Michael

In the December 2002 issue of Scientific American there is a very interesting article by Richard Taylor on the existence of fractal patterns in Jackson Pollock’s drip paintings. Dr. Taylor, a physics professor at the University of Oregon, is perhaps unusual for a hard scientist in also possessing a master’s degree in art theory.

J. Pollock, Lavender Mist, 1950 (detail) Does Art Imitate Math?

After a chance occurrence got him thinking about possible affinities between Jackson Pollock’s drip paintings and fractals, Dr. Taylor scanned a Pollock drip painting into a computer, then covered the image with a computer-generated mesh of identical squares, and analyzed which squares were occupied by the painted pattern and which were empty. As a result, he could calculate the statistical qualities of the pattern (i.e., the number of full or empty squares.)

According to Doc Taylor,

…fractals consist of patterns that recur on finer and finer magnifications, building up shapes of immense complexity.

In short, a fractal pattern looks quite similar at any scale of magnification, with a constant ratio of filled to empty areas. When Doc Taylor examined the “statistics” of the Pollock drip painting patterns under different sized-grids—from ones small enough to isolate a single speck of paint to grids a meter square—he found that the Pollock's imagery was of a fractal nature, with the same patterns appearing over the entire scale range.

The good doctor considered the possibility that all drip paintings automatically create fractals, and to check this hypothesis he analyzed a painting by an artist other than Pollock. The non-Pollock drip painting yielded no fractal patterns. On the possibility that what he had found was just a fluke, Dr. Taylor analyzed a lot of samples, including:

...five drip paintings sent to us by collectors who suspected their acquisitions might have been created by Pollock. Despite superficial similarities with Pollock’s work, none of the paintings contained fractal patterns. The fractals are the product of the specific technique that Pollock devised, and all the 20 drip paintings of his that we have analyzed have this fractal structure.

Dr. Taylor has also gotten interested in the aesthetics of fractals. For example, the more complex the fractal pattern, the higher the “fractal dimension," or "D” is:

For a smooth line (containing no fractal structure), D has a value of 1; for a completely filled area, [D’s] value is 2. For a fractal pattern, however, the repeating structure causes the line to occupy area. D then lies in the range between 1 and 2; as the complexity and richness of the repeating structure increase, its value moves closer to 2.

Dr. Taylor and his colleagues have investigated aesthetic reactions to three categories of fractals: natural fractals, such as those found in trees, mountains and clouds, mathematical fractals developed by computer simulations and artistic fractals, in this case sections of Pollock paintings. Participants in Doc Taylor’s tests consistently prefered D values in the range of 1.3 to 1.5, regardless of whether the pattern was from nature, computer imagery or art.

Interestingly, according to Doc Taylor’s analysis, the D values of the fractal patterns in Pollock's drip paintings generally rose throughout his career, ending far above the ideal 1.3-to-1.5 range. Perhaps we are viewing, in microcosm, the tendency that ultimately “exhausts” a given artistic model—as it is utilized again and again, the artists, trying to avoid exact duplication and to come up with something new, elaborate it too much and screw it up aesthetically. (The relationship between Raphael’s art production and the works of his followers comes to mind.)

I think this would be a great research topic for a young whipper snapper with plenty of ambition and computing power: to plot how various formal measurements change over time in a given genre of art. Then get back to me; I’ll be napping under the tree.

Cheers,

Friedrich

Fascinatin', thanks. And let's hear it for dynamic young people, full of energy and ambitious to make their mark. Somebody has to supply the energy, and it ain't going to be us.

The SA article and your musings about it remind me of the thoughts of the architect and theorist Christopher Alexander. And more specifically of a piece a collaborator of his, Nikos Salingaros wrote with Bruce West about ratios, beauty, architecture and nature. They claim that there's a magic ratio that all successful (as in long-lived and experienced pleasurably) architecture has conformed to. (Hilariously, and to my mind persuasively, they argue that the only way modernism can be understood is as a deliberate, programmatic defiance of this ratio.)

The article is readable here. I thought it was pretty fascinating.

