The Books: Life Stories: Profiles from The New Yorker; edited by David Remnick; ‘The Mountains of Pi’, by Richard Preston

116828

Next up on the essays shelf:

Life Stories: Profiles from The New Yorker, edited by David Remnick

Life Stories is a collection of “profiles” from The New Yorker, edited by David Remnick. The pieces span the 20th century, one of the best parts of these compilations. I also love that it’s not just celebrities who are covered, although they are represented here too. There are celebrities in certain sub-cultures (like the profile I will excerpt today), and then also a couple of people who are virtually unknown (“Mr. Hunter” from Staten Island), and yet fascinating. The best part of the profiles is that they are so in-depth and so lengthy (some of them run to 40 pages long), that you actually feel like you have met these people. There’s a huge profile of the writer behind Heloise’s Household Tips that is one of the best things I’ve ever read in The New Yorker, period.

Today’s piece is by Richard Preston, and it is enormous. It takes time and commitment to get through it, but it is so worth it. It is a famous profile of the mathematician Chudnovsky brothers, Gregory and David.

chudnovsky-math-vi

This piece called “The Mountains of Pi” was published in 1992.

I have not followed along with the saga of the Chudnovskys, so perhaps one of my more scientific-oriented readers can update me.

On the one hand, “The Mountains of Pi” is a profile of these two genius mathematician brothers from Kiev, holed up in their Upper West Side apartments, puzzling over the mysteries of Pi. In order to plumb the depths of Pi (and do a host of other calculations for other projects) they built a supercomputer from scratch, and that is another part of this lengthy essay: describing said supercomputer (which makes the apartment sweltering), and the battle to get parts Fed Ex’d to them overnight from warehouses around the country. They build the supercomputer in order to avoid having to pay for time on other public supercomputers. Preston goes into detail about how the supercomputer is set up, what it does. It has to be elevated off the floor (too hot), and fans have to blow on it, and everything. Gregory Chudnovsky is bed-ridden (he is only in his 30s at this point), and lives surrounded by books and stacks of paper. It is amazing that he is married, but he is! The other part of this essay is a profile of the Chudnovsky’s situation in America: essentially (at this point anyway) they were un-hireable. Gregory was too ill, and the two came as a team. They are sort of hired by Columbia Universiry, who give them a small stipend, but in general their genius has been ignored by the academic community and there are many advocates on their behalf who speak to Preston and say that the situation is deplorable. These men are geniuses, and it is a disgrace that America cannot find a place for them, where they can teach, and pass on their knowledge to the next generation. (That situation may have changed. Maybe because of this article? I don’t know.)

And lastly, this is an article about Pi, that mysterious eternal number. Pi is infinite, as we all know. The Chudnovskys do not ONLY work on Pi, but Pi is one of their main obsessions. They run Pi to the billions of numbers through their supercomputer, and they look for patterns in the chaos of random numbers. Many academics scorn this pursuit as a waste of time. Preston goes in detail to the history of Pi, the people who have been obsessed by it through the years, and the nature of Pi itself.

A gorgeous essay. A feast for the mind and soul. Again, you really feel like you MEET the Chudnovsky brothers. Preston hangs out with them in the hot apartment, moving stacks of paper to sit by Gregory’s bed, he records their humorous and sometimes argumentative conversations, and overall you get the sense that these brothers, who had been harassed by the KGB as children, whose parents had been beaten by the KGB, are really working on a whole other plane of intuition and intelligence.

Here is an excerpt. But seek out the whole thing! It’s not online in its entirety, but you can find the whole thing in this collection, Life Stories.

Life Stories: Profiles from The New Yorker, edited by David Remnick; ‘The Mountains of Pi’, by Richard Preston

It is worth thinking about what a decimal place means. Each decimal place of pi is a range that shows the approximate location of pi to an accuracy ten times as great as the previous range. But as you compute the next decimal place you have no idea where pi will appear in the range. It could pop up in 3, or just as easily in 9, or in 2. The apparent movement of pi as you narrow the range is known as the random walk of pi.

