Orthodiagonal Maps, Tilings of Rectangles, and their Convergence to Conformal Maps

TL;DR

This paper proves that discrete orthodiagonal maps approximating a simply connected domain with four boundary points converge uniformly to the conformal map of that domain onto a rectangle, generalizing classical tiling results.

Contribution

It establishes the convergence of orthodiagonal map-based rectangle tilings to the conformal map for domains with four boundary points, providing a discrete-to-continuous approximation framework.

Findings

Orthodiagonal maps approximate conformal maps of simply connected domains.

Rectangle tilings derived from these maps converge uniformly to the conformal map.

The results generalize classical tiling and network-based conformal mapping theories.

Abstract

A classic result of Brooks, Smith, Stone and Tutte associates to any finite planar network with distinguished source and sink vertices, a tiling of a rectangle by smaller subrectangles whose aspect ratios are given by the conductances of corresponding edges in the network. This tiling can be viewed as a discrete analogue of the uniformizing conformal map that maps a simply connected domain with four distinguished prime ends to a rectangle, so that the four prime ends are mapped to the four corners of the rectangle. \\ \\ We make this intuition precise by showing that if $\Omega$ is a simply connected domain with four distinguished prime ends $A,B,C,D$ in counterclockwise order and $(\Omega_{n})_{n\geq{1}}$ is a sequence of orthodiagonal maps with distinguished boundary vertices $A_{n}, B_{n}, C_{n}, D_{n}$ in counterclockwise order, that are finer and finer approximations of $\Omega$ with its distinguished boundary points $A,B,C,D$, then the corresponding ``rectangle tiling maps" converge uniformly on compacts to the aforementioned conformal map on $\Omega$.