Drippy Jackson Pollock Theories: Art and Mathematics!

Sometime in 1996, physicist Richard Taylor decided he'd also pursue a degree in Art History, so he's my kind of guy, obviously. He was on sabbatical and checking out Jackson Pollock's drip-art (1943-52). In 1949 Life magazine featured Pollock as the Hot New Guy Painter and then received tons of letters protesting this abstract expressionist's work, a common argument being something like, "My kid could paint that!" 50 years later, a major exhibit of Pollock at New York's MOMA had lines around the block, and a Pollock has sold for $40million. You explain it. I've got some fractal fish to fry.

                            A detail-section from Jackson Pollock's 1949 work Number 8

Taylor got the idea that Pollock, in his use of particular paints, his manipulations of the paints' viscosities, his bodily motions, his attacking the canvas by walking around it from every angle, using his famous drip technique, had produced fractals. Either Taylor intuited that Pollock had intuited the basic minute algorithmic structure of Nature, or Taylor thought it might be really interesting to pursue the idea and see where it took him, I don't know. But he began an interesting controversy that rattles and drips down to today.

Taylor pursued his hypothesis. As a physicist he was already familiar (an understatement) with Benoit Mandelbrot's book The Fractal Geometry of Nature. In order to test whether Pollock's paintings between 1943-52 contained fractals - whether they contained complex geometric patterns that could be detected at very small scales, medium scales, and larger scales, a hidden occult order that corresponded to well-defined parameters in fractal geometry - he scanned pictures of the drip paintings and put them in his computer. Then he divided the images into very small boxes that added up to five million drip patterns that were, when looked at by the naked eye, anywhere from four yards wide to less than 1/10 of an inch.

                                         Sabinoso canyons of New Mexico, aerial view
                             

If there were fractal patterns in Pollock, it didn't matter what the original size was that was placed into a tiny box in a computer image: fractals are "as above, so below": fractals at the tiniest levels are reflected in the largest levels. If we look at a tree, a twig fractally mimics the whole tree. Mandelbrot - whose work was sneered at when it first appeared (by the most eminent mathematicians of the day!), argued that he had discovered yet another non-Euclidean geometry, one that could be applied to the "roughness" of Nature. Previously, Euclid and other inventors of non-Euclidean geometry assumed smooth shapes basically governed "reality."

[Mandelbrot has won, and for further information - even though I assume nine of 10 readers of Overweening Generalist already know quite a lot about fractals - I've linked to a pretty good 50 minute Nova TV program, which includes Mandelbrot, Richard Taylor, Keith Devlin, Ralph Abraham and many others. If you're bored with my spew here and want to know more about fractals and Mandelbrot, skip to the end of this post. - OG]

                                          a computer-generated fractal image

Anyway, Taylor (1999 paper) found that Pollock's paintings were indeed filled with fractals. As Jennifer Ouellette wrote in Discover in their November 2001 issue about Taylor and Pollock, "Pollock was apparently testing the limits of what the human eye would find aesthetically pleasing." Let's be clear about why this might be so: if we contain fractal structures in our bodies, and Nature - say, a rain forest, mountains, trees, and clouds - are fractal, then we intuitively "understand" these very complex-looking images, because we evolved with them and they are in us. Studies have shown that people's stress levels can lower by as much as 50% by looking at fractal imagery.

This seems to be true at many levels. Imagine the sound of a baby pounding his fists on the keyboard of a piano. Then think of listening to someone playing major scales in every key on a piano with a metronome. Then imagine Beethoven playing variations on a simple theme.

Some evolutionary psychologists have surmised that, deep in our evolutionary ancestry, we became attuned to deviations in the visual environment that disrupted fractal patterns, and this aided in survival, thus making the argument that we are oddly soothed by looking at fractal patterns because they were primarily about survival, and predated aesthetic concerns. (Another Just-So story?)

What's weird is that Pollock did his drip paintings before Mandelbrot published his work on fractals. If we think Richard Taylor was basically correct and Pollock's drip paintings contain fractals, how did Pollock "know"? Almost every writer who's covered this story uses the words "intuition" and "instinct." Some link this to the fractal nature of our selves, but even if this wonderful idea is true, I still marvel at...how? My best guess, as of this date, is that Pollock somehow got himself into harmony with the way of Nature, what the ancient Chinese called Tao. 

"I can control the flow of paint. There is no accident." - Jackson Pollock

Taylor found, using the parameters of fractal geometry and his computer, that, as Pollock's drip technique evolved, the later, more complex paintings had even higher levels of fractal dimensionality.

The next step for Taylor was to build his "Pollockizer," which was a device that used fractal algorithms that allowed a machine to splatter or drip paint in ways that created paintings that were either fractally based, or, when he tweaked the numbers, produce just random, non-fractalized paintings. Then, to test his theory on the public, he displayed his machine-produced fractal paintings along with his non-fractal/ordinary "chaos" paintings. Of 120 people surveyed, 113 chose the fractal paintings as most pleasing.

