A Conference Panel for
IEEE Visualization '96
Best Panel Award|
Proponents of computer-aided mathematical visualization argue that visualization can help build the intuition necessary to understand and create proofs. Critics counter that the traditional pristine edifice of theorem-proof mathematics is in danger of being undermined by a dangerous lack of rigor. The acceptance, or lack thereof, of visualization as a legitimate part of mathematical inquiry has implications not only for mathematicians, but for entire visualization community. Among the questions this panel will address are:
Department of Computer Science
360 Gates Building 3B, Stanford, CA 94305
phone (415) 723-3154
fax (415) 723-0033
Tamara Munzner is currently enrolled in the PhD program at Stanford University, where she received a BS in computer science in 1991. In the intervening years she was a member of the technical staff at the Geometry Center, a mathematical visualization research group at the University of Minnesota. She is one of the authors of Geomview, a multi-platform interactive 3D visualization system freely distributed through the Internet. While at the Center she was co-director and co-animator of two award-winning computer generated mathematical videos, Outside In and The Shape of Space. These videos were shown in excerpt at SIGGRAPH and many other juried venues, including NICOGRAPH, Prix Pixel Imagina, and Prix Ars Electronica.
She was one of the developers of an exhibit which allows museum visitors to explore the connections between symmetry groups, tiling, the Platonic and Archimedean solids, and non-Euclidean geometry through interactive 3D graphics. The exhibit is on permanent display at the Science Museum of Minnesota and was shown at ``The Edge'', the interactive installation showcase at SIGGRAPH 94. She has been active in the Virtual Reality Modeling Language (VRML) standards process for specifying 3D worlds through the World Wide Web. Her research interests include interactive 3D graphics, computer animation, mathematical visualization, and information visualization.
Department of Computer Science
302 Butler Hall, Mail Stop 9637
Mississippi State, MS 39762
David Banks teaches Graphics and Visualization in the Computer Science Department at Mississippi State University. He received his MS in Mathematics and PhD in Computer Science at UNC-Chapel Hill. He organized the 1995 SIGGRAPH course on Visualizing Mathematics. His current work includes visualizing singularities of complex algebraic varieties.
Computer graphics is a useful tool for teaching simple topics in mathematics. But how helpful is such a tool for the research mathematician? A popular myth has it that the mathematician will gain new insight leading to new theorems once she is given an appropriate graphical representation of her problem. Experience shows that this paradigm is rarely upheld. Do our computer-generated images need to be more colorful? More interactive? Are the mathematicians too unimaginative to benefit from the visualization systems we create for them? Or is there an inherent flaw in the notion that mathematicians develop new ideas from geometric figures? This talk offers a few optimistic observations painted against a generally pessimistic landscape.
Department of Mathematics
University of Illinois at Urbana-Champaign
Urbana, IL 61801
George Francis is professor of Mathematics, Super-computing Applications, and the Campus Honors Program at the University of Illinois at Urbana-Champaign. His research interests include descriptive topology, numerical geometry, geometrical computer graphics, visual mathematics, dynamical systems, catastrophe and control theory, differential geometry and topology, riemann surfaces, low dimensional geometry and topology, history of the calculus and teacher education in science and mathematics. ``A Topological Picturebook'' of Francis' drawings by hand and computer has been translated into Japanese and Russian.
He is a member of SIAM and ACM Siggraph. Francis received his BSmcl from Notre Dame in 1958, an AM from Harvard in 1960, and the PhD in mathematics from the University of Michigan in 1967. We was a Woodrow Wilson Fellow in 1959, and a Lloyd Postdoctoral Fellow in 1968.
Since the donation by Silicon Graphics of the Renaissance Experimental Laboratory to the NCSA, Prof. Francis has been teaching graduate and undergraduate courses in geometrical computer graphics. Public presentations of his ``illiView'' team of students include the ``Etruscan Venus'' in the Interactive Image exhibit at the Chicago Museum of Science and Industry, ``A post-Euclidean Walkabout'' in the CAVE virtual reality theater at Siggraph'94, and the ``Laterna mathMagica'' on the GII Testbeds at Supercomputing'95.
In every age, the common definition of a geometrical primitive reflects the technology of the time. For the Greeks, the locus of points equidistant from a center suggests a land surveyor guiding a stretched rope tied to a peg. The Cartesian circle, described by the constancy of the sum of coordinates squared, represents the golden 17th century when algebra became the universal language of mathematics, science, and engineering. Seymour Papert's peripatetic turtle, whose circular "step, turn, repeat" converges to a plane curve of constant curvature, typifies the age of servo mechanisms, robots, and analog computers. Computer Geometry is the spatio-temporal science belonging to the age of digital computers and raster graphics. Its circle is the 8-fold reflection of a game of hopscotch played on a tiled floor that follows a rule saying: "always step north and sometimes also west." But the floor is tiled so finely and the moves made so quickly that the game is observed beyond the threshhold of continuity.
Surpassing as many perception threshholds as possible appears to be the true essence of virtual reality research and application. Accordingly, the experiential basis of mathematics learning and interest is changing along with information technology and praxis. In other words, it is less important to debate whether a serious preoccupation with computers is relevant to contemporary mathematics than to appreciate the fact that the very future of mathematics is predicated on the ubiquity of the computational paradigm. Just as Newtonian mechanics, optics and dynamics permanently moved mathematics away from static Euclidean geometry, so the computer dominated information revolution will ultimately move mathematics away from the sterile formalism characteristic of the Bourbaki decades, and which still dominates academic mathematics.
