The Fractal Geometry of Vision

John Haber
in New York City

Pollock's Patterns and Rembrandt's Eyes

I cringe the moment scientists come to rescue the world from the uncertainty of art. I do not mean that science fails to explain how the world works, for it does. Nor do I mean that science and art lack common interests, shared methods, and mutual inspiration. A related essay traces these in detail.

Rather, I cringe when art takes for granted its own representations, as if nature simply left its imprint unaided. I cringe, too, when science ignores the human making and unmaking of the world. Call them one culture or two, I shall argue, so long as you call them culture and not nature alone. Rembrandt's Self-Portrait with Wide-Open Eyes (Rijksmuseum, 1630)

As a warning of where the wrong claims may take art or science, I look first at a case study of one dismayingly reductive approach to Rembrandt. I then turn to a mathematical analysis of Jackson Pollock neatly avoids the pitfalls, by sticking to what science knows and does. It uses technical analysis to assist and enrich the trained eye. Yet even it can lead eager defenders of art or science to mistaken conclusions. By overlooking how technical concerns, visual experience, and understanding complement can one another, a third critical approach too readily dismisses the ambitions of modern art and science alike. As follow-ups, separate articles take the story into spring 2006, with the space between still life as installation and science experiment, thanks in part to Olafur Eliasson.


Did Jan Vermeer simply copy what he saw? Do photographs presume never to lie? Does abstraction, book art, or computer art have a single distinguishing character? Each in its own way reduces art to its subject matter or its materials. As the number of links here indicates, I have argued against them all, and I grow tired of huge installations that resemble high-school science fairs, digital art that looks like a first course in computer science, and new media that conflate video art with video games.

The authority of science and a reductive analysis of art still have enormous appeal. One can see it in such achievements as Bright Earth, a book on artist materials and their history by Philip Ball. Certainly a recent attribution to Vermeer has much to do with close technical examination. It adds another layer of irony to an artist who, like many in his time, learned so much from the camera obscura. Yet overblown claims for that reliance or for laboratory conclusions add to the ironies.

One can see where trust in the natural so often goes awry. Ball continues a longstanding project in scholarship and in teaching alike. He shows how a work, its meaning, and its claims to truth all arise not from nature in the abstract, but from the particulars of its making, the constraints on the artist, the assumptions of the viewer, and the material reality of the art object. Less happily, when David Hockney proposes that Vermeer necessarily traced a projected image, he clings to art as a transparent window onto reality, with the artist a kind of vehicle for nature's revelation. A painter of a prescientific era would have agreed, but only with a different notion of revealed truth. Talk about critical assumptions about an artist—or perhaps about an artist's Assumption.

Wonder of wonders, the miracle of art and science are back. They surfaced yet again in late 2004. Rembrandt, says an investigation by Margaret S. Livingstone and Bevil R. Conway, looks awfully "walleyed" in his many self-portraits that so define great Dutch painting as an "Age of Rembrandt." Could the disorder, which makes one unable to deal with the three-dimensional world in stereo, explain his mastery of the world in the two dimensions of paint? Maybe it also explains his collaboration with Gladly the Cross-Eyed Bear on Syndics of the Optometrist's Guild.

The quick diagnosis of a person whom one has never met sounds preposterous enough, even when Senator Frist brings his medical background to a woman on life support, for the state of Florida, and for American moral values. Presumably, Rembrandt's HMO did not require a referral and four visits to separate specialists and technicians first. My dictionary defines walleyed as abnormal whiteness, whereas the article appears to refer to asymmetric eye movements, but I cannot afford the co-payment to find out for certain.

Thankfully, in reporting the discovery, The New York Times quotes a scholar who points out that if Rembrandt's eyes appear to be looking in separate directions, they probably were. A self-portrait generally requires at least two objects, a mirror and a canvas—not to mention the medium, the mind, and an active imagination. Rembrandt also likes to disguise each component, rarely showing himself holding a brush or looking directly at the viewer. He builds on earlier artists, who portrayed themselves as the equal of noblemen rather than as mere craftsman. At the same time, he lends that model for the artist and nobility an inward dimension never seen before.

Modernism's preexisting condition

The Times also notes that most artists do not have a preexisting condition, as Rembrandt's HMO would put it. Besides, few of the many who see largely in mono become great artists. However, a daily newspaper's feature story cannot get at the real problems. Once again, an appeal from art to science ends up reducing both.

For starters, one has the old equation of the canvas with a mirror. Artists have explored and disturbed that equation at least since Las Meninas or A Bar at the Folies-Bergère. It assumes that artists just paint what they see. It ignores that understanding and seeing involve interpretation. Besides, even someone who does not see in three dimensions can learn the hoary techniques of foreshortening and linear perspective, with or without Hockney's camera obscura. Maybe even I could.

