December 10, 1999–The Sixth SIAM (Society for Industrial and Applied Mathematics) Conference on Geometric Design was recently held in Albuquerque, New Mexico, from November 2 to 5, 1999. Approximately 240 attendees met to address the most important recent advances in curve and surface design, geometrical algorithms, and solid modeling, in addition to applications in the traditional fields of automobile and aircraft manufacturing and general product design. Contributions in more modern fields, including scientific visualization, medical imaging, computer vision, robotics, and digital movie making were also discussed. Michael Trott, a physicist at Wolfram Research, organized a minisymposium entitled “Symbolic Computation and Geometric Design.”
The symposium included talks on Artlandia, a Mathematica application for creative graphic design; applications of symbolic inequality solving in geometry; photorealistic rendering; and the visualization of Riemann surfaces. Michael Trott himself gave the Riemann surfaces presentation. Of all of Riemann’s sizable contributions to mathematics, the Riemann surface is clearly the most important one. Since their initial investigation nearly 150 years ago, Riemann surfaces have been an influential concept in mathematics, and they are currently enjoying a renewed popularity in modern theoretical physics. Faithful representations of simple Riemann surfaces can be found in the form of plaster and wood models in many mathematics departments. However, using symbolic manipulation in Mathematica, it is now possible to generate visualizations of virtually every multivalued function.
Trott’s presentation included an outline of implementations of programs for the automatic generation of Riemann surfaces of arbitrary algebraic functions, arbitrary compositions of elementary functions, and selected special functions of mathematical physics. Examples of pictures of Riemann surfaces of many classes of functions were also shown. Said Trott, “Riemann surfaces are interesting not only from a scientific and educational point of view; they are also quite beautiful and represent part of the internal beauty of mathematics.”
For more detail, read Michael Trott’s presentation on the Visualization of Riemann Surfaces.