ABSTRACT
This chapter describes Thomas Banchoff project to manifest four-dimensional objects on two-dimensional computer screens, asking what it reveals about geometrical thinking. The films Banchoff and Strauss continued to produce during the 1970s were visually quite simple, appearing even primitive from the perspective of contemporary computer graphics. Banchoff and Strauss's films are devices for training mathematical imaginations, which supplement and augment human cognitive capacity, teaching mathematicians how higher dimensions might be seen, and in what ways. Banchoff was attracted to computer graphics precisely because it enabled him to escape the confines of nature, not to imitate it. While discussing Hypercube, Banchoff remarked that if a scientist wanted to study a frog, he would most likely examine it from various points of view before cutting it open to look inside. In Banchoffs films, he strives not only to visualize specific mathematical problems but also to promote a certain approach to higher-dimensional geometry, training his viewers in the enlargement of the imagination.
