Sunday, May 9, 2010

The Prime Number Model of Progress in the Visual Arts

Well, this is going to be a philosophy-for-poets sort of a post.

Just so I myself don't forget, I'm working (not very fast) on 1. two more posts about edges and edge-detection, one of which is basically just a long reply to Chris's comment on the last post and 2. two more posts about the depiction of eyes. One requires some art-historical research, and the other requires finding a bunch of images that are tough to find.

Down to business. Consider this: we can't help agonizing about whether we are doing something "new" in art. Well, mostly, we're not. Imagine being the first guy to contrast blue and orange in a picture! Wow - that must have been one hell of a thing, and that guy was probably very pleased with his genius, and his friends probably bought him dinner until he got to be all insufferable about his own awesomeness. Then he had to top himself. So what does he do? Violet and yellow!

step to this

That guy's friends resign themselves to Mr. Artistic Genius having done it again. Nobody has ever thought of this before! And when they get tired of his attitude, what stunt does he pull next? He invents Christmas, that's what he does:

...and then he lives happily ever after, because who could hate Christmas?

But now all the juxtapositions of complementary colors have been discovered. Nobody will ever again be the first person to juxtapose the big six complementary colors. So art moves on. Some other new thing is required, if new things are what we seek.

In the history of art so far, there is a finite list of discoveries of absolutely fundamental new things. Complementary color. Value. Line. Shape. The figure. The still life. The landscape. Narrative scene-making. Perspective. Anatomy. Psychology.

Notice that these discoveries gradually become more complex. The simple ones come first, the tricky ones emerge once the simple ones are available and mastered.

Now let's look at prime numbers. For those of you who have forgotten, those are the integers evenly divisible only by 1 and themselves. Of the first 100 integers, 25 are prime numbers:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

By gentleman's agreement, 1 is excluded from this list.

I am a dirty monotheist.

If you look closely, though, you'll notice that the density of prime numbers falls across this interval. Of the first ten integers, four are primes. Of the ten integers between 91 and 100, one is a prime. Two properties of the distribution of primes across the integers have been proven:

1. Primes are infinite in number.
2. Primes continue to get scarcer.

How to predict the locations of large primes in the vast universe of integers has occupied the brains of some unusually clever mathematicians, leading to formulations like this:

And this:


And yet for all this cleverness, nobody has found a road map to the large and increasingly rare primes. Computers still find big ones, but even when they do, these elusive leviathans take a while to prove their bona fides. For instance, this is a prime number:


It is the second largest known Sophie Germain prime, discovered in 2009. All ye need to know, and this you figured out for yourself, is that it lacks the grace and intuitive clarity of your 3, your 5, or your 7.

You can see where I'm going with this. I'm proposing that new, genuinely new, things in art are like the primes. There were a lot of them at the beginning. But they get rarer over time; rarer, and less intuitively beautiful. We will never run out of them, but increasingly, it will take centuries, and then millenia, to discover them. After all, we cannot assign a computer to the problem of innovation in art, because art is an artifact of the interaction of the soul and the universe. Every new discovery will demand a larger number of artists broken on the wheel of fruitless effort. Not just lousy artists either; geniuses, who had they been lucky enough to be born earlier, would have discovered the juxtaposition of complementary colors.

This presents a quandary to the artist. What should he attempt to accomplish? We have discovered so many fundamental things about art, art-primes if you like, that each art-prime is like a village, or a city. Each has a community of artists devoted to it, who live in the comfort of its fire, which is not a new fire, but has neither ceased to shed heat and light. Moreover, there are roads between these art-prime cities, well-traveled roads of familiar composites. Just as 10 is the composite product of two primes, 2 and 5, so your average Matisse or Picasso is a composite of two art-primes, line and color.

It is enough to live on these cities and to travel these roads. Our current art culture places a very high value on discovery of new art-primes, but they are rarely to be had, and many loudly-hailed discoveries turn out to be composites, or nullities. Many artists set out on the path of discovery, but few are rewarded.

My belief, in fact, is that the next art-prime will be discovered by someone who isn't looking for it. That's how it goes in human affairs. Do you think Proust sat down and said, "I'm going to invent Proust On Time?" I don't think so. I think he had a good idea, maybe he thought it was even a new idea, but I don't think he thought it was a New Idea. So he started writing, and when he got done, he said, "Well, that's a hell of a thing." And then he died.

So it will go in the visual arts. Some hard-working resident of a city of the known will strike out in a slightly new direction; he will not conceive it as a totally new direction, just something a little different to try, to set himself apart from the crowd, or to respond to that inner drive which is how most people think of art. And when he gets done, he will have changed the world, and made it larger and more glorious. But his innovation will not be so clear and beautiful as the complementary colors. It will not have the intuitive unity of a 2; it will be the artistic equivalent of a large and incomprehensible Sophie Germain prime. It will take many years for other artists to understand what this one accidental innovator has accomplished.

Now that I've written it out, I suppose this is a very dark view of innovation in the visual arts. But we are not young, as artists, any longer. If we continue to pursue the visual arts, we will either grow more mature and more comprehensive, or we will find some unanticipated way to reset the clock and become like children again.

For my part, I have painted this, and it is getting a little bit of attention as part of an art fair I am participating in this week:

Red, 60"x36", oil on canvas, 2009

A note on methodology: one of the things I like most about science and math is that it provides just wonderful similes for things in the humanities. In this respect, science and math are like the Bible. What could more perfectly capture how it is tough to enter, and tough to leave, a close relationship, than the concept of the Coulomb barrier? At the same time, while such similes are marvelous, they should not be mistaken for metaphors or identities. One thing is not the other thing. I cannot abide people who think that quantum mechanics "proves" Zen buddhism. Some things are structurally similar to other things, so that if you understand the structure of one, you can understand what is meant by comparing it with the other. This is the full extent of what I mean when I say that progress in the visual arts is like the series of prime numbers.


  1. You are so smart. S-M-R-T.

  2. Wow, thanks Amy! Perhaps what you're observing is related to my purchase recently, at Target, of a pair of S-M-R-T-P-N-T-S.


  3. I am a math major by education, actuary by profession. I came across this blog accidentally thanks to the post about wit and Aliens Werses Predators. I know absolutely nothing about art (and, generally, have no interest in it) but I can't stop reading this blog! Your ideas and observations about the technical and..should I say, psychological? emotional? aspects of art are insanely interesting. And I must say, I completely enjoy your use of math as both both a conceptual and a physical tool. Whether it's the convexity of the lines of a model's back or increasing rarity of prime numbers, you provide new and interesting ways for me to think about art.

    Awesome job.


  4. Sara -

    Thanks so much for taking the trouble to leave this note! I am very happy to be able to write about art in such a way as to engage your interest - where's the challenge in making art fun for painters to read about? When I was a kid, my neighbor Ben, who was the same age as me, wanted to be an actuary when he grew up. He was better at math than I was. He did become an actuary, but he found it was not as exciting as he thought it would be, and when last heard from was a professional gambler in Monte Carlo. I guess that's beside the point, but I am very happy you're enjoying this blog, and touched to know about it. Thank you!