A good friend taught me how to crochet on Sunday night. I started with the classic square-to-be-used-as-a-dish-rag project and moved onto the spiral-to-be-used-as-a-pot-stand project. Project number three? The Möbius Strip. Now, I understand the Möbius strip well. What kid has not taken a strip of paper, twisted it, taped the ends together and then drawn a line starting on one side of the paper only to seamlessly (or edge-lessly) reach the other side? I started crocheting the Möbius strip by creating a line and then joining the ends of the line, adding a twist to the line (a 1D Möbius strip?). I then continued crocheting along one edge of the cycle. But of course, since a Möbius strip has only one boundary, I could continue extending the thickness of the strip by spiralling outwards.

I thought: what would happen if I started dropping stitches, making this boundary shorter and shorter? Well, the Möbius strip is a non-orientable surface with one hole – that forms this boundary. I reminded myself that the Möbius strip is surface obtained by puncturing a projective plane – the non-orientable surface of minimum genus – however, I never had a good intuition of what the projective plane is. Let me tell you though, after dropping every fourth stitch on the boundary of my Möbius strip, I started having a pretty good idea of what a projective plane does.

I couldn’t continue until the boundary closed up – for obvious reasons – so I still have a Möbius strip, but the physical surface is close enough to the projective plane that I get a much better feel for what that means.

I’m not the first person to crochet a Möbius strip – apparently it is a popular scarf design – nor am I the first to explore geometry with crochet. But I have to say that *actually creating* this surface adds an intuition I’m not sure you could get elsewhere. Finally, if you haven’t seen Margaret Wertheim’s TED talk on math, crocheting and coral including how to crochet a hyperbolic geometry and in which she says

So here, in wool, through domestic feminine art, is the proof that the most famous postulate in mathematics is wrong.

I highly recommend it.

Yaroslav BulatovIs Klein Bottle crochet possible?