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Growing passion for applied mathematics 

Cornell University professor demonstrates geometry with crochet

By Karen Herland

When linguistics professor Charles Reiss introduced Cornell University mathematician Daina Taimina’s lecture on Nov. 6, the list of event sponsors hinted at the interdisciplinary treat the York Amphitheatre audience was about to receive.

Taimina’s presentation, on expressing complex geometric models through crochet, was presented by the Fibres Program, the Department of Studio Arts, the Department of Mathematics and Statistics and the Linguistics Students Association — all under the auspices of the Special Individualized Program of the School of Graduate Studies.

For Reiss, who has been involved with the SIP program since arriving here 14 years ago. “It would be hard to find a better topic to convey the spirit of SIP.”

The SIP program allows students to select three supervisors from across disciplines to guide them through complex research that does not fit within a single field, or that goes beyond the graduate programs currently available at Concordia.

Reiss, himself studied mathematics before turning to linguistics (“both involve cognitive science,” he explains). “Through SIP, I’ve supervised theses involving psychology, philosophy, computer science, mathematics and education.”

The program is increasingly popular, with 40 new students being accepted in the last year. “Everyone we accept chooses to come to Concordia.”

That spirit of intellectual curiosity was evident in the hall during Taimina’s presentation. The room was filled with mathematicians and fibres students who listened to the lecture while working on their own knitting projects.

Taimina offered a very personal presentation of her journey from Latvia to the U.S. and from mathematician to artist.

It was about a dozen years ago that Taimina conceived of using crochet to express the geometric notion of hyperbolic planes when she saw a simple illustrated model of exponential growth. The concept of exponential growth is fairly straightforward: think of that old example of having a penny on day one, 2¢ on the second day and the amount of money you collect by the end of the month by doubling each day.

Taimina has developed a more evocative example. “I just mention Bernie Madoff and Ponzi schemes and everyone understands what I mean.”

The concept of hyperbolic planes is a bit harder to grasp. In the past, mathematicians understood two basic forms for geometry, the plane and the sphere. It was in the 1820s and 1830s that two mathematicians conceived of a third structure, hyperbolic space.

It was Taimina who determined how to materially demonstrate that. “One way of understanding it is that it’s the geometric opposite of the sphere. On a sphere, the surface curves in on itself and is closed. A hyperbolic plane is a surface in which the space curves away from itself at every point.” In other words, the space is ever-increasing.

The illustrated model of exponential growth she saw reminded her of the schematic patterns she used in Europe to realize crochet projects. “I could never follow the patterns you use here, those are just words,” she said. By doubling every stitch into two, she was able to construct a hyperbolic plane. Taimina explained the early crochet rows in a project are completed in minutes, whereas the final ones can literally take hours to complete. By changing the ratio (for instance, an increase every fifth or tenth stitch, or a ratio of 2:3) the form and properties of the plane shift.

The project has been taken up by a number of groups. The Institute for Figuring in the U.S. has an online gallery of different exhibits using the basic form to create coral reefs or other displays.

She developed the idea with the help of her husband, David W. Henderson, also a Cornell mathematician. Like Taimina, Henderson was interested in developing concrete, hands-on ways for his students to learn about geometry. His book, Experiencing Geometry, offered similar approaches to mathematics education. According to Taimina, the book led to conversations, the conversations to marriage and the third edition of Experiencing Geometry was co-authored by the couple.

Henderson also spoke on Nov. 6. He presented a new curriculum for teaching geometry for the Algebra Project, an organization that teaches math skills.