Riemann mapping theorem

The Riemann Mapping Theorem states that every simply connected, open, and non-empty region in the plane (except for the entire plane itself) can be transformed into a standard shape—specifically, a unit disk—through a conformal (angle-preserving) map. In essence, no matter the complex, irregular shape of such a region, it can be "stretched" and "compressed" without tearing or overlapping so that it becomes a perfect circle, making analysis and understanding of its properties much simpler.