What Does It Mean to See?May 3, 2002
Celestial Vortices, Rene Descartes' "extraordinary illustration of the sun in the midst of its own vortex, packed within a three-dimensional system of contiguous vortices, stretches even his illustrative resources to breaking point," writes Martin Kemp. Illustration from Principia Philosophiae (1644), Wellcome Institute Library, London; reproduced in the book under review.
Philip J. Davis
Visualizations: The Nature Book of Art and Science. By Martin Kemp, University of California Press, Berkeley, 2000, 220 pages, $35.00.
President Ulysses S. Grant is said to have stated that he knew only two tunes: "One is Dixie, and the other isn't." I was once asked in the presence of some Austrian wine buffs what I thought of the vintage that we were drinking. Following in the footsteps of Grant, I answered that, apart from white wine and red, I could distinguish only two types: wine that I liked and wine that I didn't. My palate, they responded, was uneducated.
We need to know what we're hearing in order to hear, and we need to know what we're seeing in order to see. Confronted with an unidentified piece of graphics, we may find absolutely no meaning in it. Yet when amplified by a description, a piece of graphics can come alive and add much more information to the enveloping context.
Nature, a weekly journal of science, founded in 1869 and now of international repute, not surprisingly makes extensive use of color graphics in its scientific papers and reports. In October 1997, the journal introduced a series of short and popular articles in which several related graphics are accompanied by a one- or two-page explanation of what it is that we're looking at; almost all have been written by Martin Kemp, a professor in the history of art at Oxford. The book under review is based on articles from this series.
The reproduction quality of the graphics is extraordinarily good---of "coffee table" quality---and the list price is entirely reasonable given the quality. It's lamentable that with the very high-quality graphics possible today, financial restrictions force some publishers and authors to accept work of poor quality.
The book includes about 70 articles. Most of the graphics have a direct scientific context; others are glossed along lines wherein the author identifies the underlying scientific principles. The topics covered are from biology, geology, chemistry, psychology, anthropology, architecture, and so forth. About a quarter of the articles have some overt mathematical substance. The thrust of the collection is to develop answers to the question "What does it mean to see?"
Among the mathematically oriented graphics are:
- Kepler's Model of the Orbits of the Planets, i.e., the nest of regular polyhedra.
- Feynman's diagrams---the H-shaped labeled squiggles.
- Wheatstone's wave machine displaying some patterns.
- Monge's projection of a sphere on a cylinder.
- Descartes' Celestial Vortices.
- Buckminster Fuller's "buckyball" polyhedron---the geodesic dome at Expo '67 and a molecule of buckminsterfullerine.
- The Voronoi cells (explained not by Kemp but by Ian Stewart).
- Mullin and Skeldon's pictures of coupled pendulum attractors.
- Salvador Dali's Corpus Hypercubus, a Christ-like figure crucified on a 4D cube (painting at the Metropolitan Museum in New York).
Now for a few clips from the descriptive material.
The leadoff picture is da Vinci's Mona Lisa. This is a nonmathematical picture but is given a mathematical gloss. Kemp comments:
"In this painting, as in all others by Leonardo, every painted effect was, in theory, based on a natural law."
What are those laws? Among others, the law of the intensity of light falling on a surface and the law of helix formation and its relation to figures found in nature, including ringlets of hair.
Fast-forwarding to contemporary computer-generated art, William Latham's Evolving Form has a tremendous but invisible mathematical basis and a sci-fi, fractal aesthetic. It evokes this comment from Kemp:
"For the 'art world', such images still provoke general unease and meet with limited acceptance. Whether this is because of the art establishment's conservatism or the lack of aesthetic longevity in computer art remains to be seen. I suspect the former to be the case."
This reviewer suspects that both reasons are operating to "ho-hum-ize" such productions.
Monge, the founder of descriptive geometry, extolled his subject as
"a means of investigating the truth; it perpetually offers examples of passing from the known to the unknown. . . . it should necessarily be introduced into the plan of national education!" [Exclamation point added.]
Here, Kemp comments:
"His methods exercised a huge impact on . . . technical drawing . . . but few artists were predisposed to follow the demanding letter of his law."
In regard to Feynman's diagrams, Kemp quotes Feynman:
"Strange! I don't understand how it is that we can write mathematical expressions and calculate what the thing is going to do without being able to picture it. . . . There . . . is a rhythm and a pattern between the phenomena of nature which is not apparent to the eye, but only to the eye of analysis."
And Kemp's own comment is:
"The diagrams rapidly and economically explained and predicted in ways that were at once intuitive and analytical. . . . Yet . . . is it possible to wonder whether such a potent grammar of diagrams and matching equations not only provides a marvelous tool but also constrains what minds lesser than Feynman's permit themselves to envisage."
