Former SLC Professor Makes Mathematical Knitted Crafts

These are samples of Dr. Belcastro's mathematical knitting. Möbius bands, which act as both teaching tools and fashion accessories. Photo courtesy Dr. Sarah-Marie Belcastro

Former professor of Mathematics, Dr. Sarah-Marie Belcastro, recently had an article selected by the Princeton University Press for their annual series “The Best Writing on Mathematics” for this year. The article, “Adventures in Mathematical Knitting,” was originally published by the bimonthly science and technology magazine, American Scientist. In her article, Dr. Belcastro explains how she has used her knitting techniques to create mathematical surfaces and shapes such as Möbius bands and Klein bottles.

Mathematical objects like Klein bottles can be hard to visualize due to the fact they are four-dimensional objects.

“Conceptually it’s simple, you just have four axes instead of three” said Dr. Belcastro as she was explaining the concept of four-dimensional objects. “Trying to visualize it is another thing entirely.”

Klein bottles and Möbius bands are non-orientable surfaces, meaning that the two sides of the objects “flow into each other,” as Dr. Belcastro put it. “You have to make it so that the objects have neither a front nor back,”she added.

The finished objects make good teaching aids due to their flexibility, and they can be physically manipulated unlike their computer-generated counterparts. Some knitted objects can even be worn as scarves, bracelets, and other types of fashion accessories.

The idea of knitting such complex shapes came to Dr. Belcastro quite naturally while she was in graduate school. She took two things that she was passionate about, knitting and mathematics, and was able to use her craft to bring such shapes into reality.

“I was sitting in a math class and thinking ‘I wish I could feel one of these things,’” said Dr. Belcastro recounting how she first thought of the concept. “I happened to be knitting, and maybe I can knit one and I thought about how to do it.”

Dr. Belcastro admitted in her article that she was not the first person to come up with the idea of using knitting as a tool to make mathematical objects. She cited the earliest example of knitted mathematical surfaces made by Scottish chemistry professor Alexander Crum Brown in her article.

When asked if she was excited about her article being selected for the publication, she admitted that she was not particularly excited about it because in fact this was not the first time her work was selected for “The Best Writing on Mathematics.” Another article she wrote, “Dancing Mathematics and the Mathematics of Dancing,” was featured in the 2012 edition of the Princeton publication as well.

Dr. Belcastro collaborated with Dr. Karl Schaffer, a math professor and dance choreographer at De Anza College, on the article, which was published by the magazine, Math Horizons, in February 2011. The article examines the relationship between mathematics and dance by exploring how mathematical principles like symmetry, geometry and topology play a role in dance choreography.

Dr. Belcastro dances and choreographs herself. In fact, almost every year she and Dr. Schaffer meet at the annual mathematics conference, Joint Mathematics Meetings, and discuss putting math in their choreography. They decided to put together a demonstration for the entire conference about the mathematics of dance. They also incorporated demonstrations about how dance can be used to demonstrate mathematical theorems and principles.

“We did so much work to create this, and we had made a script for ourselves and we thought maybe we should turn this into an article,” said Dr. Belcastro.

These days, Dr. Belcastro is still keeping herself busy. She is currently a Research Associate at Smith College, the Director of MathILY (a residential summer program for high school students who excel at math) and an instructor at the Art of Problem Solving school. She does hope to have another opportunity to teach at SLC again if given the chance. 

by Hugh Thornhill '15
Staff Writer and Contributing Layout Editor
hthornhill@gm.slc.edu