Theorema Egregium

A consequence of the Theorema Egregium is that the Earth cannot be displayed on a map without distortion. The Mercator projection, shown here, preserves angles but fails to preserve area.

Gauss's Theorema Egregium (Latin: "Remarkable Theorem") is a foundational result in differential geometry proved by Carl Friedrich Gauss that concerns the curvature of surfaces. The theorem says that the Gaussian curvature of a surface can be determined entirely by measuring angles, distances and their rates on the surface itself, without further reference to the particular way in which the surface is embedded in the ambient 3-dimensional Euclidean space. Thus the Gaussian curvature is an intrinsic invariant of a surface.

Gauss presented the theorem in 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.

The theorem is "remarkable" because the starting definition of Gaussian curvature makes direct use of position of the surface in space. So it is quite surprising that the result does not depend on its embedding in spite of all bending and twisting deformations undergone.

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

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

Elementary applications

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.

A sphere of radius R has constant Gaussian curvature which is equal to 1/R2. At the same time, a plane has zero Gaussian curvature. As a corollary of Theorema Egregium, a piece of paper cannot be bent onto a sphere without crumpling. Conversely, the surface of a sphere cannot be unfolded onto a flat plane without distorting the distances. If one were to step on an empty egg shell, its edges have to split in expansion before being flattened. Mathematically speaking, a sphere and a plane are not isometric, even locally. This fact is of enormous significance for cartography: it implies that no planar (flat) map of Earth can be perfect, even for a portion of the Earth's surface. Thus every cartographic projection necessarily distorts at least some distances.[1]

The catenoid and the helicoid are two very different-looking surfaces. Nevertheless, each of them can be continuously bent into the other: they are locally isometric. It follows from Theorema Egregium that under this bending the Gaussian curvature at any two corresponding points of the catenoid and helicoid is always the same. 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 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 along 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, as it holds its shape long enough to be consumed without a mess. This same principle is used for strengthening in corrugated materials, most familiarly corrugated fiberboard and corrugated galvanised iron.

Notes

1. ^ Geodetical applications were one of the primary motivations for Gauss's "investigations of the curved surfaces".

References

• Carl Friedrich Gauss (Author), Adam Hiltebeitel (Translator), James Morehead (Translator), General Investigations Of Curved Surfaces Unabridged (Paperback), Wexford College Press, 2007, ISBN 978-1-929148-77-6.
• Carl Friedrich Gauss (Author), Peter Pesic (Editor), General Investigations of Curved Surfaces (Paperback), Dover Publications, 2005, ISBN 978-0-486-44645-5.