Squaring rectangles for dumbbells

TL;DR

This paper explores the limitations of rectangle-squaring methods, like those used in approximating the Riemann mapping, through the example of a dumbbell-shaped quadrilateral.

Contribution

It provides a detailed analysis of how rectangle-squaring approximations can fall short in representing classical conformal mappings, highlighting their limitations.

Findings

Rectangle-squaring offers a combinatorial approach to the Riemann mapping.

The dumbbell example illustrates specific limitations of this method.

The paper clarifies the gap between combinatorial and classical conformal mappings.

Abstract

The theorem on squaring a rectangle from a tiling of a quadrilateral (Schramm and Cannon-Floyd-Parry) gives a combinatorial version of the Riemann mapping theorem. We elucidate by example (the dumbbell) some of the limitations of rectangle-squaring as an approximation to the classical Riemamnn mapping.