Modern geometry transcends the formulas for the Pythagorean theorem and the area of a circle that you probably learned in high school. The field has branched into a variety of sometimes esoteric subdisciplines. There are now hyperbolic, projective and even tropical geometers, some of whom devise abstract constructions that even the most brilliant mathematicians have a hard time understanding without software visualizations.

Labs at the University of Maryland, College Park (U.M.), the University of Illinois at Urbana–Champaign and the University of Texas–Pan American are employing a new educational model for math in which professors mentor postdocs and graduate students, who in turn mentor undergraduates, to create visualizations of geometric structures, facilitate undergraduate research and participate in community outreach.

Bill Goldman, a mathematician at U.M., participated in the Geometry Center, a National Science Foundation–funded research organization, in the 1990s. The center produced several innovative geometry computer programs, in addition to videos that could be used for educational purposes, but political problems shut it down prematurely. Some mathematicians believed it was poorly managed, and many viewed it as too exclusive, especially given how expensive it was. For Goldman, it became a learning experience that helped him plan the Experimental Geometry Lab (EGL) at U.M. with co-founder Rick Schwartz, who is now a professor at Brown University.

They started the EGL in 2000. "In the beginning I didn't really know what was going on," Goldman says, but the lab started with outreach programs, and over time it evolved organically into a place where students produce images that illustrate concepts in a field called hyperbolic geometry.

*View a slide show of the geometry labs' projects.*

Goldman says that using computers to visualize mathematical objects does more than just create pretty pictures. In the midst of developing software to visualize objects from his own research, he found that, "the exercise of trying to explain math to something really stupid, yet obedient and disciplined," actually helped him discover new results, even though the computers themselves didn't really play much of a role in the discovery. The level of detail required to program a computer to produce geometric images can improve a researcher's understanding of the ideas in play.

One of Goldman's students, Sean Lawton, brought the idea of a geometry lab with him to the University of Texas–Pan-American where he created the Experimental Algebra and Geometry Lab (EAGL). Like the EGL, Lawton's lab emphasizes outreach to students who do not envisage science as a career path. "We're really targeting the poets and the dancers and the business majors, people outside of the traditional folds of mathematical interest," he says. Mathematics is both a science and an art, and Lawton wants to show people that it can be appreciated from an aesthetic point of view, "in much the way someone could appreciate a poem or a painting."

Members of EAGL have made presentations at middle and high schools, area colleges, and even libraries and museums on hyperbolic crochet. The visually appealing medium—colorful yarn knotted into beautiful, curly shapes—has introduced many people to non-Euclidean geometry and the idea of math as a creative art. Even Lawton's two young daughters got involved; each, in fact, owns a personal hyperbolic plane. "They both got pretty hyped about hyperbolic crochet for a little while, but they're more into biology these days," he says. "My five-year-old keeps telling me about her platelets."

Lawton often teaches future math teachers. On their way to becoming educators, these students have increased their own mathematical knowledge by joining EAGL.

Another EGL alumnus, Anton Lukyanenko, now at the University of Illinois Urbana-Champaign, founded the Illinois Geometry Lab (IGL) along with faculty member Jayadev Athreya. Outreach remains a major focus—and the initial response propelled the program forward. "We were joking, well, we hope for a positive number of people to show up, instead of zero. In the end we had people out the door of the room," Athreya says. They were able to use the IGL as a means to help undergraduates understand that they can make meaningful contributions to mathematics. "The idea is to get them really plugged in to this community of scholars and researchers."

The variety of projects supported by the IGL is impressive. In addition to more traditional lines of research, one project, led by math department budget director Wendy G. Harris, documents the hundreds of mathematical models scattered about Altgeld Hall, home of the department, some of which date back as far as the 19th century.

The IGL has also hosted enrichment activities for area schoolchildren. When students from a nearby middle school visited Altgeld Hall, they counted the vertices, edges and faces of polyhedra. Athreya recalls that some of the student groups noticed the pattern that v – e + f = 2 for convex polyhedra, a nontrivial fact known as Euler's formula. "We were very impressed," he says. Their visits also include a tour of the math building, which looks kind of like an old castle. "The carillonneur played the 'Hogwarts' theme for them," he adds.

Lukyanenko says that the undergraduates who have participated in the IGL have undergone a transformation. "At the beginning the students were really not sure what they should be doing. They were treating it like a class, but by the end of the semester they were taking more of a research role," he says. "That was really useful for them, to develop their ability to think independently."

The EGL, IGL and EAGL are hardly the only places where research and visualization of geometric structures take place; others use it both in their own research and to educate their own students.

Going forward, the labs will continue to evolve. "I'm really curious what's going to be happening in the future. [The year] 1986 is when I started doing this stuff, and things were so different in 1990," says Goldman, "and then in 2000 we started [the EGL], and now it's going to be completely different, too. It's a little hard to predict what it's going to be, but I think the underlying philosophy and general principles are going to be the same."

*View a slide show of the geometry labs' projects.*

The video below, created by Anton Lukyanenko and Platon Lukyanenko for the Illinois Geometry Lab, illustrates physics in hyperbolic space. As they describe it, "In three-dimensional hyperbolic space, things look smaller as they get closer to the horizontal XY plane, so a ball traveling straight will look like it's changing size and moving along a curved line."