Theorema Egregium

From Academic Kids

The Theorema Egregium ('Remarkable Theorem') is an important theorem of Carl Friedrich Gauss concerning the curvature of surfaces. Informally, the theorem says that the curvature of a surface can be determined entirely by measuring angles and distances on the surface, that is, it does not depend on how the surface might be embedded in (3-dimensional) space.

Gauss presented the theorem this way (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.

In more modern language the theorem may be stated this way:

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

The theorem is remarkable because the definition of Gaussian curvature makes direct use of the embedding of the surface in space. So it is quite surprising that the end result does not depend on the embedding.

Some simple applications

You can't bend a piece of paper onto a sphere (more formally, the plane and the 2-sphere are not locally isometric). This follows immediately from the fact that the plane has Gaussian curvature 0 (at all points) while no point on a sphere has Gaussian curvature 0. (It is, however, possible to prove this special case more directly.)

A somewhat whimsical application of the Theorema Egregium is seen in a common pizza-eating strategy: A slice of pizza can be seen as a surface with constant Gaussian curvature 0. Gently bending a slice must then roughly maintain this curvature (assuming the bend is roughly a local isometry). If one bends a slice horizontally across a radius, non-zero principal curvatures are created along the bend, dictating that the other principal curvature at these points must be zero. This creates rigidity in the direction perpendicular to the fold, an attribute desirable when eating pizza (since it prevents stuff from falling off and making a mess).

Corresponding points on the catenoid and the helicoid (two very different-looking surfaces) have the same Gaussian curvature. (The two surfaces are locally isometric.)

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