Nestled quietly among the 43 applications that are scheduled to run in the VROOM at SIGGRAPH 94 is the Post-Euclidean Walkabout. It is compiled from various projects that were part of a class in which students learned how to build and run their own 3D applications in NCSA's virtual environment.
"My principal goal is to teach geometry," says George Francis, UIUC professor of mathematics and long-term user of NCSA's visualization facilities. Unobtrusively labeled "Math 428--Topics in Geometry" in the UIUC timetable with the tag "Section G1 considers geometrical graph[ic]s" to denote Francis's section, this one-unit graduate-level course is at the vanguard of computational science/computer graphics. Francis teaches students how to exploit the vast opportunities that virtual reality has to offer.
Francis was recently recognized for his teaching style. He won the $1,750 Amoco Foundation Award for Innovation in Instruction.
"You start out with what you know about computer graphics and mathematics, and you work at your own pace," says Robert Schade, UIUC graduate student in mathematics who was in the class last fall. "I went in with a half-baked understanding of the [Silicon Graphics] IRIS and computer graphics. Professor Francis was very good at picking up on what I didn't know and explaining it from the ground up."
In semesters when it is offered, 10 students usually complete Francis's course. "Students continue their projects beyond the class," says Paul McCreary, UIUC teaching associate in mathematics who sat through the class each of the last three semesters. Though he never officially enrolled, McCreary says he was an active participant and used what he learned to develop a thesis topic.
"I do not know if it's me or the students' natural inclination," Francis says, "my students tend to want to do things on their own. They are not great consumers of fancy software packages. Rather, they make their own codes. They learn an appreciation for doing the geometry themselves.
"I call it post-Euclidean to avoid the odd-sounding non-Euclidean geometry, which is its official name in mathematics since Gauss and Riemann," says Francis. "The similarity to post-modern is intentional, too." Francis compares the post-modern movement in the arts to what he is doing with virtual reality.
"Since Frederick Jameson's book on post-modernism, the term [post-modern] has referred to a pastiche, a quilt or patchwork, an eclectic juxtaposition of diverse stylistic elements without necessarily exhibiting any internal logic or intended structure," says Francis. "And that fits post-Euclidean best, since, in these earliest of times in the evolution of virtual reality, we explore whatever geometrical phenomena strike our fancy and fit into our limited technical repertoire."
"Walkabout is what Australian aborigines, and also native whites, sometimes do when they 'drop out' of their society for a while and wander about the waste regions of the Australian outback," he says. "The disorientation frequently experienced by CAVE visitors is alluded to here."
The goal of this project is perfecting visual and sonic environments to exhibit time-varying multidimensional geometrical shapes that interest research geometers. It is useful in applied and pure mathematics as well.