Fractals are found in nature, but there are others that are man-made (like Pollock's dripworks), and there are still others that are computer-generated, and anyone can look on You Tube and find meta-psychedelic fractal videos that trippers during the Summer of Love could've never imagined. Taylor worked with perceptual psychologists and found that subjects preferred looking at objects that fell within the 1.3 to 1.5 level of dimensionality of fractals, irregardless whether the images were from nature, computers, or human-made. (This reminded me of my studies of why we find some people more beautiful than others.)

There's much more to Taylor's arguments about fractals in Pollock, but what he wanted to argue was that he had developed a way to quantitatively analyze the style of an abstract artist. As wild as Pollock's abstract paintings appear, Taylor said his method was objective and that he may have stumbled onto a way to authenticate and legitimate actual Pollocks, to detect the "fingerprints" of his style, and separate them from the (probably) hundreds of fakes out there. Conservators came running to Taylor, eager for him to apply his method.

The Method Gets Tested: Let the Drama Heat Up!
In 2003, Taylor analyzed 24 putative Pollocks and said they did not possess the fractal signature of Pollock.

In 2006, Alex Matter, the son of friends of Pollock, said he'd found 30 Pollocks that the famous painter had given to his parents. They were found in a storage bin in Long Island, New York. Taylor used his method and determined the paintings were not Pollocks.

Enter two physicists, Kate Jones-Smith and Harsh Mathur of Case Western Reserve University in Ohio. They argued that Taylor's methodology was flawed and that it should not be relied upon to authenticate paintings. In the journal Nature they said, "Several problems must be addressed before fractal analysis can be used to authenticate paintings." And one was the small-box counting technique Taylor used. Jones-Smith took her own simple childlike line drawing, Untitled 5:

...and subjected it to Taylor's method. She and her colleagues used the box-counting technique and found them to have the fractal dimensions of a Pollock! (?) Furthermore, Jones-Smith said Pollock's paintings lacked the range of scales needed to be considered as fractal, because the smallest speck of paint analyzed by Taylor was only 1000 times smaller than the entire canvas. Taylor responded in Nature that invoking this power law "would dismiss half the published investigations of fractals." Additionally, Jones-Smith found that, when she and colleagues analyzed Pollocks using Taylor's published methods, they found that two out of three Pollocks failed to satisfy Taylor's criteria. They charged that Taylor had kept some of his methodology unpublished. Also, Jones-Smith said Taylor only analyzed 17 of the drip paintings (other sources I've read give it as 20), which makes the sample size too small, as there were at least 180 drip paintings. This means that Matter's works may have really been Pollocks and that Taylor's idea about Pollock's stylistic fingerprints should not be used in authentication.

Taylor responded that Jones-Smith's Untitled 5 does not show fractal patterns. By late November, 2007, the idea of authentication by fractal signature seemed wildly unsettled.

Here's an interesting point for the OG: by the time physicists with the advanced ninja Mandelbrotian juju apply their numbers, I obviously lack the math chops to stick with the argument. It seems that when a story goes off in this direction - specialists clashing in a language I'm not conversant in - I retreat to the sidelines to watch the melee and see how it all plays out. And sometimes it takes years! Which is fine with me. I actually enjoy the drama of it all.

Enter: Four New Nerds
In early 2008 a paper appeared, "Multifractal Analysis and Authentication of Jackson Pollock Paintings," by Coddington of NY MOMA, Elton of Pegasus Imaging Corp, Rockmore of Dartmouth mathematics, and Wang of Michigan State mathematics. They looked at Taylor's methodology and how he'd determined that the 30 Alex Matter "Pollocks" were bogus. They extended Taylor's methods, adding the "entropy dimension," which they described as a quality related to the fractal dimension, and I'll just have to take their word for it. The four seem excited that geeks like themselves are making inroads into "Stylometry," and early in the paper cite its nascent use in literature, and this reminded me of a totally fascinating book I'd read a few years ago by Don Foster, Author Unknown.  Foster called what he did "literary forensics" (other times "forensic linguistics") and he used his computer and algorithms he'd developed about word choice, length of sentences, certain tendencies towards tropes, etc, to try to determine who some unknown author was. He was involved in the Unabomber case, he figured out who wrote Primary Colors, and showed that an incredibly literate and witty letter-writer to a Northern California newspaper, who signed as "Wanda Tinasky," was not Thomas Pynchon, as many suspected. It's a great read, but I've digressed. Again.

So yea, the four geeks. Stylometry would use advanced statistical analysis and improved digital representations to quantify works of art. They thought Taylor's box-counting method was legitimate. Hell, what do I know when they haul out their "entropy plots" and "slope statistics"? What did they conclude about Taylor's methods?

They thought Taylor was right: based on their entropy plotting, the Matter paintings were fakes. When they compared a known Pollock with one of Matter's "Pollock"s, they saw a "dramatic distinction between a secure Pollock and this drip painting found by Alex Matter."

