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.