You may never look at a pizza or an orange in the same way again. Watch as astronomer, author, and teacher Cliff Stoll explains the Theorema Egregium or the Remarkable Theorem in this Numberphile vid. More via Aatish Bhatia:
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.
Bonus: Pizza physics (New York-style).