[0,1]$, respectively, under the parametrizing homeomorphism from $[0,1]^2$. We write $\Gamma_1(Q)$ to denote the family $\Gamma(\zeta_1,\zeta_3; Q)$ of curves in $Q$ connecting $\zeta_1$ and $\zeta_3$, and $\Gamma_2(Q)$ to denote the family $\Gamma(\zeta_2,\zeta_4; Q)$ of curves in $Q$ connecting $\zeta_2$ and $\zeta_4$. More generally, for disjoint closed sets $E,F$ contained in the set $G \subset X$, the notation $\Gamma(E,F;G)$ is used to denote the family of curves in $G$ which intersect both $E$ and $F$.

\[defi:reciprocality\] The metric surface $(X, d)$ is [*reciprocal*]{} if there exists $\kappa \geq 1$ such that for all quadrilaterals $Q$ in $X$, $$\label{equ:reciprocality(1)}
        \operatorname{mod}\Gamma_1(Q) \operatorname{mod}\Gamma_2(Q) \leq \kappa$$ and $$\label{equ:reciprocality(2)}
        \operatorname{mod}\Gamma_1(Q) \operatorname{mod}\Gamma_2(Q) \geq 1/\kappa,$$ and for all $x \in X$ and $R>0$ such that $X \setminus B(x,R) \neq \emptyset$, $$\label{equ:reciprocality(3)} 
        \lim_{r \rightarrow 0} \operatorname{mod}\Gamma(B(x,r), X \setminus B(x,R); B(x,R)) = 0.$$

We then have the following result.

\[thm:uniformization\] There exists a domain $\Omega \subset \mathbb{R}^2$ and a quasiconformal mapping $f: (X,d) \rightarrow \Omega$ if and only if $X$ is reciprocal.

The necessity of each condition in Definition \[defi:reciprocality\] is immediate; standard computations show that $\mathbb{R}^2$ is reciprocal. The actual content of Theorem \[thm:uniformization\] is that these conditions are sufficient to construct “by hand” a mapping that can then 