Visit most college math classes and you'll probably see students punching numbers into high-end calculators or sketching diagrams on graph paper.
If you'd dropped in on a course at McDaniel College last month, though, you might think the new tools of the trade were crocheting hooks, scissors and brightly colored yarn.
As part of a seminar he offered during the Westminster school's three-week "Jan Term" program, math professor Ben Steinhurst had his students spend 15 hours of their class time crocheting pieces of fabric that looked part floppy doily, part curly-edged beret.
"I had no idea how this related to math when I signed up," said Daniel Cornaggio, a junior math major from Hampstead, as he looked up from his circular scarlet piece one recent morning. "Now that I'm doing it, it makes sense. And these do look kind of cool."
The projects were the focus of Crocheted Hyperbolic Geometry, an elective that used the art of crocheting to help students grasp the elusive properties of hyperbolic geometry, a variation on the field that allows modern mathematicians to deal with more of the world's irregularities than the father of geometry, the Greek mathematician Euclid, ever did.
"Euclidean geometry is ideal; it's based on principles that don't exist," said Jess Chaney, a physics and computer-science major from Carroll County who took the class. "Hyperbolic geometry is a way of accounting for what actually exists. These pieces exemplify its postulates."
If you've never heard of hyperbolic geometry, you're far from alone. The geometry most Westerners know is the kind Euclid devised and wrote about in his treatise "Elements" in 300 B.C.
Still widely taught and used in practical matters, Euclidean geometry is a system of spatial-mathematical thought based on five axioms Euclid asserted — for example, the notions that one can draw a straight line between two points and that a finite line can be extended to infinity in both directions.
The axioms, or postulates as Euclid called them, aren't objectively "true"— it's impossible to draw a perfectly straight line, and who ever drew a line to infinity? — but possessed enough internal logic to serve as the foundation for a mathematical system humans could use to design buildings, build bridges and navigate their surroundings.
"For actually going out and building things on a human scale, reality [was] as close to Euclidean geometry as we needed it to be," said Steinhurst, a Vermont native, during a break.
That, as it turned out, was good enough for just about everyone for 2,200 years. Around the turn of the 20th century, though, thinkers like Einstein shook up the system by showing that the space-time continuum we live in — far from being explainable through imaginary straight lines and planes — is curved.
"The universe is simply not Euclidean," Steinhurst said. "The geometry of that reality cannot be uniform or perfect."
Hyperbolic geometry grew out of that realization, as did the discovery of its relationship to crochet.
Steinhurst's course wasn't all about the yarn. He spent an hour each day delving into the history of the field. He recently spoke of early pioneers like the Russian mathematician Nikolai Lobachevsky, who concluded in the mid-1800s that even though Euclid's geometry was consistent and practical, it was just one of many possible geometries, each of which could shed light on reality in a different way.
What would happen, he and some contemporaries wondered, if they started out with axioms more attuned to reality than the ones Euclid claimed? He had asserted, for example, an idea called the parallel postulate — that if one draws a line, then a point beside that line, one can then draw one and only one line through that point which is parallel to the first line.
Mathematicians had debated the axiom for centuries, noting that even as it helped tie up a number of loose ends in Euclid's geometry, it limited that geometry's scope. They tossed the parallel postulate and replaced it, asserting instead that an infinite number of such parallel lines could be drawn. Hyperbolic geometry was born.
Lobachevsky and others quickly discovered there was one shape that reflected the properties of this geometry. It wasn't a flat surface but rather a hyperbolic plane — a spiral, in effect, whose outer circumference expands at a fixed rate with each circumnavigation, creating a floppy, rippling effect at the edges (The shape can be seen in nature, at the edges of kale leaves and the bodies of sea slugs.)
Though the shape did, in fact, reflect the new geometry's properties, it took mathematicians generations to discover a way to illustrate it. In 1997, Cornell mathematician Daina Taimina, who crocheted as a hobby, realized that if one used a certain hyperbolic algorithm as a template, a replica in yarn would take shape. That's what Cornaggio, Chaney and their classmates did, using Taimina's book as a guide.
None had crocheted before, but each spent dozens of hours at the craft.
Kai Jaii Gomez Wick of Columbia, a sophomore math major, found the repetitive labor a pleasant escape. Cornaggio watched a horror-movie marathon while creating his piece. And each student, having started with a small, flat, woolly circle, carefully added a fixed extra length to its circumference with each go-round, watching as their ripple-edged creations took shape.
One day during class, Steinhurst picked up a completed piece to show how the shapes are "graphs" on which hyperbolic geometry can be "drawn."
He etched lines across it in white stitches. As he stretched the fabric in different directions, one could see, for example, any number of parallel lines passing through the point Euclid described, and how, in fact, one can fit five squares around a corner, not just four.
The facts seem absurd from a "normal" point of view, but Steinhurst said that makes them no less true. Math isn't a fixed lens for viewing the world but a set of differing lenses one can choose from, depending on what one wants to see.
"The mathematics you get depends on the assumptions you make at the beginning," the professor said. "That's the big lesson to be learned."