Sunday, December 26, 2010

Hyperbolic scarf

This is the first crochet project I attempted, and I've made it several times now. It's quite easy: chain a bunch, double crochet twice in each chain stitch, and then double crochet twice in each stitch on the next row. I like it because it's funky and mathematical. See, this is a great example of negative curvature, and as a hyperbolic geometer, I dig that. Here's a website where you can read a bit more about the mathematics of crochet. (The website is maintained by a math professor friend of mine who is using crochet to help get students excited about geometry. It's pretty cool! It also has some links to other websites about crochet and math.)

(Math digression in blue because I like blue.)
In case you're interested in the math, here's a little background. In 10th grade, you probably took a geometry class. In it, you learned about triangles and writing proofs using facts about how much total angle different shapes have in them and so on. Well, that was something called Euclidean geometry. In Euclidean geometry, the plane that you kind of base everything on is flat. One feature of Euclidean geometry is that parallel straight lines remain the same distance apart forever, and there is only one line parallel to a given line that goes through a given point. (This is called the parallel postulate.) Euclid thought that this was the only consistent geometry, but it turns out that there are consistent geometries in which your "plane" is not flat. In fact, all of them involve replacing the parallel postulate with a different axiom. One non-Euclidean geometry you are probably fairly familiar with is spherical geometry. Imagine that you are living on the surface of a sphere (not too hard, right?). That is called positive curvature. You can see the differences between positive and flat curvature most easily if you imagine a plane flight between points that are very far away from each other. The shortest distance between two points on the earth looks curved on a map. You'll also notice that any two of these curves, which are called great circles, will intersect. (For example, all the lines of longitude intersect in the poles.)

Well, hyperbolic geometry is the study of negative curvature. It's a bit harder to visualize than positive, but the way I think about it is that there is "extra surface area" for a given perimeter. "Parallel" (i.e. non-intersecting infinite) lines move away from each other. There are an infinite number of parallel lines between a line and a given point not on that line. One way to visualize a portion of the hyperbolic plane is to imagine a saddle. Some curves on it go up, and some go down. (In contrast, on a sphere, if you put two perpendicular lines on the north pole, they would both go down.) I find that sewing and crochet are really great media for displaying negative curvature because you can force your fabric to have "too much area" and pucker in a pretty way. That's what this scarf does by putting "too many" stitches into the second row (two stitches for every one stitch in the first row). The fabric has to twirl and buckle to make room for all those extra stitches. People who make hyperbolic crochet like to play with different increase ratios. 2:1 is a very steep increase. 6:5 is very pretty. To do that, you make six stitches in the second row for every five stitches in the first row. No matter what your increase ratio is, though, you will learn that exponential growth is very very fast!

Hyperbolic geometry is pretty neat. And you can be employed by it! People like me make a living studying spaces of hyperbolic surfaces.

OK, that was a long math digression. Back to the crochet. I like making this gift for people. Most recently, I sent a purple one to my MIL for her birthday. The green one above is mine, and the multicolored one I sent to one of my best friends for her birthday. It's easy enough that a crochet newbie like me can get the hang of it really quickly and whip one out in just a few hours.

No comments:

Post a Comment