Knot Theory, Experimental Mathematics, and 3D Printing

Seven and a half minutes into the Distinguished Lecture Laura Taalman gave at the MAA Carriage House on February 19, six beeps interrupted her story. The beeps came from the machine that had been humming on the table beside the lectern for half an hour before the lecture even began.

“Look, it made a knot,” Taalman observed. The James Madison University (JMU) professor then continued with telling about a student who, despite knowing nothing—initially, anyway—about matrices and determinants, collected data that hastened the progress of a mathematical research project.

In her lecture, “Making Mathematics Real: Knot Theory, Experimental Mathematics, and 3D Printing,” Taalman educated her audience—which included many middle school and community college students—about the nature of the discipline, the opportunities afforded by technology, and how to get students involved, which is when she came back to the machine that made the knot.

Nature of the Discipline

“If you’re a mathematician you already know this, but if you’re not, I’m going to tell you a secret,” Taalman said as she touched on work she and students did about whether all knots have a spiral projection. “When we don’t know the answer to a question, sometimes we don’t say ‘we don’t know’ because that sounds a little stupid. So we say, ‘That’s an open problem.’ Now it’s exciting, right? Now it’s something we can figure out.”

Another project Taalman mentioned—she and her students took on the daunting task of making a table of all the singular knots—spoke to both the necessity of facing failure in the pursuit of mathematical truth and the possibility of nonexperts making lasting contributions.

“In the future people who write papers about one-singular knots are going to call that one six-four-star,” Taalman said. “And my kids named it that.”

Teaching with Technology

Taalman’s kids, it turns out, do quite a lot. The students make cufflinks and picture frames and functional, if not tuneful, ukuleles. They discover on their own that Borromean rings cannot be perfectly circular and produce tangible evidence that there are trefoil knots without tritangent planes. (That knot that was printing as Taalman began her talk? It was one of these tritangentless trefoils.)

All of this happens because Taalman recognized the pedagogical potential of the 3D printer—and took action. First she converted a closet at the university into the JMU MakerLab, which gives faculty and students in the mathematics and statistics department access to three 3D printers. Then, to expose general education students to the technology, she conceived of, outfitted, and began teaching in the eight-station JMU 3-SPACE Classroom.

Engaging Nonmath Students Too

Although Taalman has found calculus-related uses for her university’s 3D printers—she has printed Gabriel’s horn, shells and disks to illustrate integration methods, napkin rings with different shapes but the same volume—she emphasized their ability to engage non-STEM students. She showed pictures of theater and marketing majors thrilled at the results of their first 3D print and reported that, as early as the first week of class, students were worrying about what they would do at the end of the semester when they couldn’t access the equipment that allowed them to use their new knowledge.

“So if you’re feeling unappreciated,” Taalman told the educators in the Carriage House audience, “just teach this for a little while and you’ll feel great.”

Taalman doesn’t think you need to work at a well-endowed university or be terribly tech savvy to bring the wonders of 3D printing to students. The price of the technology is going down, she noted, and the major printer manufacturers have initiatives aimed at getting K-12 educators in on the fun.

As for prerequisite knowledge, Taalman assured listeners that she didn’t know anything about 3D printing a year ago and that acquiring the know-how is not hard.

“It just takes persistence and willingness to hit your head against the wall, which is what we do in math anyway, so you’re all used to it,” she said. “You don’t even have to know what you’re doing. You just have to be willing to fail and try and let the students learn with you.” —Katharine Merow