The same conception of realism has, especially today, all too often put science above suspicion and interpretation. That for me was the real lesson of Alan Sokal's infamous pseudoscience paper that fooled a postmodern journal. Once artists were skilled craftspeople and scientists a novel combination of technicians and theorists. Now both have a new standing, but it should not allow either to play god, except perhaps the trickster. Where there is a walleye, there is a way.

A "scientific" explanation of art's strangeness can even contradict itself, as in claims for El Greco's distorted vision. As first noted by Nelson Goodman, an American philosopher of art, if the painter saw people as elongated, he would have seen the canvas that way, too. As a result, his paintings would have come out pretty much normal.

Finally, the project characterizes Rembrandt miserably. It takes an artist who ran a large and at one time very successful workshop, an artist who cared so greatly for the painstaking techniques and details of Rembrandt's prints, it makes him unable to see straight. It takes an artist who helped shape modern ideas of painterly space as tactile and inner directed, and it describes him as a stereo-blind mapper of surfaces. It takes a painter of multiple, extraordinarily material layers of paint, and it describes his illusion as two dimensional. The medical diagnosis could be right, and it almost certainly is clever. But it has a lot to learn about Rembrandt and his workshop. Once again, artists become illustrators, copyists, subjects in a science experiment, or all at once.

In sum, should one fail to keep the many practices of art and science, in the plural, clear and distinct, then one explains art and science away. Conversely, fail to acknowledge the layered relationships between practices, and one empties them of meaning.

Fractal expressionism

And it happens. It happens when David Hockney treats painting as a branch of optics, when people take photography for a "natural" imprint of reality, or when digital artists themselves start to trust their data too much. It happens when a psychologist reduces van Gogh's color or Rembrandt's gaze to a pathology.

Is it Jackson Pollock? Untitled (Pollock-Krasner Foundation/Artists Rights Society, 1948–1949)It seemed to happen again in February 2006, when The Times on three different occasions celebrated the findings of Richard P. Taylor, a physicist at the University of Oregon. Twenty-four paintings had surfaced in 2003, once owned by Herbert and Mercedes Matter, friends of Jackson Pollock. The works exhibit Pollock's infamous oil and enamel drips, but did he paint them? The present owner and the Pollock-Krasner Foundation have taken sides, each backed by an expert or two—one side eager to boost his holdings, the other to protect its integrity. Now the foundation has enlisted Taylor as an ally. He has used fractal analysis to claim a stylistic signature in Pollock's drips, and he finds that signature wanting in the new stash.

Actually, Mercedes Matter met Pollock through Lee Krasner, who confided to her some of her first feelings for her future husband. (The Times spares both families the old, false rumors that Matter was carrying on with Pollock.) Matter herself painted and taught painting, rather than just grabbed her friend's work as fast as she could, and surely no artist close to Pollock emerged untouched. Her son, Alex, says that his father attributed the work to Pollock, although one might then wonder about reconciling his memory with the work's recent discovery. Conversely, Alex Matter's defenders would point to Taylor's generalization from a small sample of both Pollock's paintings and the contested work. One notes, too, the vested interests on both sides.

The saga and its high-tech conclusion again had me cringing at first, and a late 2007 study concluded that the fractal results are a sham, but it could be wrong. For one thing, the attribution process is proceeding through perfectly ordinary stages. Technical analysis is helping fallible eyes to weigh provenance and stylistic evidence, just as when one checks the pigments on an alleged old master. For another, Taylor modestly draws no firm conclusion, noting that "certainly my pattern analysis shouldn't be taken in isolation." His Web site even reproduces a paper disputing one nasty bit of scientism, Hockney's claims that Jan van Eyck used a projection to move from drawing to canvas. Besides, The Times neglects to mention, Taylor has been looking at Pollock's drips for years—as a scientist and an artist on the side.

In popular mathematics, fractals have sometimes taken on the mystique of life itself. Fractal analysis looks for "self-replicating" patterns, meaning that a broad pattern takes on greater complexity simply by its sprouting smaller versions of itself, which in turn can replicate themselves on a still smaller scale. A computer takes into account both the degrees of similarity and the number of replications. The process indeed recalls Pollock's architectonic but detailed creations, as he, too, worked from the large to the small. If Taylor finds a signature complexity in the artist, and others are challenging him entirely on the physics, he need not make a general claim for the applicability of mathematics to painting. Rather, he may have found another, more quantitative description of what Pollock did—at least before his symmetry gave way to the loose allusiveness of his later, black paintings.

