not-so-random splatter

Several years ago, I wrote a feature article for Discover on the work of physicist Richard Taylor, who used the same analysis techniques he applied in his laboratory to study several paintings by splatter master Jackson Pollock (affectionately known to Jen-Luc Piquant as "Jack the Dripper"). He found very clear fractal patterns in the seemingly random drip patterns Pollock splayed across his giant canvases.

Thanks to the enormous popularity of Jurassic Park, many people now realize that "chaos" -- a word that typically denotes utter randomness -- has a different meaning in the context of math and science. It applies to systems that only appear to be random on the surface; underneath is a hidden order. The stock market is a chaotic system, for example: a slight blip can be amplified many times over until the system "goes critical" and the market crashes. It's known as the "butterfly effect" (not to be confused with the 2004 Ashton Kutcher film by the same name, although the movie certainly plays with the implications of the concept): a butterfly flaps its wings in Brazil, and the air disturbance amplifies over time and distance, eventually causing a tornado in Texas.

Fractal patterns are the mathematical offspring of chaos theory, the remnant of chaotic motion -- wreckage strewn in the wake of a hurricane, for example. Something might appear to be haphazard on the surface, but look closer and one might realize that there is, in fact, a single geometric pattern repeated thousands of times over at different size scales, just like those nested Russian dolls.

I've always liked the concept of chaos; it made perfect sense to me, in a way that quantum mechanics never could. (In my lexicon, "quantum" = "inexplicable weirdness." The fact that it appears to be true, at least at subatomic scales, doesn't make it any less bizarre.) So I was chuffed to learn that fractal patterns pop up not just in art, but in music and literature as well. I love it when two widely divergent disciplines -- science and art, in this instance -- somehow manage to find common ground. Taylor's work particularly fascinated me because he explored not just whether such patterns occurred in Pollock's paintings, but why they might be there in the first place. When he analyzed archival film footage of Pollock in the act of creating those canvases -- shot in 1950 by Hans Namuth -- Taylor found that Pollock actually moved around the canvases in chaotic motions. So there was nothing random about Pollock's work at all, at least to Taylor's way of thinking.

Nearly five years later, Taylor (now a physics professor at the University of Oregon) is back in the news, having been asked to analyze several small paintings that were recently discovered, and appear to be original Pollocks. His findings, published in the British journal Nature on February 9, indicate that they might not be authentic after all. They don't exhibit those telltale fractal patterns that Taylor has found to typify Pollock's greatest work. In fact, there were "significant differences"  in the drip patterns -- so significant that Taylor concluded the new paintings were either due to one artist whose style was extremely varied, or to several different artists. Naturally, the owner of the paintings, one Alex Matter, is less than thrilled with this news. Pollock's work typically fetches millions of dollars at auctions, so Matter's cache of canvases, collectively, would be equivalent to winning the lottery -- if they're genuine.

There was heated dispute among art scholars on that front even before Taylor got into the act, but the physicist has added fuel to the flames. Some still contend that the paintings are more likely to be a pale imitation of Pollock's signature technique on the part of Matter's mother, Mercedes, an artist in her own right, as well as an art teacher, and also an "F.O.J." ("friend of Jack"). These naysayers point to Taylor's analysis as hard, empirical evidence that the foundling canvases can't be genuine. That argument carries some weight, especially since Taylor wasn't paid to do the analysis (although his lab was reimbursed for expenses), and hence has no financial stake in the controversy.

Matter's supporters insist that fractal analysis is far from a proven commodity when applied to the field of art authentication, which by its nature is fairly subjective, and usually comes down to a consensus judgment call. After all, they argue, how could a mere computer program possibly be capable of analyzing all the complexities of the human creative process? But in this case, there is no consensus among the usual experts.

Taylor himself is careful to insist that his analysis shouldn't be the final word on the subject, telling the New York Times that his findings "should be integrated with all the known facts -- including provenance, visual inspection, and materials analysis."  In other words, any objective, scientific data should be considered in light of the traditional, more subjective criteria typically employed by art historians. And he never said outright that the paintings weren't done by Pollock, just that the drip patterns aren't consistent with the artist's known authentic works. This places the burden of proof on Matter et al to explain why the drip patterns are so different.

I tend to favor Taylor's findings, and not just because it's a more scientific approach. A little skepticism is perfectly justified: stumbling upon 24 potentially priceless paintings so many decades after Pollock's demise seems just a little too convenient. On the other hand, coincidences do happen, especially in a random world. That's because within the context of science and math, "random" doesn't quite mean what we think it does -- just like "chaos." If something is truly random, odd convergences and coincidences should happen on occasion.

I recall reading a commentary last year by the owner of a newly purchased iPod, who complained that the music player's "random" shuffling feature wasn't truly random after all. His evidence: sometimes the iPod would play two or three songs in a row by the same artist, or repeat the same song more often than others. He wasn't the only one to complain, and Apple eventually had to incorporate a second, non-randomness algorithm into the programming to ensure that the illusion of randomness -- as it is more commonly understood -- was maintained.

