos
%D 2019
%K complex_analysis conformal_mappings graphs tilings
%T Convergence of square tilings to the Riemann map
%U http://arxiv.org/abs/1910.06886
%X 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 $Ømega$ into the unit disc $D$
converges to a Riemann map from $Ømega$ to $D$ 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.
@preprint{georgakopoulos2019convergence,
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.},
added-at = {2021-05-11T22:27:50.000+0200},
author = {Georgakopoulos, Agelos and Panagiotis, Christoforos},
biburl = {https://www.bibsonomy.org/bibtex/22fa9ff9e11349f9abb072ff882dc43b3/lukas.geyer},
description = {Convergence of square tilings to the Riemann map},
interhash = {55f110a6a8dee95ba8545b724af7a917},
intrahash = {2fa9ff9e11349f9abb072ff882dc43b3},
keywords = {complex_analysis conformal_mappings graphs tilings},
note = {cite arxiv:1910.06886},
timestamp = {2021-05-11T22:27:50.000+0200},
title = {Convergence of square tilings to the Riemann map},
url = {http://arxiv.org/abs/1910.06886},
year = 2019
}