VISUAL NUMBERS

The Style of Numbers Behind a Number of Styles


By DANIEL ROCKMORE

In May 2005, the discovery of 32 previously unknown works by Jackson Pollock in a storage bin in Wainscott, Long Island, was announced by Alex Matter, a filmmaker whose family was friendly with the artist. Investigations of authenticity were undertaken, and recently we heard a pronouncement on the provenance of six of these works from Richard Taylor, a physicist at the University of Oregon and consultant to the Pollock-Krasner Foundation. Taylor found "significant differences" between the contested works and known Pollocks, thereby casting doubt on the entire collection. Like all the experts who have been called into the fray, Taylor is asking whether the work is characteristic of Pollock's style in the relevant phase of his career — but what distinguishes him in the field of connoisseurship is that he is a specialist in fractals and chaos, and his approach replaces subjective judgment with mathematical analysis. At its core is an unsettling idea — that style can be quantified. But that concept isn't new; it's part of a larger discipline that has been around for more than a century.
Questions of style are questions of consistency, rather than equality. "Is this one like the others?" asks the connoisseur, not "Is this one the same as any of the others?" An answer requires the ability to quantify variability, and that's precisely what statistics offer. The analysis of authorial style is more straightforward in literature, and that's where the field of "stylometry" had its first notable successes. Letters, words, and sentences can be counted, and basic statistics can be gathered — such as how often a given word is used, or how frequently words or sentences of a given length appear. Count those things, and you are guaranteed to get numbers. What's surprising is that patterns emerge from those numbers: a distillation of the idiosyncratic frequencies of usage that mark a person's writing style.
The first published account of a mathematical analysis of literature appeared in 1887, written by Thomas C. Mendenhall, an American physicist. Mendenhall focused on the statistics of word lengththat is, what percentage of words have one letter, two letters, and so on. He hoped to identify an author by this list of word-length frequencies, or what he called the "word spectrum." Mendenhall tried to distinguish numerically between Charles Dickens's Oliver Twist and William Thackeray's Vanity Fair, but his method wasn't quite up to the task.
Ten years later, a Polish philologist and historian of philosophy, Wincenty Lutoslawski, had more success when he looked at 500 numerical attributes computed from each of Plato's Dialogues, in order to determine their chronology. His guiding assumption was that, over time, any one of those numerical attributes should evolve in a regular manner, so that given several works with known dates, works whose dates were in question could be dated by seeing where their numbers best fit among those of the known works. He called that mathematical approach "stylometry."
In one way or another, simple considerations of word frequency remain the heart and soul of literary stylometry. Sometimes rare words provide telltale stylometric indicators (that's how Don Foster determined that Joe Klein was the author of Primary Colors), but more often it's the usage statistics of the ubiquitous "function words" — pronouns, conjunctions, prepositions, and so on — that reveal the personal signature. Notable successes include the correct attribution of anonymously published Federalist Papers, as well as settling the disputed authorship of the 15th book in the Oz series.
The visual arts have presented a harder problem for stylometry. They don't consist of discrete, countable entities the way literary works do. Still, the traditional form of attribution used by connoisseurs actually has scientific origins, usually credited to Giovanni Morelli, a 19th-century art-loving Italian statesman. His approach to art was in large part shaped by an early education focused on the sciences and, in particular, his experiences accompanying the pioneering paleontologist and naturalist Louis Agassiz on his glacier expeditions in Switzerland.
In his magnum opus, Italian Painters: Critical Studies of Their Works, Morelli writes, "As most men who speak or write have verbal habits and use their favorite words or phrases involuntarily and sometimes even most inappropriately, so almost every painter has his own peculiarities, which escape him without being aware of them." According to Morelli, those peculiarities would find expression in the quiet corners of a work of art, and the "Morellian method" relies on the comparison of seemingly minor details in paintings: depictions of fingernails, earlobes, or drapery. In such details — in effect, visual "function words," or what he called an artist's Grund-formen (fundamental forms) — the forces of tradition or schooling would be diminished enough so that the artist's true nature could shine through.
Morelli laboriously maintained a list of Grundformen that enabled him to distinguish between such details as hands painted by Botticelli and those painted by Giovanni Bellini. But today's technology can begin to make mathematics out of Morelli. For example, using "wavelet analysis," my colleagues Hany Farid and Siwei Lyu and I have found a way to extract a statistical summary relating to the simple linear elements in a picture.
In essence, wavelets permit the reconstruction of a small patch of a picture as an accumulation of lines of horizontal, vertical, and diagonal orientation, and of varying resolutions. A useful analogy is the classic Etch A Sketch, whose two knobs can be used to move a cursor right and left, or up and down, and thereby enable the user to "draw" on a screen. The virtuoso Etch A Sketch artist can make highly nonrectilinear drawings with this medium, but nevertheless, any given region of the picture consists of, at a small scale, some amount of vertical and horizontal motion. Using this analogy, a basic wavelet analysis would be determining how much each of the knobs is turned in each of a collection of small regions that tile a drawing.
We've applied the wavelet-based method to high-resolution digital scans of drawings by the great Flemish artist Pieter Brueghel the Elder, and it seems to provide a numerical signature for Brueghel's work. This approach is in essence the Morellian philosophy pushed to a logical extreme — a quantitative comparison of the Grundformen — and in early tests, the approach has reproduced the classifications made by connoisseurs.
