Why the New York pizza fold works

Pizza has a surface with zero curvature, which means a slice resists bending both vertically and horizontally at the same time (unlike the double curvature of, say, a Pringle chip). Your brain intuits this without any assistance from geometric theory. But the theory exists anyway. In 1827, Carl Friedrich Gauss proved the Theorema Egregium—roughly “Awesome Theorem”—which, extremely simplified, says that you can’t change an object’s curvature and keep its geometry intact. An orange peel has positive curvature, and you can’t flatten it without ripping or stretching the peel. Paper has zero curvature, and you can’t fold it into the shape of an orange. A piece of pizza is like a piece of paper: Fold it horizontally, and it will not droop vertically. You’ve probably also confronted the Awesome Theorem when looking at a map. The famous Mercator Projection reduces our earthly sphere to a flat plane—or, if you roll it up, a cylinder—in which the parallel longitudinal lines are pulled apart near the poles. This has the effect of making Greenland look bigger than Africa. Why is it so hard to design a perfect map? Because of the Awesome Theorem. Just as a pizza slice, with zero curvature, cannot fold and droop at the same time, a planetary sphere, with positive curvature, cannot fully flatten without breaking its own geometric integrity.