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Vol.32 No.4 November 1998

Transforming Spheres — In Three Parts

Thomas G.West
Visualization Research Institute, Inc.

November 98 Columns
Visfiles Real-Time Interactive Graphics

Thomas G.West
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According to a recent article by Dana MacKenzie in Science, “More than 40 years ago, a University of Michigan graduate student named Stephen Smale laid down a challenge for future mathematicians. He proved an abstract theorem that had a startling corollary: An elastic sphere can be turned inside out, or ‘everted’ without tearing or creasing it — providing the sphere can pass through itself, ghostlike. . . .”

“Demonstrated in a 6-1/2-minute video tour that [premiered in the August] International Congress of Mathematicians in Berlin and was shown [first] in an abbreviated form [in July at the] SIGGRAPH 98 convention in Orlando, Florida, [a new solution] provides the most satisfying answer yet to Smale’s challenge. It also shows how topologists are turning to computer graphics to solve some of their hardest problems. ‘In a real sense, [according to one mathematician,] the eversion question has become the benchmark for the use of computer technology in attacking problems of surfaces in three-dimensional space.’ ”[1]

Spheres Coming Together — Returning to Images

Many of us are aware that the annual SIGGRAPH conference often scoops other conventions and conferences in other fields. But it is not too often the case that we scoop (however briefly) the mathematicians — who seem always to be, in one way or another, way out in front of us all. Years ago, I first became aware of the annual conference when I noted that the best computer graphic work I saw elsewhere had appeared six months or 12 months or 18 months before at a conference called, strangely, “SIGGRAPH.” After my first brief exposure in Boston in 1989, I quickly realized something really important was happening at this meeting and I have attended every meeting since. As Jim Blinn keenly observed in his keynote this past July, the area of interest that was once an exotic academic curiosity has indeed in recent years become a mighty engine of change and transformation in field after field, sphere after sphere — as most of the world understands little about it or its unfolding power.

Many of us will recall an earlier, more complex version of the everted sphere, Outside In, shown at SIGGRAPH 94. (It is noteworthy that SIGGRAPH seems to have a continuing relationship with such things.) Several remarked that it was “best of show” for the Electronic Theater that summer. Others will recall a year later detailed discussions of the making of the video —when mathematicians, in typical SIGGRAPH style, spoke of how much they relied on artists, choreographers and filmmakers to find ways to effectively communicate complex mathematical ideas to a wide audience, with effective motion, pacing, color, form — and some humor.

As other fields, disciplines and spheres of knowledge become progressively more and more specialized and mutually unintelligible, SIGGRAPH seems to have harnessed a powerful counter current, bringing together a range of diverse talent unthinkable in most spheres. In this way and others, SIGGRAPH has helped to redefine over 25 years, again as Jim Blinn noted, the range and content of the technical meeting as well.

It is perhaps not surprising that SIGGRAPH and certain forms of higher mathematics seem to have a special relationship. Both have been going through major changes in recent years, in somewhat parallel fashion. Some observers note that it is only fairly recently that the real nature of mathematics has come to be fully appreciated. Fundamental developments in mathematics during the last two or three decades have changed the discipline itself as well as the mathematicians’ view of themselves. According to one professional observer, Lynn Arthur Steen, mathematics should no longer be seen as the study of number, or even of space — it should now be seen as the science that studies patterns.

“Mathematics is often defined as the science of space and number, as the discipline rooted in geometry and arithmetic. Although the diversity of modern mathematics has always exceeded this definition, it was not until the recent resonance of computers and mathematics that a more apt definition became fully evident. Mathematics is the science of patterns. The mathematician seeks patterns in number, in space, in science, in computers, and in imagination. Mathematical theories explain the relations among patterns. . . . Patterns suggest other patterns, often yielding patterns of patterns. . . . “ [3]

In time, the tools that serve the discipline begin to transform the discipline. “To the extent that mathematics is the science of patterns, computers change not so much the nature of the discipline as its scale: computers are to mathematics what telescopes and microscopes are to science. They have increased by a millionfold the portfolio of patterns investigated by mathematical scientists,” says Steen.

