Sarah has been knitting some pretty scarfs lately. She was showing off her latest project to me, and lo and behold it turned out to be mathematical! As she was knitting successive rows, she added incremental stitches to give it a ruffled appearance. I told her I thought this was especially cool because the extra stitches were giving the scarf a negative curvature. It was hyperbolic! That is what happens when you try to put extra material into what would otherwise be a flat 2 dimensional surface. She lovingly rolled her eyes :)
Every time I think I have a new idea, it turns out someone has already been there, done that. On the plus side, the article I then tracked down already contained developed information and instructions for crocheting your very own hyperbolic plane. Following the directions will result in a hyperbolic surface of constant negative curvature (Sarah's scarfs don't adhere to the constant part). I also found this site with some nice pictures of completed crochetings (the site mentions the work of Daina Taimina, who is one of the authors of the paper above).
It is interesting to note that if you build a constant negative curvature surface large enough, it will necessarily end up intersecting itself in our 3D world. Models living in our physical universe are limited in their representation. This is in contrast to models of constant positive curvature surfaces, which do fit nicely into the world. The surface of any ball will do.
Sarah and I just returned from Hill Country Weavers, where Sarah bought me a crochet hook, so I'm now off to attempt creating my own hyperbolic plane!
update: Sarah did not aprove the cuteness factor of my first picture, so I've uploaded an improved version.