MAGNIFY Computer-generated prints with arrow showing the technique of magnifications of details. Signed by Benoît Mandelbrot. From the Collection Aliette Mandelbrot Image: Courtesy Bard Graduate Center
The symbolic language of mathematics can be beautiful—but even more striking are some of the patterns and forms that arise from the visual representation of math.
The power of mathematical images attracted Nina Samuel, an art historian, to investigate how visual information can inform scientific discoveries. "Images are not only a by-product but at the core of science," says Samuel, a visiting assistant professor at the Bard Graduate Center in New York City. She gained access to hundreds of papers and images in the office of mathematician Benoît Mandelbrot soon after his death in 2010. "I found images I didn’t expect to find," she adds.
A selection of those images, courtesy of Aliette Mandelbrot, the late mathematician's wife, make up an exhibit, The Islands of Benoît Mandelbrot: Fractals, Chaos and the Materiality of Thinking, now at the center's Focus Gallery in Manhattan. The display opened early this fall and will run until January 27, 2013.
The exhibit features a wealth of previously unpublished work: printouts of experiments on polynomial equations, found images such as magazine ads and photographs that reminded Mandelbrot of fractals as well as drawings from other mathematicians. Samuel curated the show based in part on her PhD thesis in art history at Humboldt University of Berlin.
Mandelbrot, a Polish-born French and American mathematician, invented the word "fractal" in 1975 to describe complex shapes that remain so at different levels of scale and show self-similar patterns. Fractal geometry is often found in the jagged edges of nature—ferns, clouds and mountains, for example—but the artificial fractal images that most resemble these forms are produced by trial and error. Some computer-generated fractal mountains do not resemble any rocky spires of earthly geology—they are crazed pinnacles of implausibility. As Samuel explains, “You can make the connection to nature if you want to, but you really need to want to.”
Images from the field of fractal geometry entered popular culture in the mid-1980s via wildly colored, psychedelic images of a fractal that Mandelbrot investigated and now bears his name, the Mandelbrot set.
In his memoir, published after his death, Mandelbrot wrote, "I don't feel I ‘invented’ the Mandelbrot set: like all of mathematics, it has always been there, but a peculiar life orbit made me the right person at the right place at the right time to be the first to inspect this object, to begin to ask many questions about it, and to conjecture many answers."
To demonstrate the process of discovery, Samuel intends viewers to first see the back wall of the small exhibit room, which is covered with "The White Series." The series is Mandelbrot and his team's effort to visualize complex polynomial sets. While working on these problems, the team was able to see solutions by running the equations through a computer. A small selection of those printouts is arrayed on the back wall of the exhibit. The exhibit brochure explains, “Seeing patterns in these pictures helped Mandelbrot decide how to adjust the input of data. Forming an important part of Mandelbrot’s thought process, these changing forms provoked new questions and led to new discoveries.” The spread hints at this, and with a little imagination the viewer can see the process—some sheets show cloudy, bubbling forms boiling up from the white page. Others show hints of shapes that foreshadow the iconic Mandelbrot set. Still others are barely freckled with dots.