If you watch enough TV, you probably remember an old sitcom plot where the characters are at a viewing of abstract expressionist art, and somehow a 3-year-old’s paint scribblings get mixed in with the famous works. Most of the characters, clueless about art, pretend to like the ‘bad’ painting as much as the real paintings, trusting that whatever is on display must be officially blessed as good by the important people. However, one wise art aficionado spots the fake, pointing out how it is obviously garbage compared to all the real art in the room. Therefore the many pseudo-intellectuals in the audience get affirmation that their professed fandom of “officially” respected art has a valid basis. I’ve always considered this kind of plot a mere fantasy, until I read about physicist Richard Taylor’s apparent success in showing that Jackson Pollock’s most famous paintings actually involve mathematical objects called fractals, and this analysis can be used to distinguish Pollock artworks from lesser efforts.

Before we talk about Taylor’s work, let’s review the idea of fractals, which we have discussed in some earlier podcasts. A simple definition of a fractal is a structure with a pattern that exhibits infinite self-similarity. A popular example is the Koch snowflake. You can create this shape by drawing an equilateral triangle, then drawing a smaller equilateral triangle in the middle third of each side, and repeating the process on each outer edge of the resulting figure. You will end up with a kind of snowflake shape, with the fun property that if you zoom in on any local region, it will look like a partial copy of the same snowflake shape. Other fractals may have a random or varying element in the self-symmetry, which makes them useful to create realistic-looking mountain ranges or coastlines. The degree of self-similarity in a fractal is measured by something called the “fractal dimension”.

Taylor’s insight was that Pollock’s paintings might actually be representing fractal patterns. This idea has some intuitive appeal: perhaps the abstract expressionists were a form of savant, creating deep mathematical structures that most people could understand on an intuitive level but not verbalize. Taylor created a computer program that would overlay a grid on a painting and look for repeating patterns, reporting the fractal dimension resulting from the analysis. After examining a large sample of these, his research team announced that Pollack’s paintings really are fractals, tending to almost always fall within a particular range of fractal dimensions. They also claimed that these patterns could be used with high accuracy to distinguish Pollock paintings from forgeries. Taylor even claimed at one point that, due to the various changes in technique over Pollock’s career, he could date any Pollock painting to within a year based on its fractal dimension. Abstract art critics and fans all over the world felt vindicated, and Taylor became the toast of the artistic community.

However, the story doesn’t end there. When first reading about Taylor’s work, something seemed a little fishy to me— it reminded me a bit of the overblown fandom of the Golden Ratio, which we discussed back in episode 185. You may recall that in that episode, I pointed out that any ratio in nature roughly close to 3:5 could be interpreted as an example of the Golden Ratio, by carefully choosing your points of measurement and level of accuracy. Similarly, it seems to me that the level of fine-tuning required for Taylor’s type of computer analysis would make it inherently suspect. Taylor isn’t claiming an indisputable point-by-point self-similarity, as in an image of the Koch snowflake. He must be inferring some kind of approximate self-similarity, with a level of approximation and tuning that is built into the computer programs he uses. Furthermore, there is no way his experimental process can be truly double-blind: all Pollock paintings are matters of public record, and I suspect everyone involved in his study were Pollock fans to some degree to begin with. I’m sure most Pollock paintings exhibit some kinds of patterns, and with the right definitions and approximations, just about any kind of pattern can be loosely interpreted as a fractal. With all this knowledge available as they were creating their program, I’m sure Taylor’s team was able to generate something finely tuned to Pollock’s style, even if they were not conscious of this built-in bias.

Of course, many people in the math and physics world also found Taylor’s analysis suspicious. A team of skeptics, led by well-known physicist Laurence Krauss, developed their own fractal-detecting program and tried to repeat Taylor’s analysis. They found that attempting to analyze their fractal dimensions was useless for identifying Pollock paintings. Several actual paintings were missed, while lame sketches of random patterns by lab staff, such as a series of stars on a sheet of paper that could have been written by a child, were given Pollock-like measurements. When issuing their report, this team claimed to have conclusively proven that fractal analysis is completely useless for distinguishing Pollocks from forgeries. In some sense, this may not be such a bad result for art fans— as Krauss’ collaborator Katherine Jones-Smith stated, "I think it is more appealing that Pollock's work cannot be reduced to a set of numbers with a certain mean and certain standard deviation.”

So, are Pollock paintings actually describable as fractals, or not? The jury still seems to be out. Krauss’s team claimed that they had definitively disproven this idea. However, Taylor responded that this was merely an issue of them having used a much less sophisticated computer program. Active research is still continuing in this area, as shown by a 2015 paper that combines Taylor’s method with several other mathematical techniques, and claims a 93% accuracy in identifying Pollocks. My inclination is that we should still look at this entire area with a healthy skepticism, due to the inability to produce a truly double-blind study when famous artworks are involved. But there are likely some underlying patterns in abstract expressionist art, at least in the better paintings, which may be a key to why some people find them enjoyable. So lie back, turn on your John Cage music, and start staring at those Pollocks.

And this has been your math mutation for today.

References:

- http://discovermagazine.com/2001/nov/featpollock
- http://physicsbuzz.physicscentral.com/2009/01/facts-on-pollocks-fractals.html
- https://www.sciencedaily.com/releases/2007/11/071125114847.htm
- http://www.scientificamerican.com/article/can-fractals-spot-genuine/
- http://blog.case.edu/case-news/2006/11/30/pollock
- https://en.wikipedia.org/wiki/Jackson_Pollock
- http://vfacstaff.ltu.edu/lshamir/publications/wm_pollock.pdf
- https://en.wikipedia.org/wiki/Koch_snowflake
- https://en.wikipedia.org/wiki/Fractal