On the Visualization of Riemann Surfaces

Abstract

The graphs of complex-valued functions f :  Ø  or functions of the type f : 2 Ø 2 are in general two-dimensional manifolds in the space 4. The article presents a method for the visualization of such a graph. The graph is first projected to three-dimensional space with parallel projection and the image~the surface in three-dimensional space~is rendered on the screen in the usual way. The visualization can be improved in two ways: the graph can be rotated in four-dimensional space or the direction line of the projection can be changed, which means that the observer flies around the graph in four dimensions. The animation and manipulation capabilities of Mathematica are appropriate tools for the purpose.