Surfaces of positive curvature (top), negative curvature (middle) and vanishing curvature (bottom) [image credit: NASA].

One of greatest achievements of Carl Friedrich Gauss was a theorem so startling that he gave it the name

*Theorema Egregium *or outstanding theorem. In 1828 he published his “Disquisitiones generales circa superficies curvas”, or

*General investigation of curved surfaces*. Gauss defined a quantity that measures the curvature of a two-dimensional surface. He was inspired by his work on geodesy, surveying and map-making, which involved taking measurements on the surface of the Earth. The total curvature — or Gaussian curvature — depends only on measurements within the surface and Gauss showed that its value is independent of the coordinate system used. This is his

*Theorema Egregium*. The Gaussian curvature

characterizes the intrinsic geometry of a surface.

Continue reading ‘Gaussian Curvature: the *Theorema Egregium*‘