As of now Matter and his cohorts are going ahead with plans for a major exhibition of the controversial canvases later this year. They still stand to make a tidy profit from the venture, but the paintings' questionable provenance means they won't profit quite as much as they'd originally hoped. Were there not so much money at stake, it could all be chalked up to a tempest in a teapot, of no interest to anyone outside the fine art community -- except to science-minded people like me, who are keen on seeing how the drama plays out. Will Taylor's method one day be accepted in art authentication circles, or will art historians continue to view the encroachment of scientific analysis  on evaluation in their discipline as a threat to human critical judgment?

Only time will tell.

Comments

[yagwara]

One quibble: The butterfly effect is not that the butterfly causes the tornado in texas. It is that the butterfly causes unpredictable changes everywhere.

That is, take two copies of the earth, identical in every detail except that one earth has one extra flapping butterfly in the amazon. Wait a sufficient time (probably a couple of months for this system), and the two earths will have completely different weather. They will be the same in overall climate, and in the big, general features (which we also see repeated each year), but all the day to day details will be different.

In particular, one earth may have a tornado in texas at a particular time and place; the other earth may have a tornado on a different day in a different town. The two Texases will probably suffer pretty much the same severity of tornado season that year.

[JenLucPiquant]

Thanks for the clarification! Quibbling corrections and additional insighs are always welcome, especially from those with specific expertise.

[inwit]

At the risk of stating the obvious, artists' techniques change over time. (Not all Seurat's paintings are pointillist and the difference between early Beethoven and late Beethoven is as night and day.) So, the absence of the expected scaling behavior in these new-found paintings may be attributable to the fact that they were executed by Pollock at a different period from that of the paintings that Taylor had analyzed earlier.

With regard to the apparent non-randomness of the iPod's "shuffle play" feature, here is the link to the original article by Steven Levy that you seemed to be referring to:

http://msnbc.msn.com/id/6854309/site/newsweek/

Here is my letter to him explaining why finding 3 or more songs from the same album when using the iPod's "shuffle play" feature is not unreasonable:

------------

Sent: Saturday, January 29, 2005 11:31 PM
To: [email protected]
Subject: 3 songs from the same album is more likely than not with your iPod Shuffle

From Steven Levy's "Does Your iPod Play Favorites?":
"[...] The question will be even more important to owners of the new tiny iPod Shuffles. These use a new feature called autofill to load the one-ounce players with a supposedly random selection of 120 or so songs from much larger collections. The first few times I tried this, I found some disturbing clusters in the songs chosen. More than once the "random" playlist included three tracks from the same album! Since there are more than 3,000 tunes in my library, this seemed to defy the odds. [...]"

Let's assume there are exactly 3000 songs in Levy's library and that they come from 300 albums, each with exactly 10 songs each. Further, let's assume that we choose exactly 120 songs at random from this library of 3000 songs. There are Binomial[3000, 120] ways of doing this. The number of ways of choosing 120 songs such that no two come from the same album is just the number of ways of choosing 120 albums out of the 300 (Binomial[300, 120]) multiplied by the number of ways of choosing one of the 10 songs from each of the 120 albums chosen (10^120). Therefore the probability that no two songs come from the same album is just (Binomial[300, 120] * 10^120)/Binomial[3000, 120] = 9.46 * 10^–12. This means that the probability that at least two songs of the 120 chosen come from the same album is 1 – 9.46 * 10^–12 = 0.99999999999054, practically a dead cert.

With slightly more involved calculations (Mathematica helps here), it can be shown that the probability that at least three songs of the 120 chosen come from the same album is 0.858, more than 85 % and far more likely than not.

Levy need not be concerned—the odds are not being defied.

------------

And here is Levy's reply (I trust he won't mind my reproducing it here):

Thanks. Of several mathematical breakdowns I got on this, yours is the clearest. Plus, your 85% probability seems more in line than the I one got saying that the certainty of three songs from the same album is so close to 100% that if autofills were atoms, only a few in the universe would not have three songs from the same album.

In any case, a decent percentage of my tunes are not part of a complete-album set, so my milage varies.

Steven

[JenLucPiquant]

Have to agree with Levy that this is the clearest mathematical breakdown on the topic. Thanks for sharing it! I, for one, found it extremely edifying. If anyone can explain the math behind Amazon rankings or Google search results, feel free to share that as well. :)

As for Taylor, he is himself an artist (has a Master's degree in addition to his physics PhD) so he's well aware of such subjective criteria as changing artistic styles. I didn't mention in my post that he's analyzed Pollock's paintings over time, and found that the fractal dimension actually increases in later paintings -- but the fractal nature of the splatters is always there, regardless of the period(s) he looked at. So I'm quite sure he took such considerations into account.

I was planning on contacting him for a brief update as to his latest science/art stuff. If I find out anything interesting on this front, I'll write a follow-up post.

[JenLucPiquant]

Just ran across an article in the New York Times about an art forger who has become so notorious, he is able to openly sell his fakes: http://www.nyyimes.com/2006/03/04/international/europe/04myatt.html

John Myatt is his name, and for years he painted fake masterpieces that sold for a pretty penny and helped him keep his kids fed and clothed. He now gives lectures on art forgery, and says he can't believe his fakes fooled anybody since they were so poorly done.

It's a great argument for more objective, scientific methods of evaluation/authentication of fine art... :)