Solving a math problem to create art - PNAS

Solving a math problem to create art Stephen Ornes, Science Writer

Mathematician Robert Bosch never intended to become an artist. But 15 years ago, he started looking for ways to engage his students in optimization research, his field of expertise. Optimization is the mathematical quest for the best way to do something, from finding the shortest distance between two places to figuring out the best way to pack a suitcase. It often involves calculating the highest or lowest value of something. The applications are farreaching. To Bosch, they also offered a pleasing aesthetic. “I wanted to convince my students that this material I teach is beautiful and incredibly applicable,” says Bosch, who teaches at Oberlin College in Ohio. “My mission was to show them that pretty much any field you could think of has optimization applications.” So Bosch went looking for examples in areas that seem as far from mathematics as one can get. He settled on visual art. There may not be any obvious overlap between the two pursuits, but Bosch figured that if he could show his students how optimization methods could produce art, then maybe he could convince them the field is applicable almost anywhere. “It became an obsession,” he says. And it paid off: Bosch is now known among mathematicians—and the math–art subset of that community—for his line drawings, mosaics, and sculptures created using solutions to optimization problems. He’s also organized showings of mathematical art at galleries and conferences, and has given lectures to audiences ranging from elementary school children to the mathematically curious at the Museum of Math in New York.

Finding the Path Most of Bosch’s artwork is derived from the Traveling Salesman Problem (TSP), the best known and most thoroughly studied example in optimization research. Like all good problems, it’s easy to state and difficult— perhaps impossible—to solve. Imagine a traveler who has to visit each city on a given list only once, returning to the starting point at the end. What’s the shortest route? If you marked targeted cities on a map, they’d look like a dot-to-dot puzzle that asks you to draw the shortest possible total line and without the numbers provided. For a short list of cities, the TSP is easy to solve: simply measure all possible paths and pick the shortest. As the list grows, so does the difficulty. Finding the shortest journey through just 10 cities requires comparing hundreds of thousands of possibilities; that pick-theshortest-trip strategy becomes untenable when the

www.pnas.org/cgi/doi/10.1073/pnas.1617584113

Robert Bosch’s optimization artworks have included a rendering of the Mona Lisa, which he starts by identifying the points or “cities” (Left) through which the Traveling Salesman route will run (Right). Original image courtesy of Shutterstock/ Oleg Golovnev and modifications by Robert Bosch.

city list reaches into the hundreds or thousands. Mathematicians can’t solve the problem in a reasonable amount of time for any given number of cities, and they don’t know such a strategy could ever be made. Mathematicians can, however, check that one solution is better than another; and pretty good solutions suffice in most fields. The TSP might give mathophobes a headache; to Bosch, it leads to artistic inspiration. To create his pictures, Bosch uses a computer program to convert a drawing (or painting or photograph) into a scattering of dots. Densely packed black dots might be used to represent darker areas of a drawing, whereas lighter areas have dots space farther apart. Gray hues are achieved with a technique called half-toning. Bosch has worked with collaborators—first his student Adrianne Herman, and later Craig Kaplan, a computer graphics expert at the University of Waterloo, in Canada—to refine this approach. Then comes the math. For that, Bosch uses an algorithm that traces an optimal, nonoverlapping TSP path through the dots. The algorithm doesn’t necessarily find the best route, but it finds a pretty good one. The line segments complete the art, and Bosch’s goal is that the line segments draw a convincing representation

PNAS | December 27, 2016 | vol. 113 | no. 52 | 14873–14874

SCIENCE AND CULTURE

SCIENCE AND CULTURE

uses math to create art by designing sculptures and puzzles generated on 3D printers. “And if you produce something interesting and beautiful,” he says, “it makes everybody else interested.” Bosch’s work has implications beyond aesthetics, says Segerman. When computer scientists write algorithms to solve optimization problems, they need big datasets to test out their strategies. Bosch’s method shows how a work of art can be turned into such a dataset. “Take the Mona Lisa,” says Segerman. “Those points are a new set of data to run your algorithms on. There’s some value even there, in having new datasets and problems to attack.”

Picture Perfect Here, Bosch used the TSP optimization approach to create a rendering of the eye of his wife, Kathy. Image courtesy of Robert Bosch.

of the original. As a rule, geometric optimal TSP routes never cross themselves, which means that Bosch’s pictures comprise a single line that ends where it started, in what mathematicians call a simple closed curve. One of Bosch’s first pieces was a TSP-derived replica of Leonardo da Vinci’s Mona Lisa. Bosch has also reproduced works by Andy Warhol, including the pop artist’s self-portrait and paintings of a Campbell’s soup can. Bosch is currently challenging himself to create the same quality of TSP art, but using fewer and fewer dots, as a way to do more with less, something like a music composer who limits himself to only a few notes. He wants to create art that’s less reliant on fast computers and more reliant on mathematical modeling. That desire, Bosch says, is universal. “What’s the best we can do with the constraints we have?” he asks. “It’s a fundamental human problem.” Mathematician Henry Segerman, of Oklahoma State University in Stillwater, says art gives people a reason to engage with the math, especially if they typically shy away from the subject. An artistic objective provides a tangible goal for an abstract quest, says Segerman, who

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Bosch has also used optimization problems to arrange hundreds of dominoes into portraits of subjects, including German mathematician Carl Friedrich Gauss and, more recently, Frankenstein’s monster. Bosch is finishing a book, tentatively titled Optimization Constraints and Design, about the optimization field that introduces readers to problem-solving methods, discusses current applications, and outlines his approach to creating art from math. He has also begun designing TSP-derived sculptures for 3D printers. One recent piece was a sculpture of his wife’s eye, visible only if an observer is looking at it directly. Otherwise, it looks like an abstract shape (see figure). Bosch says his efforts have paid off in the classroom. TSP art provides an access point to start thinking about optimization, and many of his former students have gone on to make their own art. In August, three former students presented original mathematical artwork in Jyväskylä, Finland, at the Bridges conference, an annual celebration of the intersection of mathematics and the arts. Former student Hank Guss, who now works with the Museum of Mathematics, lauds Bosch’s teaching approach. “His optimization work—even without the art—demonstrated the applicability of the coursework,” recalls Guss. “But seeing the art alongside the math gave me the inspiration to create my own.”

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