Theorema Egregium

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.
Cylindrical equal-area projections such as the Behrmann projection instead preserve area but not angles.
Each orange spot is a Tissot's indicatrix showing how identical infinitesimal circles are distorted at each point.

Gauss's Theorema Egregium (Latin for "remarkable theorem") is a major result of differential geometry, proved by Carl Friedrich Gauss in 1827, that concerns the curvature of surfaces. The theorem says that Gaussian curvature can be determined entirely by measuring angles, distances and their rates of change on a surface, without reference to the particular manner in which the surface is embedded in the ambient 3-dimensional Euclidean space. In other words, the Gaussian curvature of a surface does not change if one bends the surface without stretching it. Thus the Gaussian curvature is an intrinsic invariant of a surface.

Gauss presented the theorem in this manner (translated from Latin):

Thus the formula of the preceding article leads itself to the remarkable Theorem. If a curved surface is developed upon any other surface whatever, the measure of curvature in each point remains unchanged.

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.

In modern mathematical terminology, the theorem may be stated as follows:

The Gaussian curvature of a surface is invariant under local isometry.

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