I hate to admit it, but my own eyes bear Taylor out. In reproduction at least, one disputed painting has two rather awkward, flanking people, facing in, clearly painted from one edge of the canvas, whereas Pollock integrated his allusions to human form with the entire design and also tended to stare down the viewer's gaze. The disputed painting allows enamel to form a colored ground, above which the drips spread and thicken like mineral deposits. Pollock, in contrast, circled the work in progress, allowing his compositions to become denser and more delicate over time, without ever eradicating the bare canvas or the shallow space between canvas and paint. If Pollock's painting approaches life itself—or at least an extension of paint itself—the disputed work more closely resembles a precious stone. The head critic for The Times, Michael Kimmelman, says that Pollock, unlike Willem de Kooning (and de Kooning's Women), could have no disciples, but it might be wiser to say that Pollock's trace is everywhere in his generation, which is why the original still seems so unique.

Mind over splatter?

Still unsure whether higher mathematics may help to resolve some controversial attributions? Tired of hearing that a three-year-old could paint a Jackson Pollock, or that one might turn up in a thrift shop? Even more tired of scientists, who have enough trouble explaining children, who think they can tell you why not? Soon after the topic made The Times, the Sunday OpEd pages reminded one that similar issues arise not just with drips, but also with some highly self-conscious literary achievements. Unfortunately, Don Foster, an English professor at Vassar, makes the artist, attributions, and the cultural status of art more muddled than ever.

English literature may not seem as laden with uncertainties as art or as ripe for quantitative analysis. True, unlike traces of oil and enamel, one can count words with reasonable precision. Still, attributions rarely raise the same perplexities as in art. Juvenalia turn up in an attic, but an author's reputation survives. Literary identities, too, seem relatively invulnerable to forgeries and hoaxes, even if blogs get occasional mileage out of an anonymous novel. At least one author, however, retains both the aura of genius and an elusive identity, William Shakespeare. Foster had found Shakespeare's hand behind a forgotten elegy, and computer-assisted analysis helped swing opinion—not all that divided in the first place—more firmly against it.

From his experience with computing, Foster should have something unique to contribute now, but he insists on asking all the wrong questions. Indeed, he asks too many questions, enough to lose the thread that might help him reach a conclusion. Thanks to the casual mix of open questions, questions as signposts to guide the reader, and rhetorical questions meant to set things straight, the concept of truth already grows more slippery than necessary. More important, he seems so eager to debunk both scientism and the religion of art that he makes attribution itself a pointless game. But listen to his questions. Are they really as provocative as he thinks?

Would uncovering a forgery diminish Pollock or Shakespeare? It might, but not by making his art appear replicable, since it would point to differences worth exploring. If he turned out to have created an overlooked lemon, would that change our opinion of work that many admire? It might, but only by filling gaps in a body of work and its creative evolution. Do attributions make art less symptomatic of the society that produced it? Not unless human beings along with their "cultural artifacts" emerge unscathed from the real world.

Does market value necessarily correspond to worth as art? Of course not, but both stand subject to change, and so does the slippery, unexamimined idea of intrinsic worth. Does Taylor's mathematical physics trump visual inspection? No, but he has the sense not to claim otherwise, and his contribution carries weight only because it helps support and clarify the work of others. Does an "anti-figurative aesthetic" make a "faux Pollock" the artist's "quintessential" work. Certainly not, in part because every term of Foster's in quotation marks here makes no sense.

The greater puzzle of art attribution compared to literature may say little more than the difference between a book, which leaves an obvious paper trail, and a painting, which leaves only a physical object subject to visual and chemical inspection. It may also reflect the aura of an image, which people tend to accord a life of its own almost akin to that of an artist—and that, too, invites both debunking and interpretation along with the art. Appreciating any work involves understanding it. With art, however, understanding gets caught up in a complex web of stylistic, historical, and technical concerns, as when a committee rules on Rembrandt. And all play a role in attribution, as recently the case for that alleged Vermeer. If Pollock joins their ranks, if a physicist keeps enough of an open mind to join the chemists and art historians, and if their eye doctor keeps a safe distance, more power to them all.

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Articles titled "Was Rembrandt Stereoblind?" by M. S. Livingstone and B. R. Conway appeared September 16, 2005, in The New England Journal of Medicine and August 14 in Journal of Vision. I first read a September report in The New York Times. The Times reported twice on the disputed Pollocks and added Don Foster's commentary, all in late February 2006. Some of Taylor's publications on Pollock and other matters for the past decade appear on his Web site at the University of Oregon, and an article in Nature, December 2006, challenges Taylor's choice of grid size. My related articles have given particular attention to art and science as visions of nature and humanity, science and Postmodernism, the materials and methods of art, when photography can deceive, how digital art can lie, high-res scans and Leonardo, and Christina McPhee's digital landscapes.


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