The relationship between art and mathematics is a long one, joined by overlapping elements like scale and perspective and form that were recognized by both artists and mathematicians at least as far back as the Renaissance.

Today, the advent of powerful, advanced computer graphics is making the connections between math and art more intriguing than ever before and, quite naturally, a longtime Beckman Institute researcher occupies a unique place in the middle of that nexus. George Francis was putting abstract concepts into tangible form long before the computer age, but his vocation (mathematics professor) and avocation (art) became truly intertwined when computer advances enabled the development of mathematical visualization in the 1980s. Francis began using computers in the late 80s, about the same time his book featuring his mathematical drawings called A Topological Picturebook was published.

"I have always drawn pictures, all my life," Francis said. "That (book) is where I figured out how to draw mathematical pictures by hand. I actually started on the computer at about the same time. I'm still learning stuff about the computer."

Paralleling Francis' computer knowledge growth has been the development of highly advanced computer graphics and high-end venues utilizing those graphics like Beckman's six-sided immersive virtual reality environment, the Cube. Francis has been both a participant and a pioneer in this burgeoning world of enhanced graphics displays. He has been an adviser to students developing new technology in the field and to creators of virtual reality environments. He is also a leader in a campus initiative aimed at bringing the arts and technology together.

Francis is a Professor in the Department of Mathematics at the University of Illinois in the areas of geometry and topology and a member of the Beckman Institute's Image Formation and Processing group. He has been at Beckman from the beginning, working with the National Center for Supercomputing Applications (NCSA) when that unit was here and later with the Integrated Systems Laboratory (ISL) which operates the Institute's immersive virtual environments (IVEs) like the Cube and the CAVE™.

In addition to teaching subjects such as Hypergraphics and Advanced Aspects of Euclidean Geometry, Francis has been or is currently involved in numerous projects with his students and collaborators, including developing mathematical graphics programs for use by scientific researchers and by artists.

All of Francis' work revolves around mathematical visualization, a field of mathematics that has gained acceptance only fairly recently. In 1995 he co-authored a paper that he said sums up his ideas on geometry and graphics: "Mathematical visualization is the art of creating a tangible experience with abstract mathematical objects and concepts. While this process has been a cornerstone of the mathematical reasoning process since the times of the ancient geometers, the advent of high-performance interactive computer graphics systems has opened a new era whose ultimate significance can only be imagined."

Now, that significance is being realized in three ways, Francis said.

"The virtual environments (is one way)," he said. "The other thing that has been realized is that on your desktop you now have the power of a supercomputer."

The third thing is that the cost of doing high-end graphics work is coming down and the computing tools required to do that type of work are becoming more accessible. Francis said he is working with ISL on ways to make high end graphics systems like the Cube available to more users, especially artists.

Francis is one of the principal investigators on a proposal called VITA, short for Visual Information Technology for the Arts. The proposal includes a host of present and future projects revolving around using large-scale, interactive, immersive visualization and sonification displays that have proved their worth in scientific research for projects in the arts and humanities.

Francis was a curator and contributor to the Calculart exhibit held in 2006 at the Krannert Art Museum. Calculart featured a variety of media focusing on art and mathematics and was the first curated exhibit of the CANVAS project, a joint effort between Beckman and Krannert exploring the relationships between art and science. CANVAS featured a three-screen, portable, virtual-reality lab based on CAVE technology for creating and presenting computer-assisted multi-dimensional projects. Francis said Calculart was a partial answer to the question of what is mathematical art.

"The idea was to build a virtual environment for artists," Francis said. "It was in the basement of Krannert and it worked. Artists used it for their own exhibits.

"It was not the first mathematical art show; I curated two previous ones in earlier years. But this one was built around the CANVAS and then we curated a small number of other pieces. Calculart was an obvious thing to do: a virtual environment studio for artists in an art museum."

Francis is also involved in a project with New York artist Tony Robbin, a self-taught mathematician, as well as sculptor, painter, and, like Francis, an exponent of the power of computers to create art. In the 1990s Robbin installed a large display called Coast at a university in Denmark. Coast was a 17-foot tall sculpture of quasicrystal colored and clear geometric shapes that produced remarkable visual effects of light and shadows. The display was later taken down and destroyed, but Francis suggested to Robbin that he recreate the piece - in silico.

"I said 'Tony this is really good art' and he said 'I'm too old to do this again' and I said 'no you don't have to do this again, we can do it in the Cube,'" Francis said. "Then he perked up, so we've been working on this for three years."

This is often where the expertise of Francis and his students comes in. The students work with already developed template programs which they then modify to create the software that makes a psychology experiment or artists' concept come to life in the virtual world.

"In order to reproduce this, somebody has to build a tool but that tool itself can become a product," Francis said. "So we envision a tool that Tony can use to design not one, but many of these displays. And because it's all virtual, he can place the latitude; it doesn't have to be in Denmark, it can be in Florida. And he can pick the season of the year when the sun goes over. This can all be simulated in the Cube.

"This is what that VITA proposal is all about. This is to enable artists to take high level information technologies, the stuff we have here at Beckman, and use it for themselves. How long is this going to take? Well it depends on a lot of lucky breaks and I need some students who are going to do this."

Over the years Francis' students have created perhaps 800 programs for use in platforms such as Silicon Graphics that power immersive virtual environments, as well as Linux and wiNTel platforms used at the U of I and on the Internet.

One of his former students, Ben Schaeffer, wrote the code (Syzygy) that powers the Cube and that can also be used for other platforms for PC and PC-cluster-based virtual reality environments as well as other graphical applications. Francis is proud of the fact that Syzygy, which continues to be updated by Beckman staff members, is free for the asking.

Schaeffer is but one of Francis' former students who have left deep footprints on the information technology landscape. Names like Mosaic and Netscape co-inventor Marc Andreessen, and PayPal co-inventor Max Levchin not only learned from him but were essentially student collaborators in Francis's nearly four-decade long scholarly advocacy of the power of computers to illuminate our world through education, research, and art.

As a student, Andreessen set up one of Francis' courses in an early computer graphics laboratory and Levchin started Francis' graphics lab in the math department. Francis said his former students are sprinkled throughout academia and industry.

"Many of them are still here in town, working at NCSA or working here at Beckman and other places," he said.

Francis utilizes what he calls associated mentors - former students, collaborators, and colleagues both in the United States and abroad - who work with his students on projects.

"That means that what students do is very often determined by what the mentor is interested in," Francis said. "For example, the mentor might provide data and students work on visualization of that data. I see to it that everybody learns a lot of mathematics in doing it."

In the summer he has a Research Experience for Undergraduates program that gives his students real-world experience.

"I like to teach," Francis said. "I feel it's very important to involve undergraduates if they are bright enough and also graduate students if they are interested in what I do."

Francis plans to take a sabbatical next year, though it's hard to imagine he will be taking any time off. His sabbatical proposal states that he plans to work on four grant proposal projects, continue his work with ISL and other IVEs, and continue working on a scholarly piece on real-time interactive computer animation (RTICA) in mathematical visualization.

In his proposal Francis writes that his "research over the past two decades has been intimately connected with his interest in mentoring undergraduates to create RTICAs that visualize mathematics in cutting-edge IVEs" and that this complements his "long-time interest in creating mathematical and computational tools for artists engaged in realizing deep mathematical ideas."

Sharing those ideas either through art will continue to be a driving motivation for Francis.

"I think these exhibits invariably speak to some people as finally somebody is showing me something that I think I understand about a subject that people are rarely willing to talk about," he said.