No longer is mathematics seen mainly as the domain of number and rigorous logical proof, with just a little geometry on the side for historical interest. The long-term trend is reversing, as geometry and related visual-spatial forms move back into the spotlight once more.

“Historically, geometry, the study of space, has been one of the major pillars of core mathematics. For various reasons, its role in the mathematical curriculum has declined over the . . .years, so that even those with university degrees in mathematics often have little acquaintance with geometry beyond . . . high school [levels]. In sharp contrast to this curricular decline is the renaissance of geometry in research mathematics. In a very real sense, geometry is once again playing a central role on the stage of mathematics, much as it did in the Greek period.” [4]

Steen points out that an indicator of this reversal of trend is that two out of the three Fields Medals (the “Nobel Prize” of mathematics) awarded in 1986 went to researchers for fundamental advances in mathematics related to geometry. A similar trend in mathematical research seems to have continued to the present day. The Fields Medals, given once every four years, were presented this past August at the same International Congress of Mathematicians meeting in Berlin mentioned above.

Of the four medals presented, one was awarded to researcher who “investigated the Mandelbrot set, which compactly summarizes the chaotic behavior of certain equations, including those often used to describe phenomena such as flowing liquids and changing weather systems. His research provided a precise characterization of the Mandelbrot set’s convoluted boundary.” Other awards involved elliptic curves, string theory and mathematical knots. [2]

Without Cover — Spheres Intersecting

An inevitable consequence, perhaps, of the vast diversity encompassed in SIGGRAPH is the occasional clash of differing conventions and expectations. In the midst of high technical proficiency, a certain zaniness is expected —indeed, insisted upon. (Certainly a contender for “best of show” in 1998 was Jim Blinn’s chart of name tag ribbon length over the years.) However, one can see that occasionally a super straight engineer and a way out artist may have trouble figuring out what to make of each other — although it is frequently the case that the two may come to rely on each other and respect the other’s contributions in ways that would rarely happen elsewhere.

However, in the Orlando conference, the introduction of the first live nudes in the Digital Pavilions with Enhanced Realities and Touchware Art seemed to test somewhat the tolerances and interactions of the various groups. Those from the worlds of art and design were unaffected. But technical folks and others unfamiliar with fine art life drawing, performance art and 500 years of the nude in painting and sculpture were not entirely sure what to make of the new innovations.

The “bar code guy” was more serene, just lying there in his plastic foam form (with a small electric heater on the side) as observers swiped the various bar codes on Plexiglas over different parts of his uncovered anatomy — bringing up corresponding parts of visible human digitized medical data and other more whimsical images on monitors nearby.

However, “Project Paradise” from The Centre for Metahuman Exploration — with its two distant line-them-up booths and strangely familiar telephone controls, its Garden of Eden in a metal drum room, the uncovered models with robot camera arms and the twin television monitors open to passersby — generated a quiet but continuing buzz of puzzlement. “Is someone else is controlling this?” “What’s this for?” “Is that her boy friend?” “Where are they?” “Is that weird?” “Can they hear us?” “They will never believe me if I don’t take a picture of it.” “He looks really bored.” “This has got to be hard work.”

The arts folks managing the Digital Pavilions had wondered whether there might be complaints — but they were surprised to find that the only one they did receive finally was from a staff member of a computer company who complained of a nude in a slide show during an artist lecture. That was all.

The “collective” of five who devised and performed the Paradise Project were on the whole much encouraged. After development in the Robotics Institute at Carnegie Mellon, and several gallery and television performances, they were readying themselves for a showing at Arts Electronica in Europe in the autumn. They felt that they had succeeded in using the familiar, simple, “transparent” technologies of the telephone and television to make their art accessible to a broader audience. Too often, they explain, the technology itself is the piece being shown, not what it does. It is akin to putting a radio transmitter in a glass display case rather than having it do radio broadcasting.

