Gaussian curvature and Gauss’s “Remarkable Theorem”

Gaussian curvature is a two-dimensional surface’s intrinsic measure of how it is curved.

A bug living inside a one-dimensional curve, the sort of curve you draw when you draw two-dimensional graphs, cannot tell if it is curved or not; all the bug can do is walk forward and backward, measuring distance.

But an intelligent bug living on a surface can, by walking around, never leaving the surface, measure its Gaussian curvature. And so if the surface is bent in three dimensions without stretching it, the Gaussian curvature remains invariant (the same).

Carl Friedrich Gauss, who proved many great theorems in his life, gave this invariance result the special name “the Remarkable Theorem”.

Gaussian curvature turns out to be the product of the eigenvalues, and thus the determinant, of a matrix defined by the surface. It is also connected to Euler characteristics.

Here’s a simple explanation

and a slightly more worked-out one

and here’s how the theorem can be demonstrated using pizza.