I came across this article on Gaussian curvature recently, which was written excellently. The author has the knack of explaining things as well as anyone.
Here’s the link to the same:
Here’s the Money quote.
A surprising consequence of this result is that you can take a surface and bend it any way you like, so long as you don’t stretch, shrink or tear it, and the Gaussian curvature stays the same. That’s because bending doesn’t change any distances on the surface, and so the ant living on the surface would still calculate the same Gaussian curvature as before.
This might sound a little abstract, but it has real-life consequences. Cut an orange in half, eat the insides (yum), then place the dome-shaped peel on the ground and stomp on it. The peel will never flatten out into a circle. Instead, it’ll tear itself apart. That’s because a sphere and a flat surface have different Gaussian curvatures, so there’s no way to flatten a sphere without distorting or tearing it. Ever tried gift wrapping a basketball? Same problem. No matter how you bend a sheet of paper, it’ll always retain a trace of its original flatness, so you end up with a crinkled mess.
What does any of this have to do with pizza? Well, the pizza slice was flat before you picked it up (in math speak, it has zero Gaussian curvature). Gauss’s remarkable theorem assures us that one direction of the slice must always remain flat — no matter how you bend it, the pizza must retain a trace of its original flatness. When the slice flops over, the flat direction (shown in red below) is pointed sideways, which isn’t helpful for eating it. But by folding the pizza slice sideways, you’re forcing it to become flat in the other direction – the one that points towards your mouth. Theorema egregium, indeed.
Must read. Highly recommended.