Posted by: Michael Blowhard on November 26, 2002 10:45 PM

This all sounds familiar somehow...I seem to have some vague memory lingering from my long-ago college days of an art-appreciation class professor claiming that Piet Mondrian's geometric shapes had been found to have some resemblance to biological cell structures. Why it was important and what it proved, I can't remember.

Posted by: Dwight Decker on November 26, 2002 11:32 PM

Fractals ... schmactals. Talk about scientific pollution. Or more simply put: Pollocks to you!

I guess this is why y'all need "language pathology". I don't know where you guys get this stuff? Do y'all live in a derelict trailer park somewhere out in Area 51? What happened to research into cures for the common cold, the big 'C' or a host of other equally potent mysteries?

Its way past time all the remaining sane people left the planet.

Posted by: André on November 27, 2002 12:11 AM

Michael--Thanks for your link to Nikos A. Salingaros' and Bruce J. West's piece on scaling laws in art and architecture. Wow, that's one of the most impressive things I've EVER read on the mathematics of aesthetics, as well as an astonishing link between aesthetics and the rest of the natural world. I would really, really recommend that EVERYBODY check this out (see the link in Michael Blowhard's link above.) It reminds me of a saying by an early Greek sculptor: "Slowly the Beautiful comes about, as the result of many numbers."

Posted by: Friedrich von Blowhard on November 27, 2002 1:13 AM

Christopher Alexander has perhaps had more influence in software design (which is what supports my efforts at poetry) than in architecture. Here and here are introductions to his theoretical and architectural work written for object-oriented developers;here is probably the best starting point for how his ideas about patterns and pattern languages have been incorporated and extended by the software design community.

Posted by: Michael Snider on November 27, 2002 9:20 AM

Just read that paper, on Friedrich's urging. Having not read any of the papers cited, I'm not particularly familiar with exactly how these things are applied to the real world. But I think that blaming modernism for impersonal cities, buildings etc is pushing things a bit too far.

It seems to me, from reading the paper, that the real problem lies in top-down planning rather than bottom-up growth. As I understand it, the problem is generally too much large-scale stuff (big public buildings, Robert Moses freeways) and too little human-scale stuff (bike paths, footpaths, small local shops). In other words, any "designed" city (Canberra, Brasilia, even Paris and New York north of 14th Street to some extent) is going to suffer from being designed.

It's just not possible to pour the majority of your resources into little projects and only do one or two big ones: the World Bank has been wanting to do this for years, and still, with the best of intentions, hasn't worked out a way of doing it. The only way to have an "organic" city (like London, say) is if it grows organically, without any overarching design. So you can't really blame the designers, only the fact that there were designers to begin with.

Posted by: Felix Salmon on November 27, 2002 12:32 PM

Anyone who's interested in the themes and questions Felix has raised (top-down? small scale? organic?) will probably enjoy two classic books on cities, Jane Jacobs' "The Death and Life of Great American Cities," and William Whyte's "City." Both are likely to get your head spinning about cities, planning, and growth, as well as art and aesthetics more generally.

Posted by: Michael Blowhard on November 29, 2002 8:08 PM

Distinguished colleagues;

I read some of the comments about Christopher Alexander, and also on my paper with Bruce West. I'm very pleased to see an interest in these topics. As to Christopher's work, let me give a link to one of my papers on Pattern Languages http://applied.math.utsa.edu/~salingar/StructurePattern.html

It is important to let your group of readers know that a significant convergence is now taking place in our view of the world. Christopher's new book "The Nature of Order" will soon be out (check with natureoforder.com), which will set the tone for an overhaul of current thinking about art, architecture, urbanism, aesthetics, and many other human endeavors.

I am pleased to be a part in all of this, having prepared the way with some publications linking human creations to scientific laws. Some of my papers are mentioned occasionally on this site.

Let me also mention the forthcoming issue of the webzine Katarxis, which will be released in January http://luciensteil.tripod.com/katarxis02-1/id51.html and of which I happen to be co-editor.

Best wishes to all,
Nikos Salingaros

Posted by: Nikos Salingaros on December 4, 2002 3:05 PM






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