On the other hand, it is absurd to expect computational simulation and computer experimentation, even at a level of ``infinite precision,'' to replace the rigor mathematics has achieved for its methodology over the past two centuries. Rather than change the nature of mathematics, the computer will change the content of mathematics. And that expectation alone suffices to motivate at least some intrepid mathematical pioneers to brave the scorn and disapproval of their colleagues as they venture into increasingly alien worlds of object oriented languages, photo-realistic rendering and virtual environments.
Andrew J. Hanson
Department of Computer Science
Bloomington, Indiana 47405 USA
Andrew J. Hanson is a professor of computer science at Indiana University. Previously, he worked in theoretical physics and then with the perception research group at the SRI Artificial Intelligence Center. His research interests include scientific visualization with applications in mathematics and physics, machine vision, computer graphics, perception, and the design of interactive user interfaces for virtual reality and visualization applications.
Hanson received his BA in chemistry and physics from Harvard College in 1966 and his PhD in theoretical physics from MIT in 1971. He is a member of AAAI, American Mathematical Society, American Physical Society, ACM Siggraph, IEEE Computer Society, and Sigma Xi.
Visualization in general embodies a transformation between a body of knowledge and a picture, or perhaps an interactive animation, capable of representing features of the data to the viewer. In general, the hope is that the displayed features will stimulate associations in the mind of the user that will lead to further insights, suggest new hypotheses to test, and thus advance the progress of science more rapidly than without this methodology.
Mathematics and mathematical physics (like most other visualization domains, in fact) have potentially direct relations between the concepts in question and the choice of representation: it seems obvious to try to represent a geometric object by a faithful picture of the geometry. Several situations typically arise when we ask whether such a picture can assist mathematical research.
In general, the idea of simply seeing global, holistic features of an object that has never been depicted before can potentially suggest associations that algebraic analysis alone might miss.
We can easily present examples of each of the above situations in 19th century mathematical subject areas that have proven to be, at least, amusing to 20th century mathematical audiences. Subject to continuing debate, however, is the question of whether such techniques will in fact directly contribute numerous new insights to 21st century mathematics, or whether, for the most part, mathematical visualization will be principally of pedagogical value.
Research Manager, Centre for Experimental & Constructive Mathematics
Simon Fraser University
Burnaby, British Columbia, CANADA V5A 1S6
Loki Jorgenson is co-founder and research manager of the Centre for Experimental & Constructive Mathematics where he is involved in numerous inter-disciplinary projects involving visualization, mathematics, philosophy, information and network technologies. The CECM was established in 1993 by Dr. Jonathan Borwein, his brother Peter and Dr. J\"orgenson to explore the interplay between the emerging computer technologies and mathematics.
He has a B.Sc. in theoretical astrophysics from Queen's University and M.Sc. and Ph.D. in condensed matter physics from McGill University. He has been working in computer simulation and visualization since 1984. He co-founded the visualization research and development company Eminence Grise in Montreal before moving to Vancouver, B.C. to work at Simon Fraser University.
Mathematics is experiencing a tremendous upsurge of new activity. Like so many other fields, this is primarily due to computers and related technologies. However there is a notable difference: Mathematics is being fundamentally challenged by these new modes of thinking and discourse, some of which threaten to upset long standing traditions. A good example is experimental mathematics which represents a recent movement towards empirical and heuristic research practices. While similar methodologies have been in effect for centuries in the sciences, there have rarely been opportunities to perform classical experiments in mathematics. Rather, mathematical knowledge has typically been authenticated on the basis of strict adherence to theorem-proof constructions that are (in general) unreproachable. For many, these new possibilities threaten to undermine the rigor which most mathematicians have hold dear. Some doomsayers have even predicted the trivialization and subsequent demise of mathematics in the face of such influences.
Scientific visualization hits very close to the heart of the matter. While there is an anecdotal history for the use of mental imagery in mathematics and science, it has no established role save for notable exceptions like geometry, graph theory and most recently the study of chaos and nonlinear systems. Speculatively, this is due to the assumed nature of perception; that it is a largely subjective mode of expression and understanding which is difficult to separate from its ``human failings". Typically an insight arrived at in graphical fashion must be transformed into a corresponding analytical result before it can be accepted into the common body of mathematical knowledge. However it may be the case that this is not always possible or even desirable. Some might even suggest that something valuable is irretreivably lost in such a translation.
To a greater or lesser degree, this is the case for scientific visualization in most of the established sciences. However mathematics poses a unique challenge: In no other field is the language so carefully formalized or the basis for knowledge so rigorously defined: Mathematics is perceived as the ultimate form of reason (Plato) and the inviolable laws through which we see our world (Kant). For visualization to find a place in mathematics, the nature of its contribution to the process of discovery will need to be understood. Further, the boundaries of mathematical knowledge will need to be stretched beyond their current limits to accomodate the information and relationships that heuristic and empirical practices will provide. The question which needs to be asked is a concern for both the visualization and mathematics communities: What can visualization contribute to mathematics and how might that effect the nature of mathematical knowledge?