A map is called conformal when it preserves angles. Consider a sequence of simple closed curves Ct in the plane that get squished to a path. That is, Ct consists of a pair of curves Et and Wt that converge to the same curve in Fréchet distance. By the Riemann Mapping Theorem, there is a conformal map from the interior of Ct to the upper half-plane. Preserving angles comes at the cost of distorting scale, and as Ct gets squished, this map becomes highly distorted. I will show in this talk that, for appropriate choice of conformal map, the image of any arbitrarily small neighborhood of a single point covers the entire upper half-plane in the limit.