Alex's Adventures in Numberland - Alex Bellos [68]
The symbol indicates a series of values all added up, starting with the value when n equals zero, added to the value when n equals one, and so on to infinity. Even without understanding the notation, however, one can appreciate the drama of such an equation. The Ramanujan formula races towards pi with remarkable speed. From the very start, when n is 0 the formula has one term and gives a value of pi accurate to six decimal places. For each increase in the value of n, the formula adds roughly eight new digits to pi. It is an industrial-strength pi-making machine.
Inspired by Ramanujan, in the 1980s the Ukrainian-born mathematicians Gregory and David Chudnovsky devised an even more ferocious formula. Each new term adds roughly 15 digits.
The first time I saw the Chudnovsky formula I was standing on it. Gregory and David are brothers and they share an office at the Polytechnic University in Brooklyn. It consists of an open-plan space with a sofa in the corner, a couple of chairs and a blue floor decorated with dozens of formulae for pi. ‘We wanted to put something on the floor and what else can you put on the floor other than stuff which relates to mathematics?’ explained Gregory.
In fact, the pi floor pattern was their second choice. The original plan was to lay down a giant reproduction of Melencolia I by Albrecht Dürer (reproduced on chapter 6). The sixteenth-century woodcut is beloved of mathematicians since it is full of playful references to numbers, geometry and perspective.
‘One night, when there was nothing on the surface, we printed 2000 or more pages of [Melencolia I] and we laid it on the floor,’ said David. ‘But if you walked around it, you wanted to throw up! Because your point of view changes extremely abruptly.’ David began to study the floors of the cathedrals and castles of Europe in order to work out how he could decorate the office without inducing nausea in anyone walking through. ‘I discovered they are mostly laid out in a –’
‘Simple geometric style,’ interrupted Gregory.
‘Black, white, black, white squares…#8217; said David.
‘You see, if you really have a complex picture and you try to walk on it, the angle changes so abruptly that your eyes don’t like it,’ added Gregory. ‘So the only way you can do something like that is to –’
‘Hang from the ceiling!’ David shouted in my left ear, and both men lost themselves in guffaws.
Talking to the Chudnovskys was like wearing stereo headphones with an erratically alternating connection to both ears. They sat me on their sofa and sat on either side of me. Constantly interrupting each other, they finished each other’s sentences, speaking in a highly melodic English thick with Slavic tones. The brothers were born in Kiev, when it was in the Soviet Republic of Ukraine, although they have lived in the United States since the late 1970s and are American citizens. They have collaborated on so many papers and books together that they encourage you to think of them as one mathematician, not two.
For all their genetic, conversational and professional homogeneity, however, the men look very different. This is mostly because Gregory, who is 56, suffers from myasthenia gravis, an auto-immune disorder of the muscles. He is so thin and frail that he spends most of his time lying down. I never saw him get up off the sofa. Still, the energy his limbs lacked was compensated for by a brilliantly expressive face that burst into life as soon as he talked about maths. He has pointed features, large brown eyes, a white beard and wispy unkempt hair. David, who has blue eyes, is five years older, rounder of body and fuller of face. He was clean-shaven and his short hair was hidden under an olive-green baseball cap.
The Chudnovskys are arguably the mathematicians who have done the most to popularize pi in recent years. In the early 1990s they built a supercomputer in Gregory’s Manhattan apartment out of mail-order parts that, using their own formula, calculated the number to more than two billion decimal places – a record at the time.
This amazing achievement was chronicled