The computer-aided discovery of new embedded minimal surfaces

In 1984, Bill Meeks and I established the existence of an infinite family of complete embedded minimal surfaces in R 3. For each k > 0, there exists an example which is homeomorphic to a surface of genus k from which three points have been removed. Figure 30-1 is a picture of the genus-one example. The equations for this remarkable surface were established by Celsoe Costa in his thesis, but they were so complex that the underlying geometry was obscured. We used the computer to numerically approximate the surface and then construct an image of it. This gave us the clues to its essential properties which we then established mathematically. The programming expertise of James T. Hoffman, who is mainly responsible for the quality of the illustrations here, was a central ingredient in our research use of computer graphics. Without the use of a new programming environment, of which he is the principal creator, we would not have made the discoveries I will attempt to describe.