Other exhibits in the Touchware area became favorites with repeated visitors, each promising to transform spheres of their own. A source of instant recognition and fun was “Thirteen Sketches for an Incompetent User Interface.” Those who sat on the blue screen sofa wondered whose hands were behind them, although not really there. A compact and hidden marvel was the “Lost Worlds” exhibit with moving images projected on tiny ruins in a dark corner. Others wondered at the cones on the floor that when touched, released flights of birds and other marvels in patterns that were hard to predict.

Thomas G.West is author of In the Mind's Eye. From a family of artists and engineers, he has long been interested in the connections between mixed abilities, technological innovation and visual thinking in various occupational and cultural settings. He is conducting research for a new book with the working title, Insight--Computer Information Visualization and the Visual Thinkers Who are Reshaping the Future of Technology and Business.

Thomas G.West
Visualization Research
National Dyslexia Research Foundation

Tel: +1-301-654-5828
Fax: +1-301-654-0987

The copyright of articles and images printed remains with the author unless otherwise indicated.

Spheres in Flux — Cells Locating by Smells

In a column dealing partly with early trail blazing work, I would like to take the liberty of mentioning, very briefly, some work by a user of scientific visualization and self-described strong visual thinker, William J. Dreyer, a biologist from Cal Tech, who may soon be recognized as being responsible for a scientific scoop of his own. Like SIGGRAPH itself, this work might at first seem academic, remote and esoteric. But, in time — if it is in fact proven to be correct in the next few months — this work may come to be seen as a major factor in biology, genetics and embryology. Accordingly, I want to take this opportunity to say (at least for many of you) that “you heard it here first” — as I don’t expect to have too many opportunities like this in the future.

Dreyer’s radical new theory was published August 4th in the Proceedings of the National Academy of Sciences and summarized in an August 15th article in Science News. According to Science News writer John Travis, “Dreyer lays out the provocative idea that the cell surface proteins in the nose that detect odors also help assemble embryos. He argues that these olfactory receptors and related proteins act as identifiers, much like the last few digits of a telephone number, that help cells to find their intended neighbors in a developing embryo.” [5]

“ ‘I’ve been searching for these last digits for 20 years,’ says Dreyer. . . .’No one can say for sure [the new theory] is true, but I’m up to 90 percent confident. . . . ‘ [One scientist] who studies development of the vertebrate olfactory system . . . admits he was ready to reject Dreyer’s theory but found he couldn’t. ‘One experiment could tell us that the theory is totally impossible, . . . but I haven’t found that experiment yet. I don’t know of any one thing that unequivocally says it’s wrong,’ he says. As for Dreyer, he has faced skepticism before . . . but later [has been] proven correct. [He] is now hoping history will repeat itself.” [5]


  1.  MacKenzie, Dana. “Sphere Does Elegant Gymnastics in New Video—A tour de force of computer graphics gives the simplest solution yet to the venerable problem of turning a sphere inside out,” Science, vol. 281, July 31, 1998, p. 634, Website.
  2.  Peterson, I. “Math Prizes: Moonshine to Quantum Logic,” Science News, vol. 154, August 22, 1998, p. 117.
  3.  Steen, Lynn Arthur. “A Science of Patterns,” Science, 1988, p. 616. Lynn Arthur Steen is professor of mathematics at St. Olaf College and has served as chairman of the Conference Board of the Mathematical Sciences. (Part of this discussion appeared in different form previously in “Forward into the Past, Computer Graphics, November 1995.)
  4.  Steen, “Science of Patterns,” Science, 1988, p. 616.
  5.  Travis, John. “Dailing Up An Embryo—Are olfactory receptors digits in a developmental code?” Science News, vol. 154, August 15, 1998, pp. 106-107 and Dreyer, William J. “The Area Code Hypothesis Revisited—Olfactory receptors and other related transmembrane receptors may function as the last digits in a cell surface code for assembling embryos,” Proceedings of the National Academy of Sciences, vol. 95, no. 16, August 4, 1998, pp. 9072-9077,