
Exponential TetraviewA series of images in "Surfaces Beyond the Third Dimension" centered around the idea of a "tetraview," which is a mechanism of trying to understand the graph of a complex function as an object in real fourspace. There are four natural viewpoints for the graph that correspond to looking along the four positive coordinate axes. The projections into threespace from these viewpoints produce the graphs (in threespace) of the real and imaginary parts of the function, and the real and imaginary parts of the inverse relation. These appear in the four corners of our "tetraview". A viewing direction corresponds to a point on the 3sphere in fourspace, and these four points form the vertices of a spherical tetrahedron on the 3sphere (thus the term "tetraview"). Each point within the tetraview gives a different viewpoint and so a different view of the complex graph being studied. The view at the center of the picture corresponds to the viewpoint at the center of the tetrahedron. Moving from one viewpoint to another is equivalent to keeping a fixed viewpoint but rotating the graph. For example, moving along an edge of the tetrahedron is like rotating the graph in one of the coordinate planes. Such a rotation can transform one of the standard views into another, and helps to give the sense of connection between the two views that is not apparent from the separate graphs. This idea became the subject of several student research projects at Brown University, one using an immersive virtual environment to navigate the views of a complex function using the spherical tetrahedron as a control device. Again, my role in this was to go from Banchoff's initial instructions ("I'd like an image using the four viewpoints to form a tetrahedron with an 'averaged' view at the center") to the final product, which involved considerable work to develop the understanding of the fourdimensional views necessary to produce the image. This idea for viewing complex function graphs has yet to be explored fully from a mathematical standpoint, and may lead to new insights. The latest version of the 3D Graphing Calculator on the Macintosh includes a 4D exploration module that is based directly on these ideas. 