Score one for Taylor.

2009: Jones-Smith and Mathur Team Up with Big-time Physicst Lawrence Krauss
They have done their best to debunk Richard Taylor. HERE is a short writeup on their doings.

2011: Fluid Dynamics Leaks In
Check out this article by Lisa Grossman from Wired. When I read the title I thought, okay, more of the math-physics-art geek wars and Richard Taylor, but noooo. What we get here is two geeks from Boston College and a Harvard mathematician claiming the first quantitative analysis of Jackson Pollock. The nerve! Grossman doesn't mention Taylor, but links to his 1999 correspondence to Nature about fractals in Pollock, co-written with Micolich and Jonas. There's no mention of Coddington, Elton, Rockmore and Wang. Why? Because they're not at Harvard or Boston College? What? If you're at Dartmouth it's a denigration?

Herczynski and Cernuschi of BC and Mahadevan of Harvard "believe they've done the first quantitative analysis of drip painting." Yea, hoo-kay.

Well, these fluid dynamics guys had just published an equation about how Pollock spread paint on a canvas, in Physics Today. So...why are they "the first?" Well, they're drawing on novel findings in the area of physics called fluid dynamics. How honey coils when you pour it onto a conveyor belt, the behavior of a dripping faucet, the movement of nanofibers and rope. Some whole other thing that ain't fractals. No, they've done a "quantitative analysis" of "inertial, gravitational, and viscous coiling regimes." Alright, but I think Taylor, Micolich and Jonas did a quantitative analysis in 1999, and the four nerds in 2008 did so also.

I will chalk all this up to the dimension of scientist's lives perhaps best shown in Carl Djerassi's books of what he calls "science-in-fiction," in which an anonymous author's short bio of Djerassi, we read, "illustrates, in the guise of realistic fiction, the human side of scientists and the personal conflicts faced by scientists in their quest for scientific knowledge, personal recognition, and financial rewards."

But just as Taylor thought Pollock was doing fractals before anyone knew about them - even himself - the fluid dynamics dudes say Pollock was doing experiments in fluid dynamics before any physicists had known about the way certain fluids of certain viscosities moved. (This all reminds me of a stunningly imaginative speculative work by a real Leonardo-level Generalist: Leonard Shlain's Art and Physics: Parallel Visions In Space, Time and Light. Here's a thick book on how, throughout history, artists seem to stumble upon "knowledge" about Nature before the physicists, astronomers, chemists, and biologists "discover" the same thing, and then quantify it. Shlain was a surgeon. Yet another digression. Sorry!)

Pollock was obsessive about finding new paints and pigments, and loved to alter their consistencies. Maybe the fluid dynamics guys have something here.

Meanwhile, Benoit Mandelbrot (who died in 2010) thought Taylor was right: Pollock drip paintings were fractal. He supported Taylor in a 2007 Science News article.

Harsh Mathur, Kate Jones-Smith's colleague and Richard Taylor's nemesis, is impressed by the fluid dynamics guys' work. But I find Mathur a tad glib when he says, "Either Taylor is wrong or Kate's drawings are worth $40 million. We'd be happy either way." Why? I'd still like to figure out whether Jones-Smith and Mathur had an ulterior motive for taking on Taylor.

As of 2011, Richard Taylor says that fractals are "just one key to authentication, and should be used with other methods. It's not a red-light green-light method." He's gone on to investigate neurobiology and fractals, using MRIs, EEGs, and skin conductance to measure stress levels of people looking at fractal imagery.

Tangentially, Frances Stonor Saunders, in The Cultural Cold War showed how the CIA covertly backed Western abstract expressionists like Pollock in order to win the war with the Soviets. The idea was that we were way ahead, because, well...just look how "advanced" someone like this Pollock character is! This is what a "free society" looks like!  A wonderful work of "hidden history" that almost reads like a byzantine spy novel, yet it's true. What a weird, weird world some of the CIA guys live(d) in.

Late in my investigation, I discovered Sarah Everts's article in Central Science. It serves as a recapitulation and extends my little ditty here. I like how she points out that fractals soothe, the debate rouses, the math and physics sooth, presumably due to their elegance.

"Technique is just a means of arriving at a statement." - Jackson Pollock

Here's a good overview documentary of fractals and their history, Mandelbrot, etc:

Other Works Consulted

• Chaos: Making A New Science, James Gleick, 1987
• Complexification: Explaining a Paradoxical World Through the Science of Surprise, John L. Casti, 1994
• Computers, Pattern, Chaos, and Beauty: Graphics From An Unseen World, Clifford A. Pickover, 2001
• Nature's Numbers: The Unreal Reality of Mathematics, Ian Stewart, 1995
• Patterns In Nature, Peter S. Stevens, 1974
• Trialogues At The Edge Of The West, Ralph Abraham, Terence McKenna, Rupert Sheldrake, 1992
• Jennifer Ouellette's home page
• Jackson Pollack site at Artsy