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