Alex's Adventures in Numberland - Alex Bellos [162]
CHAPTER ELEVEN
The End of the Line
A few ye
ars ago Daina Taimina was reclining on the sofa at home in Ithaca, New York, where she teaches at Cornell University. A family member asked her what she was doing.
‘I’m crocheting the hyperbolic plane,’ she replied, referring to a concept that has mystified and fascinated mathematicians for almost two centuries.
‘Have you ever seen a mathematician do crochet?’ came the dismissive response.
The rebuff, however, made Daina even more determined to use handicraft in the course of scientific advancement. Which is just what she did, inventing what is known as ‘hyperbolic crochet’, a method of looping yarn that produces objects as intricate and beautiful as anything produced by the WI, and that has also contributed to an understanding of geometry in a way that mathematicians once never thought possible.
I’ll come shortly to a detailed definition of hyperbolic and the insights gleaned by Daina’s crochet models, but for the moment all you need to know is that hyperbolic geometry is an utterly counterintuitive type of geometry that emerged in the early nineteenth century in which the set of rules that Euclid laid out so carefully in The Elements are taken as being false. ‘Non-Euclidean’ geometry was a watershed for mathematics in that it described a theory of physical space that totally contradicted our experience of the world, and therefore was hard to imagine, but nevertheless contained no mathematical contradictions, and so was as mathematically valid as the Euclidean system that came before.
Later that century an intellectual breakthrough of similar significance was made by Georg Cantor, who turned our intuitive understanding of the infinite on its head by proving that infinity comes in different sizes. Non-Euclidean geometry and Cantor’s set theory were gateways into two strange and wonderful worlds, and I’ll visit them both in the following pages. Arguably, together they marked the beginning of modern mathematics.
Hyperbolic crochet.
The Elements, to recap from much earlier, is easily the most influential maths textbook of all time, having set out the basics of Greek geometry. It also established the axiomatic method, by which Euclid began with clear definitions of the terms to be used and the rules to be followed, and then built up his body of theorems from them. The rules, or axioms, of a system are the statements that are accepted without proof, so mathematicians always try to make them as simple and self-evident as possible.
Euclid proved all 465 theorems of The Elements with only five axioms, which are more commonly known as his five postulates:
1. There is a straight line from any point to any point.
2. A finite straight line can be produced in any straight line.
3. There is a circle with any centre and any radius.
4. All right angles are equal to one another.
5. If a straight line falling on two given straight lines makes the interior angles on the same side less than two right angles, the two given straight lines, if produced indefinitely, meet on that side on which the angles are less than the two right angles.
When we get to number 5, something does not feel right. The postulates start briskly enough. The first four are easy to state, easy to understand and easy to accept. Yet who invited the fifth to the party? It is long-winded, complicated and not especially, if at all, self-evident. And it is not even as clearly fundamental: the first time The Elements requires it is for Proposition 29.
Despite their love of Euclid’s deductive method, mathematicians loathed his fifth postulate; not only did it go against their sense of aesthetics, they felt that it assumed too much to be an axiom. In fact, for 2000 years many great minds attempted to change the status of the fifth postulate by trying to deduce it from the other postulates