And once you start to consider a picture as a data source — as an agglomeration of numbers or frequency patterns — it actually makes sense to ask whether the landscapes of Brueghel or the drip paintings of Pollock have more informational content than a Kazimir S. Malevich Suprematist composition or a Barnett Newman zip painting.
The mathematical notion of information was developed in the 1940s in the context of communications systems and is generally attributed to Claude Shannon. It is a measure of the randomness in a collection of numbers produced by some process. In Shannon's original work, that process was language, but with digitization, any artistic output is ripe for measurement, and hence comparison. In particular, the Dutch information theorist Paul M.B. Vitanyi has pushed this outlook to its extreme, developing a stylometry that depends only on this mathematical definition of information content. Through Vitanyi's work, it is now possible, by using only the digital representation of a work, to make sense of a question like, "Is the painting of Rothko more like the music of Phillip Glass or the writing of Samuel Beckett?" For him, the medium no longer matters, just the message.
Richard Taylor is by training and temperament both artist and scientist; in the mid-1990s, he took advantage of a gap between physics fellowships to go to art school and try to recreate on canvas the fascinating patterns he had witnessed in his physics research on the conductive properties of nanomaterials. And so it was a felicitous intersection of art and science that led to Taylor's research into the works of Jackson Pollock. The "chaos" for which one critic famously denounced Pollock's work when it first appeared in the 1950s is something that, in 1997, Taylor saw, quite literally, as the mathematics of fractal geometry.
The word "fractal" was coined in 1967 by the IBM mathematician Benoit B. Mandelbrot to describe the geometry of natural objects. While the perfect lines, planes, and spheres of Euclidean geometry are abstractions, good for a first approximation to things like coastlines, landscapes, and clouds, they clearly fall short in describing the true variation of the natural world. Mandelbrot noticed that the character of such natural phenomena was a similarity in scale — that at each increase in magnification, the structures of nature, complicated though they are ("fractal" is a neologism derived from the same root as fragment and fracture), repeat themselvesmaybe not precisely, but to a degree that can be quantified. The crags of a mountain range are replicated in the nooks and crannies of the stones that compose them. The eddies of a turbulent river flow are themselves composed of eddies within eddies.
It's one thing to say something is fractal; it's another to measure it. Fractal images have the property that the closer you look, the more there is to see, and furthermore, there is a regularity in this increase. For a black picture on a white background, this can be quantified according to the way in which the number of square boxes of a given size necessary to contain the image grows as the size of the box used shrinks.
For simple structures, this relationship is, well, simple. For a black line on a white canvas, the number of square boxes doubles every time you use square boxes that are half their original size, while to cover a black square, halving the size of the boxes requires that you square the number of boxes used. The former reflects that a line has "fractal dimension" of one, and the latter that a square has fractal dimension of two. It turns out that nature — coastlines and mountain ranges, the branches within branches of a fern leaf, or our own circulatory or pulmonary system — has a fractal dimension somewhere in between one and two. This is the style, a fractal signature, and it seems that it's the same as the one that is to be found in the skeins of paint in a Pollock drip painting. That is Taylor's discovery.
Moreover, there appear to be fractal dimensions that are specific to particular periods of Pollock's work. Within those periods, he reliably reproduced in his work a small range of fractal dimensions. In fact, Taylor's examinations provided evidence for two distinct fractal dimensions, as might be predicted by a documented two-step working style in which Pollock would lay down a broad "anchor layer" of paint to which he would later add detail. Pollock's early drip work has a fractal dimension near 1.4 and rises past 1.7 late in his career. By comparison, the fractal dimension of the coastline of England is about 1.24, your pulmonary artery system weighs in at around 1.6, and various fern leaves give a number around 1.8. In more recent work, Taylor has linked these fractal findings directly to the physical dance through which Pollock created his poured paintings. These new ideas relate the rhythms of his canvases to recently discovered patterns found in the quaverings of motion produced by a body trying to maintain a delicate point of balance.
In 1999 an e-mail message from the Museum of Modern Art's chief conservator, Jim Coddington (a Pollock scholar and strong supporter of computational techniques in art authentication), spurred Taylor to consider the idea that his work might be useful for identifying fakes. Taylor was put in touch with the Pollock-Krasner Foundation and in recent years has applied his fractal approach to 46 poured works of "unknown origins." (According to his agreement with the foundation, Taylor cannot disclose the outcome of those tests.) But the most recent, and highest profile, case of disputed authorship was presented by the six poured paintings of Alex Matter's cache.
In a phone call with Taylor, he said to me: "People often ask me, 'Why would you bring mathematics in to look at art?' My response is, 'Well, why not?' When I got into it, I didn't realize how close art and math are, but in retrospect, it's pretty clear. Both mathematics and art are all about pattern. I mean, it would be unusual that you would not apply mathematical analysis to the question."
For some people, stylometrics clearly threatens to reduce art to a number, but that's not how things look to those who pursue it. Fractal analysis doesn't diminish Pollock's athleticism and movement, nature and turbulence, chaos and beauty; it reveals and amplifies it. When Pollock said, "My concern is with the rhythm of nature ... the way the ocean moves," he was probably closer to the truth than he knew.

Daniel Rockmore is a professor of math and computer science at Dartmouth College and the author of Stalking the Riemann Hypothesis: The Quest to Find the Hidden Law of Prime Numbers (Pantheon, 2005).