Experiments in the Theory of Surfaces

The purpose of this article is to show the advantage of using Mathematica in the theory of surfaces. The examples in the classical differential geometry, namely the theory of curves and surfaces, have been confined to a small group of calculable objects, because of the difficulty in evaluating geometrical quantities and solving differential equations explicitly. But Mathematica has made it possible to deal with a wide range of objects and to perform experimental treatments of them, based on the power of numerical calculation--the NDSolve and NIntegrate commands in particular--and graphical visualization by the ParametricPlot command. I would like to introduce here several animations and figures that I use in my differential geometry class for undergraduate students. First of all, these graphics help students understand the basic notions of differential geometry. Secondly, we can experiment on geometry through these graphics. This article is one of the serial talks given by the author at 1995 Developer Conference, IMS '97 in Rovaniemi, and WMC 98 in Chicago. They are all targeted for the experimental usage of Mathematica on differential geometry. This time the topics are focused on the surfaces in the three-dimensional Euclidean space. The commands are contained in six notebooks. If the commands in other notebooks cause any trouble, please restart Mathematica. See also the explanation of the notebooks.