Knitted math

About a year ago, I conquered my fear of DPNs forever: I knit a cross-cap projective plane. This was an insane challenging decision for a not-too-experienced knitter, since the trick to knitting a projective plane is that it intersects itself -- parts of the fabric you're creating actually pass through each other. This requires some fiddly knitting on the intersection row. It's a little like double knitting, except with four or five needles to keep track of.

I don't like the finished look of most knitted mathematical objects, like Klein bottle hats, but it is fun to see how knitting makes it possible to create tangible models of things that can't be made correctly from paper or clay. You can't get paper to intersect itself without cutting it up, but two knitted pieces can pass through each other and still be distinct, solid pieces.

This projective plane is even kind of cute -- a colorblocked square -- and if you don't sew all the seams, it has a couple of pockets. My boyfriend uses the tiny one I knitted to hold little notes on his office wall. A bigger one might make a fun potholder. (Since a projective plane, like a Moebius strip, has only one side, your pots are technically always inside such a potholder, but the double layer of fabric would also be convenient in more practical terms.)

I don't seem to have a photo of my projective plane handy, but here's a picture of one created by Mark Shoulson, with a link to his instructions and more cool information about this mathematical object.