Mathematics
The Curvature of Surfaces and the Gauss-Bonnet Theorem
Quick fact
A classic result from the Gauss-Bonnet theorem: if you take any smooth closed surface and add up its Gaussian curvature over the entire surface, you always get the same number for a given 'hole' count—specifically, 2π times the Euler characteristic. For a sphere, that's a full 4π, no matter its size or shape.
Why this is interesting
Think of a crumpled paper ball, a doughnut, and a flat sheet. Even if you smooth them out, something about their 'curviness' remains—a hidden number that never changes. What is it?