Mathematics and Art

Nikki Graziano’s intriguing integration of mathematical curves into her photography sparked a Radar discussion about the relationship between mathematics and the real world. Does her work give insight into the nature of mathematics? Or into the nature of the world? And if so, what kind of insight?

Mathematically, matching one curve to another isn’t a big deal. Given N points, it’s trivial to write an N+1 degree equation that passes through all of them. There are many more subtle ways of solving the same problem, with more aesthetically pleasing results: you can use sine functions, wavelets, square waves, whatever you want. Take out a ruler, measure some points, plug them into Mathematica, and in seconds you can generate as many curves as you like. So finding an equation that matches the curve of an artfully trimmed hedge is easy. The question is whether that curve tells us anything, or whether it’s just another stupid math trick.

It's not a simple question. A few weeks ago, I came across a brilliant essay from 1960, The Unreasonable Effectiveness of Mathematics in the Natural Sciences. The point of the essay is that while mathematics is an incredibly abstract discipline, with no apparent contact to the real world, against all reasonable expectation mathematical results have proven to be invaluable in studying the physical world. Why should purely mathematical results be relevant in everything from population studies to quantum physics? Why do the Fibonacci series, fractals, and other abstract mathematical concepts show up in nature?

My favorite example is the square root of -1. In grade school, we’re first told that there’s no such thing. Then, sometime in high school, we’re told “well we lied, but the square root of -1 is imaginary, call it i”. Except it isn’t imaginary. The square root of -1 makes a profound connection between the world of exponentials and the world of circles. And the world of circles is the basis for every every wave phenomenon we know about. Much of physics, and just about all of electrical engineering wouldn’t exist without i–or would be so hampered by grossly complex mathematics that we couldn’t explore it. But even though it’s what makes the world go ’round, you can never point me to the square root of -1. It has extraordinary power, both mathematically and physically, but it’s “imaginary.”

So–back to the connection between art and math. Is she showing a deep connection between the objects she photographs and the mathematical world, or is it just a math trick? And is that deep connection necessary? A number of years ago, I went to a concert of the works of Gy├Ârgy Ligeti. Ligeti gave a talk in which he said the works were inspired by Benoit Mandelbrot’s work–though he had not attempted to replicate Mandelbrot curves in the music itself. You didn’t need to understand fractals to appreciate the music, and upon hearing the music you wouldn’t say “damn, is that a fractal I just heard?” The music didn’t attempt to represent the math, and the math doesn’t define the music.

In Graziano’s photography, as presented in Wired, I find the relationship more troubling. The math is imposed on (both in the sense of “superimposed” and in the sense of a power relationship) the natural scenes. Does the math inform the natural world? Or does it usurp it? I think this ambiguity is both healthy and intentional. But what’s less healthy is that I don’t find the math that convincing. In the first of the Wired pictures, the mathematical curve is (log(x)/2)^sin(x). (Or so I think. The equation is typeset poorly; this interpretation comes closes to matching the picture). It’s difficult to believe that this log^sin relationship corresponds to nature. And it blows up outside of the domain used in the photograph–an almost certain sign of ham-fisted curve fitting run amok. The second photo, some clouds that match a Gaussian bell curve, is much more convincing: I can believe that there’s a deep relationship between a Gaussian curve and cloud formation. I find the fifth and sixth photos particularly interesting. The fifth shows a canopy of vines, with a hyperbolic paraboloid mapped on top of it (z = y^2 -x^2). Wikipedia says that this curve is useful in the design of large roofs because it is inherently strong and allows the use of lightweight buildng materials. I’m more than willing to guess that a canopy of vines would take this form. The final picture in the series is similar: another vine or bush (right now, I wish I were better at horticulture), overlayed with a surface that looks like a paraboloid, but is really sin(3*y*x^2/4). An interesting choice, but it’s hard for me to believe that there’s any intrinsic relationship between this surface and the way plants grow. Ligeti’s music was “inspired” by Mandelbrot; Mandelbrot didn’t force his equations into an analysis of the score. In Graziano’s case, the math frequently appears to be “inspired” by the photography in a way that feels false to me.

But ambiguities abound. Math has been “unreasonably effective” throughout history; could that be happening here? The first picture doesn’t look like a wild scene; I think it’s a garden, and that the plants were possibly pruned by a gardener. So this odd log/sine combination might not be a statement about how plants grow, but a statement about human aesthetics (like the Golden Ratio). And the final photograph–while I don’t believe the sin(xy^2) relationship says much about how a tangle of vines hangs together, plant growth depends strongly on sunlight. The sun travels in a more-or-less circular path, and in a large, overgrown tangle, it’s at least plausible (though I’m not convinced) that this curve says something about exposure to the sun, given the sun’s path through the sky and the effects of shadows.

Are mathematical complex effects hiding in plain sight? In the curve trimmed by a careful gardener, or in the seemingly random growth of a tangled vine? Absolutely. I’m not convinced that these effects are always the ones Graziano is showing us. Is Graziano’s math arising out of the natural world, or is it imposed upon it? I don’t know whether she’s doing something profound, or just being clever, and this ambiguity bothers me. But it has made me think, and that’s certainly the function of art.

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