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Interacting with surfaces in four dimensions using computer graphics

High-speed, high-quality computer graphics enables a user to interactively manipulate surfaces in four dimensions and see them on a computer screen. Surfaces in 4-space exhibit properties that are prohibited in 3-space. For example, non-orientable surfaces may be free of self-intersections in 4-space. Can a user actually make sense of the shapes of surfaces in a larger-dimensional space than the familiar 3D world? Experiment shows he can. A prototype system called Fourphront, running on the graphics engine Pixel-Planes 5 (developed at UNC-Chapel Hill) allows the user to perform interactive algorithms in order to determine some of the properties of a surface in 4-space. This dissertation describes solutions to several problems associated with manipulating surfaces in 4-space. It shows how the user in 3-space can control a surface in 4-space in an intuitive way. It describes how to extend the common illumination models to large numbers of dimensions. And it presents visualization techniques for conveying 4D depth information, for calculating intersections, and for calculating silhouettes.