Hyperbolic Planes

Yesterday I went to the talk by Diana Taimina about how to crochet hyperbolic planes. The gallery was in a not so great neighborhood in Echo Park, and I almost didn't go because it was hard to find parking, but I did, and I'm glad I went. The crowd was an interesting mixture of mathematicians (or people who knew big math words) and crocheters. I don't know much about math myself, and unfortunately didn't pay much attention in Physics (though I wish I did), so the only concept which was not new to me was the explanation of parallel lines.

A hyperbolic plane, or at least the way I understood it, is pretty much the opposite of a sphere. If you were to draw a line around a balloon the line would curve into itself and then it would create a ring. With hyperbolic space the line curves out, and instead of creating a circle the line continues to grow infinitely. For a long time there wasn't a very good tangible representation of hyperbolic space, and the few that existed were very fragile so you couldn't really maneuver it. Taimina, who learned to knit when she was a child and new how to crochet, one day realized that you could manipulate yarn, using a crochet hook, to make something with the same features as a hyperbolic plane. Essentially all it is is increasing at an exponential rate. The outcome, depending on how fast you increase sometimes looks like a brain, which Taimina thought wasn’t a coincidence. This same wavy shape occurs elsewhere in nature, including some in some sea creatures. There is a short description of how to make hyperbolic planes at: http://www.math.cornell.edu/~dwh/books/eg00/supplements/AHPmodel/index.html . You can read more about it on http://www.theiff.org/lectures/05a.html .

Actually, the most interesting thing I heard in this lecture wasn't related to crocheting or even really about the concept of hyperbolic planes, but is a simple fact that stares you in the face every day, the fact that we live in three dimensions but can see in only two dimensions. An apple, although it exists in three dimension (up/down, left/right, forward/backwards), can only be seen in two dimensions (up/down, left/right). You cannot actually see the other side of an apple. And the only way for us to see something in three dimensions is to exist and live in four dimensions. But seriously, how could I have not known this? It's not actually that hard of a concept, and I feel like this is one of those mind-expanding things I should have learned in sixth grade. Was I just not paying attention?