Faced with the challenge of expressing a highly technical math idea called negatively curved space, Barbara Nimershiem discovered her solution in an unlikely medium.
The Franklin & Marshall associate professor of mathematics created a quilt featuring three mutually interlocking rings, a distinctive symbol that has been part of the Ballantine beer brand since before World War II and appeared on the inside cover sleeve of classic rock band Led Zeppelin's fourth album in 1971.
The rings are known in mathematics as Borromean rings, named for Borromeo, a centuries-old aristocratic family in Northern Italy. What is distinctive about the rings is that no two are linked; yet all three are linked together.
The quilt is a physical way of illustrating negative curvature -- or hyperbolic space -- in three dimensions, Nimershiem said. She first presented the quilt at the American Mathematical Society's 2014 Joint Mathematics Meetings in Baltimore, the largest such meeting in the world. She presented it again in February 2014 at the F&M and Millersville University Joint Math Colloquium. To demonstrate how it works, she tugs on a series of strings that pull the fabric into a three-dimensional shape, revealing the hyperbolic space around the rings.
"The field of knot theory is over a century old, but only in the last 40 years have we adopted a geometric perspective," said Nimershiem, who studies hyperbolic geometry. "With geometry, instead of looking at the knots and links, we look at the spaces around them, which gives us a richer understanding."
Professor of Mathematics Annalisa Crannell, chair of the Mathematics Department, said the quilt is an artistic and mathematic marvel and makes for a unique teaching tool.
"Artists often refer to negative space -- the gaps between and behind things -- and the quilt shows exactly that; everything else in the world that is not the Borromean rings," she said. "And by knowing what that 'everything else' looks like, we understand the links between the rings better than before."
Nimershiem was inspired to make the quilt by the late mathematician William Thurston, a pioneer in the field of low-dimensional topology -- the study of shapes and spaces with four or fewer dimensions. Thurston showed that with certain exceptions, knots and links have complements that are hyperbolic.
In June 2014, Nimershiem attended the Thurston Legacy Conference at Cornell University, where she displayed the latest version of the quilt in a curated exhibit of digital artwork and mathematic models. She plans to use the quilt in her classes to demonstrate geometric properties and spatial relations of shapes and figures.