This image, generated from a relatively simple mathematical formula, has become iconic and permanently connected with the man who identified it: mathematician Benoit Mandelbrot. But its iconic nature has ended up leaving it an awkward no-man’s land, dismissed by art critics as kitsch, and divorced from the underlying mathematics that generated it. Now, a small exhibit in New York City is attempting to place the Mandelbrot set and other mathematical constructs back in their original context: one that’s part of a long history of visualizations playing a key part in the creative process of math and science.
Even though that sounds like a tall order, it’s all handled in a small space on an upper floor of the Bard Graduate Center in Manhattan, where the exhibit will be on display until January. “The Islands of Benoit Mandelbrot” has been curated by Nina Samuel, a visiting professor who has a background in the histories of science and art.
Enlarge / Some of the many fractal islands on display at the exhibit. Samuel told Ars that she was first attracted to the topic precisely because the images generated from Mandelbrot sets and their relatives have been derided as a cliché. What was supposed to have been a short research project ended up a PhD thesis. The exhibit is filled with works she’s found in Mandelbrot’s own papers, those of German mathematician Otto E. Rössler, and other material from Edward Lorenz held by the Library of Congress.
From pens to programmers
The works on display from the three mathematicians capture a time when visualization itself was changing dramatically. The oldest come from Lorenz, and it consists of simple line drawings on graph paper. Lorenz is famous for his role in the development of chaos theory, which was popularized through ideas like the butterfly effect. But Lorenz found that, while chaotic systems are extremely sensitive to initial conditions, many had a tendency to gravitate towards a limited set of conditions. For example: it’s impossible to predict the weather in New York on a given July day, it’s safe to expect that it will be warm.
These likely states are called “chaotic attractors,” and the exhibit contains Lorenz’ sketches of their three-dimensional representations, complex mixes of straight lines and curves, with dashed lines attempting to represent their three dimensionality.
Enlarge / An attempt at representing a chaotic attractor in two dimensions. Rössler’s work contained no shortage of sketches, though he tended towards colored pens and would sometimes perform revisions using bits of white tape to block over things that no longer satisfied him. But Rössler was working during the dawn of the computer age, and the exhibit shows off one of his early efforts: a simple visualization, done with an analog computer—but in stereo to attempt to capture the three-dimensionality of the underlying subject.
Mandelbrot did sketch things, but he was part of the first generation of mathematicians who relied heavily on computers. This was no small challenge, given that he didn’t actually know how to program them himself. That’s one of the reasons he started a long-term collaboration with IBM’s Watson Research Center, where he had access to people who could turn his ideas into computer code, and high-end output devices to bring them to life. There’s a wall full of that output on display in the exhibit, from a wall full of rough drafts of the classic Mandelbrot set to IBM ad copy of some artificial fractal landscapes.
Visualization as science and math
The fact that the need for visualization transcended a change in technology should probably speak to its central role. But both Samuel and the material she’s gathered made that point explicit. Samuel described how some prints of Mandelbrot sets had small, individual pixels that might have been the result of an imperfect printing process. But, when zoomed in, each of these pixels represented a fractal world of its own. Not only did these visualizations reveal one of the central features of Mandelbrot sets, but they implied something about the mathematical system itself. Mandelbrot thought these speckles were islands, unconnected to some of the larger patterns around them and (erroneously, as it turned out).
(These disconnected dots were very important to him. In one book, the editor removed some of the apparent noise, assuming it was an artifact of the printing process. Samuel told Ars that, in copies he shared with colleagues, Mandelbrot actually added the speckles back by hand).
The visualizations also can form a key bridge between math and science. Mandelbrot also worked on patterns called Lévy flights, which describe paths where a random number of short hops are interrupted a single large leap. These Lévy flights create a very distinctive pattern when visualized (a number are on display in the exhibit). And, as people have studied foraging behavior in various animals, they’ve found that they also display the sort of series of hops and leaps typical of the Lévy flight. These parallels can be formally demonstrated mathematically, but the impetus for performing this demonstration was undoubtedly someone noticing the visual resemblance between the two patterns.
In Rössler’s case, the link to science was explicit. There’s a letter from him on display in which a series of hand sketches of three-dimensional surfaces are interspersed with typed notes. At some point, the text makes it clear that these complex shapes represent potential solutions to the Prisoner’s Dilemma, a key example that’s often used in game theory experiments.
Works in progress
The other thing that the exhibit makes clear is that the visualizations were part of the creative process. An entire wall is given over to a series of prints made from the famous Mandelbrot set. The zoom and location of the print relative to the entire image are varied in each one and, in many cases, it’s a visual disaster—only a few random specks appear, with no pattern discernible. Samuel said these prints were like rough drafts, helping Mandelbrot understand what the equations he was working with said in various ways, and seeing how they varied in space and scale in a way that couldn’t be done easily (if at all) by sketching on paper.
Enlarge / Mandelbrot mixed computer output and hand-written notes when trying to come to grips with the math. Is any of it art? There’s clearly a lot of technical skill behind the material, and a lot of thought and passion went in to producing some of it. With a few exceptions—like a dramatic, colored graph that ended up in the cover of Scientific American—it wasn’t intentionally produced as art, but it’s clear that’s less of a barrier to things than it might otherwise be.
Since art is so subjective, it’s probably best that anyone who wants to have that debate visit the exhibit itself. It’s a small one, so it won’t take long to digest the whole thing. But it could leave you considering the intersection between math, science, creativity, and visual arts for quite a bit longer than it takes to admire the displays.