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Solving a math problem to create SCIENCE AND CULTURE Stephen Ornes, Science Writer

Mathematician Robert Bosch never intended to be- come an . 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 far- reaching. To Bosch, they also offered a pleasing aes- thetic. “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 Robert Bosch’s optimization artworks have included a rendering of the Mona Lisa, could produce art, then maybe he could convince them which he starts by identifying the points or “cities” (Left) through which the the field is applicable almost anywhere. “It became an Traveling Salesman route will run (Right). Original image courtesy of Shutterstock/ obsession,” he says. Oleg Golovnev and modifications by Robert Bosch. And it paid off: Bosch is now known among — – mathematicians and the math artsubsetofthat city list reaches into the hundreds or thousands. — community for his line drawings, mosaics, and sculp- Mathematicians can’t solve the problem in a reason- tures created using solutions to optimization prob- able amount of time for any given number of cities, lems. He’s also organized showings of mathematical and they don’t know such a strategy could ever be art at galleries and conferences, and has given lec- made. Mathematicians can, however, check that one tures to ranging from elementary school solution is better than another; and pretty good so- children to the mathematically curious at the lutions suffice in most fields. of Math in New York. The TSP might give mathophobes a headache; to Bosch, it leads to artistic inspiration. To create his pic- Finding the Path tures, Bosch uses a computer program to convert a Most of Bosch’s artwork is derived from the Traveling drawing (or or photograph) into a scattering of Salesman Problem (TSP), the best known and most thoroughly studied example in optimization research. dots. Densely packed black dots might be used to Like all good problems, it’s easy to state and difficult— represent darker areas of a drawing, whereas lighter perhaps impossible—to solve. Imagine a traveler who areas have dots space farther apart. Gray hues are has to visit each city on a given list only once, returning to achieved with a technique called half-toning. Bosch has — the starting point at the end. What’s the shortest route? worked with collaborators first his student Adrianne If you marked targeted cities on a map, they’d look Herman, and later Craig Kaplan, a computer graphics like a dot-to-dot puzzle that asks you to draw the shortest expert at the University of Waterloo, in Canada—to possible total line and without the numbers provided. refine this approach. For a short list of cities, the TSP is easy to solve: simply Then comes the math. For that, Bosch uses an al- measure all possible paths and pick the shortest. As the gorithm that traces an optimal, nonoverlapping TSP list grows, so does the difficulty. Finding the shortest path through the dots. The algorithm doesn’t neces- journey through just 10 cities requires comparing sarily find the best route, but it finds a pretty good one. hundreds of thousands of possibilities; that pick-the- The line segments complete the art, and Bosch’sgoalis shortest-trip strategy becomes untenable when the that the line segments draw a convincing

www.pnas.org/cgi/doi/10.1073/pnas.1617584113 PNAS | December 27, 2016 | vol. 113 | no. 52 | 14873–14874 Downloaded by guest on October 1, 2021 uses math to create art by designing 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 , says Segerman. When computer scientists write algo- rithms to solve optimization problems, they need big datasets to test out their strategies. Bosch’smethod shows how a work of art can be turned into such a dataset. “Take the Mona Lisa,” says Segerman. “Those points are a new of data to run your algorithms on. There’s some value even there, in having new datasets and problems to attack.”

Picture Perfect Bosch has also used optimization problems to arrange Here, Bosch used the TSP optimization approach to create a rendering of the eye hundreds of dominoes into portraits of subjects, in- of his wife, Kathy. Image courtesy of Robert Bosch. cluding German mathematician Carl Friedrich Gauss and, more recently, Frankenstein’smonster.Boschis of the original. As a rule, geometric optimal TSP routes finishing a book, tentatively titled Optimization Con- never cross themselves, which means that Bosch’s pic- straints and Design, about the optimization field that tures comprise a single line that ends where it started, in introduces readers to problem-solving methods, dis- what mathematicians call a simple closed curve. cusses current applications, and outlines his approach to One of Bosch’s first pieces was a TSP-derived replica creating art from math. He has also begun designing of ’s Mona Lisa.Boschhasalso TSP-derived sculptures for 3D printers. One recent ’ reproduced works by Andy Warhol, including the pop piece was a of his wife s eye, visible only if an artist’s self-portrait and of a Campbell’ssoup observer is looking at it directly. Otherwise, it looks like can. Bosch is currently challenging himself to create the an abstract shape (see figure). same of TSP art, but using fewer and fewer dots, Bosch says his efforts have paid off in the classroom. as a way to do more with less, something like a TSP art provides an access point to start thinking about composer who limits himself to only a few notes. He optimization, and many of his former students have wants to create art that’s less reliant on fast computers gone on to make their own art. In August, three former and more reliant on mathematical modeling. That desire, students presented original mathematical artwork in Bosch says, is universal. “What’sthebestwecandowith Jyväskylä, Finland, at the Bridges conference, an annual the constraints we have?” he asks. “It’s a fundamental celebration of the intersection of mathematics and the human problem.” . Former student Hank Guss, who now works with Mathematician Henry Segerman, of Oklahoma State the Museum of Mathematics, lauds Bosch’steaching University in Stillwater, says art gives people a reason to approach. “His optimization work—even without the engage with the math, especially if they typically shy art—demonstrated the applicability of the course- away from the subject. An artistic objective provides a work,” recalls Guss. “But seeing the art alongside the tangible goal for an abstract quest, says Segerman, who math gave me the inspiration to create my own.”

14874 | www.pnas.org/cgi/doi/10.1073/pnas.1617584113 Ornes Downloaded by guest on October 1, 2021