During the seventies, Thomas Banchoff and his collaborators pioneered a mathematical visualization program at Brown university. They achieved interactive, real-time geometrical visualizations of surfaces projected from 4D to 3D with a custom-built matrix multiplier and a fast-refresh vector graphics display for wire-frame modelling. They also explored techniques such as enhancing depth perception by rotating the resulting 3D object at a constant angular velocity, and produced many animations of classical objects such as projective planes and tori projected from 4D to a 3D graphics depiction. In Figure 1, we show an image of a classic Klein bottle generated by this group that exhibits two separate visualization methods: the use of a color code to indicate 4D depth and a surface rendered with alternating transparent ribbons to reveal some internal structure of the self-intersecting surface.
At the University of North Carolina/Chapel Hill, the Fourphront system by David Banks is another interactive system for the study of surfaces in 4D . This system, which runs on the high-speed massively parallel Pixel-Planes 5 graphics engine, provides user control of transparency, depth cueing, intersection highlights, and 2-sided paint as well as supporting user control of rotation and translation. An illustration of Fourphront's alternative approach to the Klein bottle display is shown in Figure 3.
Another influential development was Apéry's presentation in 1984 of a long sought parametric equation for Boy's surface, a classic immersion of the projective plane in 3D whose purely topological description dates to the turn of the century. Apéry also formulated the ``Romboy homotopy'' a smooth deformation between his own parametrization of Boy's surface and Steiner's Roman Surface, another projective plane whose parametrization as a 4D surface had been known for a century. George Francis and his collaborators Donna Cox and Ray Idaszak exploited this work to create an animation, ``The Etruscan Venus,'' whose title refers to the suggestive shape of a singular Klein bottle that appears in the deformation. An additional deformation leads to a new, highly symmetrical immersion of the Klein bottle dubbed ``Ida,'' which graced the cover of the August 1989 issue of IEEE Computer. The animation editor used to produce the Etruscan Venus and Ida has evolved into the highly interactive real-time illiView collection of mathematical animators, produced principally by students taught by George Francis in the Renaissance Experimental Laboratory at the National Center for Supercomputing Applications (NCSA) at the University of Illinois. Figure 4 shows an example of a current illiView visualization.
In 1987, the Geometry Supercomputer Project was created at the University of Minnesota, and subsequently evolved to become the Geometry Center, a National Science and Technology Research Center, in 1991. The Center has served as a focal point for a number of efforts in mathematical visualization; a widely used system distributed by the Geometry Center is Geomview , a very general surface viewer developed by Stuart Levy, Tamara Munzner, and Mark Phillips. Its built-in functionality can be extended by customized user programs, called external modules. While Geomview is fundamentally a 3D viewer, the 4DView external module by Daeron Meyer supplied with Geomview accepts 4D data points, allows the user to change the 4D viewpoint, and includes tools for creating 4D slices. Another external module, NDView by Olaf Holt and Stuart Levy, interacts with objects of dimension 4 and higher using multiple projections into families of 3D subspaces. A typical Geomview surface display is shown in Figure 16.
A 4D viewer with a different philosophy is the MeshView program designed by Hui Ma and Andrew Hanson of Indiana University. This system supports the Geomview 4D surface mesh data format, but in addition provides a high-speed mouse-driven 4D rotation interface and a utility for locating particular points on a projected surface relative to the abstract parametric mesh coordinates. Figure 5 illustrates a MeshView display of the case of a closed-form construction developed by Andrew Hanson for representing the complex ``Fermat'' equations ; the surface is projected from 2 complex dimensions to 3 real dimensions from any desired viewpoint. Large families of complex surfaces can be displayed interactively in MeshView using this technique.