Art And Mathematics Pattern A Professor's Career

UW's Branko Grunbaum Receives High Honors For Work On Polytopes

By Desiree Willis

"Have you seen these?" he asks, reaching for a brown plastic shopping bag. He pulls out several colored orbs that look like oversized Christmas tree ornaments. "They're called Tamari balls," explains the mathematics professor, turning them over in his hands.

Each sphere is decorated in geometric patterns from yards of silk thread wound in different colors. Elaborately woven, they are the size of grapefruit. As he rotates them in his hands, he says, "They are a Japanese tradition; Japanese and Japanese-American friends gave me these as gifts."

Grunbaum holds up two smaller ornaments, even more decorative. They remind me of a frosted wedding cake. Looking down at the jewels in his hands, he says, "This is an example of how art relates to mathematics."

Branko Grunbaum, professor of mathematics at the University of Washington (UW), has devoted his academic interests to the melding of art and science since the 1960s.

Light streams into Grunbaum's office, illuminating shelves of old math and physics titles together with newer art textbooks. Stuffed here and there in corners around the room like cobwebs are geometric shapes made from ball-and-peg chemistry sets. On his desk are more paper shapes clustered in corners. He picks up a paper shape and says, "A polytope is something of any dimension. All of the polyhedron is open to one side of any plane." He runs his hand along one plane of the shape, demonstrating that there are no concave crevices anywhere on its surface.

These shapes have applications in nature, he explains. Fullerenes, specific forms of carbon, are, at the molecular level, convex polytopes that look like tiny soccer balls. Their facets are composed of 12 pentagons and many hexagons.

He looks up, pulls a small tan book from the shelves, and shuffles through the pages. He rifles through pages of theorems and black-and-white drawings. Grunbaum's first book, Convex Polytopes, was published in 1967.

Fast forward to 2005, when Grunbaum was honored with an American Mathematical Society Steele Prize for this book, recently published in its second edition. The award recognizes Grunbaum's book as a standard mathematical reference in use for over three decades.

Grunbaum's research on convex polytopes has carried him from Hebrew University in Jerusalem to universities around the U.S., including the Michigan State University at Lansing and the University of Washington.

He springs up from his chair, grabs another title from the shelves. "You've heard of M.C. Escher?" he asks. Opening a page to Escher's famous black-and-white lithographs, he says, "Escher was interested in tiling–how one can fill the plane with different shapes."

While Escher had an interest in organic tilings, and often used animal or plant designs, Grunbaum's interests lie in geometric tilings like those woven into the Tamari balls. There is geometry behind the patterns of tilings in many cultures, and Grunbaum is fascinated with the mathematics behind art.

Like Escher, many tilings in cultures, from Moorish art to Native American pottery and European tapestries, use common symmetries in their designs. Interlace patterns in many tilings are linked to weaving practices in cultures, while some patterns have religious significance within a society.

Cultures around the world employ Escher's type of organic tilings, using repeated designs of animals or plants. Repeated designs using symmetry and interlacing can be explained using mathematics.

There are patterns that people in different professions know about, Grunbaum says–from archaeologists to zoologists. But there is not much communication between fields. Patterns familiar to archeologists within a culture often remain virtually unknown to mathematics researchers.

Grunbaum first became interested in patterns and tilings when he attended a meeting in England. He spent a large amount of time in the chemistry library and art library, flipping through back issues of journals. Many issues contained detailed patterns and tilings he had never seen before.

Grunbaum closes the Escher volume, then rises quickly again from his office chair. "If I can show you a pattern," he asserts, reaching for another title, "it is better than showing a bunch of calculations." He finds a small, battered book with a light blue cover, and cracks open the yellowed pages.

Flipping through chapters of calculations and proofs, Grunbaum notes that tilings have an educational value. They aid in visually teaching geometry in a way that equations and words cannot. He has taught mathematics at five universities, and came to the University of Washington in 1966, where he has settled and continues to teach classes today.

His interest in mathematics and physics has been lifelong. Born in Yugoslavia in 1929, Grunbaum grew up in Europe during World War II. Grunbaum's own experiences are cross-cultural, and it is these worldwide influences that drive his work. Like the cardboard shapes cluttering his office and the tamari balls in the plastic shopping bag, mathematics can be thought of as elucidating patterns. He has culled examples from across continents in order to display similarities of design that mathematics can explain.

He thinks for a moment, his eyes lost in thought. He leans back easily in his chair and stares out the window. "People should be concerned with ideas rather than details," he says finally. "Very often, it is the simplest things that people get wrong."

And this is perhaps the overall message of Grunbaum's body of work. He believes that modern mathematics can often be too focused on details, and in so doing researchers miss the patterns and possibilities open to them. By focusing on ideas, Grunbaum's work transcends cultures and detects the common thread that links us all.

Desiree Willis is a senior majoring in both neurobiology and technical communication at the University of Washington.

Images

Top: Branko Grunbaum is a professor of mathematics at the University of Washington. His main interests are in the mathematics behind patterns, tilings, and geometric shapes called convex polytopes.

Middle: Grunbaum's hand-made models of convex polytopes display their unique properties.

Bottom: Escher-type rug patterns use repeated designs of animals or plants to create an organic motif. These types of designs can be found in cultures around the world.

Photos: Branko Grunbaum/UW