Doilies & Chaos Theory

Posted by jns on March 28, 2005

Kriston, at Grammar.police posted a fantastic picture of a crocheted sculpture in yarn: “Crocheted Model of Hyperbolic Plane” (1970s) by Daina Taimina. (He references this original article: “Crocheting the Hyperbolic Plane: An Interview with David Henderson and Daina Taimina“)
His reaction:

When I saw the images of Taimina’s crocheted hyperbolic figures, I was immediately struck by how instructive it could be as an applied tool to teach non-Euclidean geometry, because–well, I don’t know anything about crochet, but I get the sense that this is true–one could viscerally experience ultraparallel lines or even space curvature. It turns out that Taimina, in fact, invented the first workable model of Lobachevskian, i.e., hyperbolic geometry by abandoning paper and turning to crochet. Certainly makes a great deal of sense after the fact, doesn’t it?

Now, I’m quite serious in what follows, although it may not appear so.
He’s quite right about Taimina’s crocheted geometrics — they are fascinating and instructive as well. They suggest that there indeed could be more mathematical possibilities along the lines he mentions, regardless of whether one knows anything about crochet or not.

People who know me know that I know a bit about crochet, although I prefer working in thread rather than yarn. I crochet doilies. Obsessively. It serves the purpose of keeping my hands busy and productive when I’d otherwise just fidget. These days, since we watch television so rarely, I make most of my pieces in the car, when Isaac is driving. The problem is that, after doing this for some 10 years, one ends up with a lot of doilies — let’s say several hundreds — which is really more than one household can make use of. (Some of my work is displayed, for sale, at The Pansy Forest; there are still lots more for me to put up, however.)
Anyway, I’d never thought about making crocheted hyperbolic figures, although it’s a brilliant idea. I have, however, designed some of my own doily patterns, and the experience gave me the idea for a book about it. (This started several years ago now.)
The tentative title for the book is Doilies, Chaos Theory, and the Origin of the Universe. Seriously.
I don’t want to go into the entire story here, but I discovered what I felt were interesting and illuminating connections between chaos theory (concerning which I did some reasearch in my early graduate-student and post-doctoral days) and creating doily patterns.
Most doilies are chrocheted “in rounds”, worked successsively in thin rings from the center to the outside. Traditional doily patterns, particularly those that make use of a motif called a “pineapple” (sometimes “acorn”), typically develop their patterns over many rounds — that is to say, the patterns emerge one round at a time over the course of completing, say, 10 or 20 rounds.
Now, at the same time the pattern is emerging, it is necessary that the number of stitches in each round increase (relative to the previous round) in fairly strict geometric ratios. There can be a bit of fudging for a round, maybe two further from the center, but one cant’t get away with it for long.
Doily patterns, therefore, are highly constrained systems, and to set out to create a pattern over the course of many rounds requires planning, good luck, and a cooperative pattern. They don’t always go the way one wants.
Another way to put it is to say that doily patterns can show extreme sensitivity to initial conditions: what is allowed to happen on round 25 can depend critically on what happened on round 6. Sensitivity to initial conditions is a defining characteristic of some “chaotic systems” (at least it characterizes the motion of the systems through its phase space, but that’s a longer version of the story.)
Nevertheless, doilies do not look chaotic. Instead, they are amazingly developed mathematical patterns in many cases. How they can look so organized and yet share these characteristics with certain types of chaotic systems interests me. As for the origins of the universe: contemplating the constraints on how doily patterns emerge brings one pretty easily to considering the anthropic cosmological principle (which I tend to think is mostly bunk) and such topics.
I will be the first to admit that there may not be a big cross-over audience for a book that covers antimacassers and modern ideas about dynamical systems and self-organized complexity and such, but that may not stop me. If only I could figure out how to type the manuscript while I crochet the doilies.