Theorema Egregium

Gauss presented the theorem in this manner (translated from Latin): The theorem is "remarkable" because the definition of Gaussian curvature makes ample reference to the specific way the surface is embedded in 3-dimensional space, and it is quite surprising that the result does not depend on its embedding.

Thus isometry is simply bending and twisting of a surface without internal crumpling or tearing, in other words without extra tension, compression, or shear.

An application of the theorem is seen when a flat object is somewhat folded or bent along a line, creating rigidity in the perpendicular direction.

This is of practical use in construction, as well as in a common pizza-eating strategy: A flat slice of pizza can be seen as a surface with constant Gaussian curvature 0.

This creates rigidity in the direction perpendicular to the fold, an attribute desirable for eating pizza, as it holds its shape long enough to be consumed without a mess.

A consequence of the Theorema Egregium is that the Earth cannot be displayed on a map without distortion. The Mercator projection preserves angles but fails to preserve area, hence the massive distortion of Antarctica .
Gauss's original statement of the Theorema Egregium, translated from Latin into English.
Animation showing the deformation of a helicoid into a catenoid . The deformation is accomplished by bending without stretching. During the process, the Gaussian curvature of the surface at each point remains constant.