How crochet solved an age-old maths problem

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 Feminine handcrafts are not known for their contribution toscientific progress. Nuclear physics has been untouched by advancesin knitting. The understanding of molecular biology does notrequire embroidery skills. Yet basic crochet is giving new insightsinto centuries-old maths problems.
For this we can thank Daina Taimina, a Latvian mathematician with atalent for crochet. If you had to draw a Venn diagram with onecircle representing women, one circle representing mathematiciansand another celebrities, Taimina is about the only person in theintersection of all three.
When she picked up her crochet hook a decade ago, Taimina had amore ambitious goal than making a scarf or hat: her aim was tocreate a model of hyperbolic space, a strange world where parallellines do not stay the same distance apart but curve away from eachother. It is so conceptually challenging that for a centurymathematicians were unable to visualise what this type of spacemight actually look like. In fact, there is no formula thataccurately describes hyperbolic space, so computers cannot model iteither.
Yet Taimina achieved results. Using craft skills learnt whilegrowing up in Latvia, she invented “hyperbolic crochet”- a testament to her mathematical nous and single-mindedness. Whenwe meet in the Greenwich flat she is staying in for the opening ofthe hyperbolic crochet coral reef exhibition at the SouthbankCentre, she tells me that when she started to use crochet in maths,the reaction from some family members was dismissive. “Theysaid, ‘You are not serious. You should do something moreserious',” she recalls. This was the provocation she needed.“It is serious. And I did what I set out to do.”
It turned out that Taimina's models were not just abstract andmathematical. They looked surprisingly like many natural phenomenasuch as lettuce leaves, sea slugs, kelp and coral. This helped tocreate a movement that has spread from academia to the craft andenvironmental communities. The Southbank exhibition, inspired byher models, aims to highlight the destruction of coral reefs.
Taimina made her discoveries shortly after moving from Latvia toCornell University in Ithaca, New York a decade ago, after beinginvited to be a visiting professor in mathematics. She still speakswith a distinctive Eastern European accent.
Hyperbolic crochet might have originated from complex mathematicalideas, but in reality it is very simple. Taimina started with aline of crochet, and for each subsequent line she increased thenumber of stitches. For example, adding an extra stitch in thesecond line for every five stitches in the first. And for everyfive stitches in the second line, adding an extra one in the third.The number of stitches increases at an exponential rate. As thelines are longer, but joined together, the material quickly startsto fold in interesting ways.
Once she got the hang of it, she realised that you can see just howparallel lines behave on the hyperbolic plane. On a blue model, shecrocheted a white line. She then took a point that was not on theline and, through folding and tracing the contours with her finger,could find several new lines that went through the point, wereparallel to the white line and also curved away from it. “Iwas excited that I could do this,” she says, since this was aphysical representation of something that had until then been onlyconceptual. “Making something with my hands that cannot bemade by computer was also a thrill.”
Even more excited was her husband, David Henderson, professor ofmathematics at Cornell, who she met at a maths conference.“His dream was to see a hyperbolic pair of pants,”Taimina says, referring to the properties of an octagon inhyperbolic space. When you fold the sides of an octagon in thehyperbolic plane, the shape looks like a pair of trousers. And hercrocheted octagon worked. She has an example and shows me exactlyhow it folds up to be like a woolly pair of toddler's shorts.
Word quickly got round the Cornell maths faculty. One colleague, aworld expert on the hyperbolic plane, was mesmerised by Taimina'scrochet models. He said that for the first time he could visualisea certain type of hyperbolic curve. “It was a happyexclamation. He had been writing about them all his career, but itwas all in his imagination. The difference with the crochet is thatyou can touch it. You can experience it.”
Hyperbolic crochet has given new insights into old geometricalproblems. One way to think about it is that the hyperbolic plane iswhere lines curve away from each other, in contrast to what happenson a sphere, where all lines curve towards each other. It is stillnot known if the geometry of the universe is spherical, flat, oreven hyperbolic - although using data from satellites, scientistshope to have the answer within the next few years.
A hundred years ago, it was proved that there is no formula todescribe the hyperbolic plane, so mathematicians all but gave uptrying to create a physical model of one. In the 1970s, BillThurston, a US mathematician who went on to win the Fields Medal,the most highly regarded prize in maths, made an attempt usingstrips of paper and Sellotape. Yet while this demonstrated some ofthe properties of the hyperbolic plane, it was finicky to make, andflimsy. “We have one. It sits on our piano,” saysTaimina. “You are not going to fold and investigate it. Ihate glueing paper. It drives me crazy. So that is when I came upwith the idea to make crochet.”
In fact, her first attempt was with knitting. But this didn't work.“If you lose a stitch, everything unravels. With crochet youcan do one stitch at a time.”
David Henderson says the crochet has made deep contributions to thefield: “For the first time it allows mathematicians andstudents to actually experience the hyperbolic plane. The genius ofit is connecting craft with mathematics.”