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Starting from a crocheted hyperbolic plane, you can go in all kinds of directions. Holding one in your hand—or making one—can spark many lines of inquiry. Here are some examples.

Math
In what ways is this object math? Why didn’t I learn that this was math when I was in school? What mathematical questions can I ask, or answer, by starting from hyperbolic crochet—both the object itself and the process of making it? What is non-Euclidean geometry (of which hyperbolic geometry is one kind)? How can there be other geometries than the one we learned in school? What is the shape of the universe anyway?

Computing and Systems
To make a crocheted hyperbolic plane you follow a “procedure;” can the procedure be expressed as an algorithm, which is at the heart of computing? Could you write a program to create a hyperbolic plane, and then variations on that plane? What other aspects of computational thinking might be found in hyperbolic crochet—both the artifact and the activity?

In systems theory and in other domains, “emergence” is an important idea. The term refers to the way in which something novel and complex can arise from the interaction of one or more simple units. Examples are coral, flocking of birds, the stock market, city form, and even the emergence of consciousness from the brain. In hyperbolic crochet, the repetition of a single simple stitch, with periodic increases, creates a surprising and complex shape (see How To for directions.) The complex shape of the hyperbolic plane “emerges” from a very simple process. What other aspects of systems ideas, such as “feedback,” might be found in one or many crocheted hyperbolic planes?

Beauty and Engagement
What accounts for the deep appeal of hyperbolic crochet to many people? When people see crocheted hyperbolic planes, they can’t keep their hands off them. Sometimes they ask to keep them overnight—people with Ph.Ds, fiber artists, and plenty of other folks. They are “attractive” in two senses of the word: they draw people to them, and they are beautiful. I’ve heard many times that “mathematics is beautiful,” but it was not until I started hyperbolic crochet that really understood what that might mean. How is beauty linked to  engagement and persistence? Does the aesthetic play a role in deep discovery in math, science, and other domains? If it does, what role?

Gender
Crochet and knitting are strongly associated with women in the United States in the 21st century (this has not been true at all times and places.) One of the attractions of hyperbolic crochet is the collision of math (still considered male despite the many recent successes of women) with a relatively devalued form of “women’s work.” When Daina Taimina’s book was awarded the 2009 Diagram Prize for the Oddest Book Title, the group giving the award said, “On the one hand you have the typically feminine, gentle and wooly world of needlework and on the other, the exciting but incredibly un-wooly world of hyperbolic geometry and negative curvature. In Crocheting Adventures with Hyperbolic Planes the two worlds collide—in a captivating and quite breathtaking way.” What role can hyperbolic crochet, and the fiber arts more broadly, play in attracting girls and other under-represented groups to Science, Technology, Engineering, Mathematics, and Computing?

Comfort and Touch
Crocheted objects are tactile and comforting. The feel of the yarn, particularly yarns made from natural fibers, can be relaxing and pleasant. In a pilot study we conducted at University of Massachusetts Lowell, female students who watched a math lecture, then took a short math test, reported less math anxiety when they held a crocheted hyperbolic plane during the lecture and test. They also were more likely to agree with the statement, “I like math.” How can we take advantage of these benefits of hyperbolic crochet? Could we reproduce these results with other under-represented groups using culturally relevant objects? “Worry beads,” prayer beads, and other tangibles are used across cultures and are shown to lower general anxiety. Is the effect of crocheted hyperbolic planes similar?

Conversation
When they see me making crocheted hyperbolic planes, some people ask, “what do you do with them?” I say, “I use them to start conversations.” I hope you will join the conversation on my blog or by sending me email.