From: jrl16@po.cwru.edu (Yossi Lonke)
Subject: Gauss's Wonderful Theorem
Date: 4 Jun 1999 08:25:44 -0400
Newsgroups: sci.math
Keywords: Theorema egregium

Hi All,

A famous theorem due to Gauss,
states that the Gaussian curvature
of a surface remains invariant under
surface-isometries. The Gaussian
curvature is computed by finding at
each point the principal curvatures, and
then multiplying all of them together.
The sphere has a constant positive Gaussian
curvature. A plane- clearly has zero
curvature. Also a cylinder has zero
Gaussian curvature, because one of
its principal curvatures (at every point)
is always zero. That's why it is possible
to put any plane figure on a cylinder,
but it is impossible to lay a plane-figure
on a sphere, becuase the Gaussian
curvatures do not agree. In particular,
it is impossible to lay a plane triangle on a sphere.

Gauss was so pleased with his theorem, that he named it
"Theorema Egregium", which translates from Latin to : The wonderful
theorem. And indeed it is, because
prior to that, there was no single quantity attached to surfaces, that
was
known to remain invariant under isoemtries.


Best Regards,

Dr. Yossi Lonke
Dep of Math
Case Western Reserve University
Cleveland, Ohio 44106