Alex's Adventures in Numberland - Alex Bellos [168]
Daina showed the hyperbolic crochet model to her husband – and he was as excited as she was. David Henderson is professor of geometry at Cornell. His specialism is topology, which Daina claims to know nothing about. He explained to her that topologists have long known that when an octagon is drawn on the hyperbolic plane it can be folded together in such a way as to resemble a pair of pants. ‘We have to construct that octagon!’ he told her, which is just what they did. ‘No one had seen a hyperbolic pair of pants before!’ Daina exclaimed, and opened a sports bag she had with her, took out a crocheted hyperbolic octagon and folded it to show me the model. It looked liht liery cute pair of woollen toddler’s shorts:
Word spread around the Cornell maths faculty about Daina’s threaded creations. She told me she showed it to one colleague who is known for writing about hyperbolic planes. ‘He looked at the model and started playing with it. Then his face lit up. “This is what a horocycle looks like!” he said,’ having recognized a very complicated type of curve that he had never been able to picture before. ‘He had been writing about them all his career,’ added Daina, ‘but they were all in his imagination.’
It is no exaggeration to say that Daina’s hyperbolic models have given important new insight into a conceptually punishing area of maths. They give a visceral experience of the hyperbolic plane, allowing students to touch and feel a surface that was previously understood only in an abstract way. The models are not perfect, however. One problem is that the thickness of the stitches makes the crocheted models only a rough approximation of what in theory should be a smooth surface. Still, they are a great deal more versatile and accurate than a Pringle. If a piece of hyperbolic crochet had an infinite number of lines, it would theoretically be possible to live on that surface and walk for ever in one direction without ever coming to an edge.
One of the charms of Daina’s models is that they are unexpectedly organic-looking for something conceived so formally. When the relative line-by-line increase in stitches is small, the models look like leaves of kale. When the increase is greater the material naturally folds itself into pieces that look like coral. In fact, the reason Daina came to London was for the opening of the Hyperbolic Crochet Coral Reef, an exhibition inspired by her models to promote awareness of marine destruction. Thanks to her mathematical innovation, she has unwittingly spawned a global movement of crochet activism.
Over the last decade, Daina has crocheted more than a hundred hyperbolic models. She brought her largest to London. It is pink, uses 5.5km of yarn, weighs 4.5kg and took her six months. Finishing it was an ordeal. ‘As it got bigger it took a lot of energy to turn.’ A remarkable property of the model is that it has an incredibly large surface area – 3.2 square metres, which is twice as much surface area as Daina herself. Hyperbolic surfaces maximize area with minimal volume, which is why they are favoured by some plants and marine organisms. When an organism needs a large surface area – to absorb nutrition, as is the case with coral – it grows in a hyperbolic way.
It is unlikely that Daina would ever have thought up the idea of hyperbolic crochet had she been born a man, which makes her inventions a noteworthy artefact in the cultural history of mathematics, where women have long been under-represented. Crochet, in fact, is just one example