My question is: intuitive to whom? The impact of graphics may be a private matter, intelligible only to an individual or an elect group.
Of Dali's Corpus Hypercubus, Kemp comments:
"Dali's painting does stand effectively for an age-old striving in art, theology, mathematics and cosmology for access to those dimensions that lie beyond the visual and tactile scope of up-and-down, left-and-right, and in-and-out that imprison our common-sense perceptions of the physical world we inhabit."
Johann Erdman Hummel, Polishing of the Granite Bowl, 1831. The curvature of the bowl was itself "a stiff task" for the painter, Martin Kemp comments in an essay titled "Monge's maths; Hummel's highlights," "but the geometry of the reflections presented problems that only the most dedicated investigator would attempt to resolve." (c) Bildarchiv Preußischer Kulturbesitz 2000, Nationalgalerie Staatliche Museen, Berlin; reproduced in the book under review.
For the most part, the historical, biographical, or descriptive material accompanying the graphics is accessible to an educated readership. Also included is a nine-page introduction and five pages of concluding remarks, in which Kemp provides a philosophical overview derived from his selections and shapes the insights he has garnered into a system. I found this material a bit turgid and targeted toward the author's art-historical colleagues.
I have noticed that museums of art provide much more by way of verbal description of their art objects than formerly. Sometimes the descriptions are attached to a nearby wall. Sometimes they are embedded in the recorded tapes that visitors rent or, in the case of "virtual museums," in a Web site. I have often found this accompanying material tedious, and in my occasional art-for-art's-sake moments, I would have said that art objects should be able to stand on their own, making their impression on the viewer without verbal intervention.
In a recent letter to the editor of the Providence Journal, art historian James Trilling pointed out in cogent terms the necessity for such auxiliary material. The principal reason is that most of us no longer have any knowledge or experience with past cultures, their lifestyles, their philosophical or religious beliefs. Viewing art in the absence of such knowledge is an incomplete experience. What can we see when we don't know what we're looking at? I was convinced---but not entirely---by Trilling's letter.
Of course, art historians, critics, museum directors all along have made it their business to give an object a context within the culture that evoked it. Kemp's book, whether he is explicating scientific graphics or giving paintings a scientific slant, fits in with Trilling's point of view.
But there are counter arguments:
1. Are we somehow hardwired for aesthetics? Many people find the Saint Louis Gateway Arch breathtakingly beautiful. Is it important to know that the arch is in the shape of a catenary? This rewarms the ancient question about the existence of pure aesthetic impulses. Despite a social constructivist outlook, Kemp seems to think that we have pure aesthetic impulses and that aesthetics have survival value:
"Why do we gain pleasure from seeing a 'beautiful' tree and its surrogate in a painted landscape? My suspicion is that what we call the 'aesthetic impulse' is part of the feedback mechanism that reinforces our hugely demanding attempts to make coherent sense of those natural orders with which we can and must work if we are to survive. Our pleasure in pattern, in symmetry, in order and its judicious breaking . . . provides a system of gratification and reward."
2. What is the meaning of an art object for a contemporary viewer? We view a graphical object today, in a museum, in a book, on the Web, on the street, and neither at the time nor the place of its creation.
Consider the Parthenon on the Acropolis. Designed by Ictinus and Callicrates, ornamented by Phidias, the Parthenon has served, over the millennia, as the housing for a cult statue, a place for honoring Greek athletes, a treasury, a church, a mosque, a powder magazine, a source of friezes to be removed and exported. Its meaning for us is that of a major tourist attraction and an icon to be rescued from acid air.
Yet the aesthetic experience persists and is independent of what the structure has been used for by succeeding generations. Surely this persistence is partially behind the reproduction of the Parthenon in Nashville, Tennessee (which has its own salad bar of uses).
3. Is a visual image enhanced by a background---whether historic, cultural, or scientific---so complex that only specialists understand what's going on? Would Kemp's comment on Corpus Hypercubus, given above and amplified by a mathematical discussion of a 4D cube tacked to the wall, add to the tremendous impact the painting has on viewers? Or would it merely confuse them?
Despite all the counterarguments that can be mustered, we are immersed increas-ingly in multimedia. I'm all for more visuals in mathematical exposition, recognizing that they may be useful or they may be superfluous. I'm all for verbalizations of what we see, recognizing that they also may be useful or superfluous.
The late George Wald, Nobelist in the biology of vision, once told me that the problem of what it means to see had perplexed him for years. Martin Kemp has provided us with some pertinent observations on this difficult question.
Philip J. Davis, professor emeritus of applied mathematics at Brown University, is an independent writer, scholar, and lecturer. He lives in Providence, Rhode Island, and can be reached at email@example.com.