Pi does not move; pi is a fixed point. The algebra wobbles around pi. There is no such thing as a formula that is steady enough or sharp enough to stick a pin into pi. Mathematicians have discovered formulas that converge on pi very fast (that is, they skip around pi with rapidly increasing accuracy), but they do not and cannot hit pi. The Chudnovsky brothers discovered their own formula in 1984, and it attacks pi with great ferocity and elegance. The Chudnovsky formula is the fastest series for pi ever found which uses rational numbers. Various other series for pi, which use irrational numbers, have also been found, and they converge on pi faster than the Chudnovsky formula, but in practice they run more slowly on a computer, because irrational numbers are harder to compute. The Chudnovsky formula for pi is thought to be “extremely beautiful” by persons who have a good feel for numbers, and it is based on a torus (a doughnut), rather than on a circle. It uses large assemblages of whole numbers to hunt for pi, and it owes much to an earlier formula for pi worked out in 1914 by Srinivasa Ramanujan, a mathematician from Madras, who was a number theorist of unsurpassed genius. Gregory says that the Chudnovsky formula “is in the style of Ramanujan,” and that it “is really very simple, and can be programmed into a computer with a few lines of code.”

In 1873, Georg Cantor, a Russian-born mathematician who was one of the towering intellectual figures of the nineteenth century, proved that the set of transcendental numbers is infinitely more extensive than the set of algebraic numbers. That is, finite algebra can’t find or describe most numbers. To put it another way, most numbers are infinitely long and non-repeating in any rational form of representation. In this respect, most numbers are like pi.

Cantor’s proof was a disturbing piece of news, for at that time very few transcendental numbers were actually known. (Not until nearly a decade later did Ferdinand Lindemann finally prove the transcendence of pi; before that, mathematicians had only conjectured that pi was transcendental.) Perhaps even more disturbing, Cantor offered no clue, in his proof, to what a transcendental number might look like, or how to construct such a beast. Cantor’s celebrated proof of the existence of uncountable multitudes of transcendental numbers resembled a proof that the world is packed with microscopic angels – a proof, however, that does not tell us what the angels look like or where they can be found; it merely proves that they exist in uncountable multitudes. While Cantor’s proof lacked any specific description of a transcendental number, it showed that algebraic numbers (such as the square root of two) are few and far between: they poke up like marker buoys through the sea of transcendental numbers.

Cantor’s proof disturbed some mathematicians because, in the first place, it suggested that they had not yet discovered most numbers, which were transcendentals, and in the second place that they lacked any tools or methods that would determine whether a given number was transcendental or not. Leopold Kronecker, an influential older mathematician, rejected Cantor’s proof, and resisted the whole notion of “discovering” a number. (He once said, in a famous remark, “God made the integers, all else is the work of man.”) Cantor’s proof has withstood such attacks, and today the debate over whether transcendental numbers are a work of God or man has subsided, mathematicians having decided to work with transcendental numbers no matter who made them.

The Chudnovsky brothers claim that the digits of pi form the most nearly perfect random sequence of digits that has ever been discovered. They say that nothing known to humanity appears to be more deeply unpredictable than the succession of digits in pi, except, perhaps, the haphazard clicks of a Geiger counter as it detects the decay of radioactive nuclei. But pi is not random. The fact that pi can be produced by a relatively simple formula means that pi is orderly. Pi looks random only because the pattern in the digits is fantastically complex. The Ludolphian number is fixed in eternity – not a digit out of place, all characters in their proper order, an endless sentence written to the end of the world by the division of the circle’s diameter into its circumference. Various simple methods of approximation will always yield the same succession of digits in the same order. If a single digit in pi were to be changed anywhere between here and infinity, the resulting number would no longer be pi; it would be “garbage”, in David’s word, because to change a single digit in pi is to throw all the following digits out of whack and miles from pi.

“Pi is a damned good fake of a random number,” Gregory said. “I just wish it were not as good a fake. It would make our lives a lot easier.”

Around the three-hundred-millionth decimal place of pi, the digits go 88888888 – eight eights pop up in a row. Does this mean anything? It appears to be random noise. Later, tex sixes erupt: 6666666666. What does this mean? Apparently nothing, only more noise. Somewhere past the half-billion mark appears the string 123456789. It’s an accident, as it were. “We do not have a good, clear, crystallized idea of randomness,” Gregory said. “It cannot be that pi is truly random. Actually, a truly random sequence of numbers has not yet been discovered.”

No one knows what happens to the digits of pi in the deeper regions, as the number is resolved toward infinity. Do the digits turn into nothing but eights and fives, say? Do they show a predominance of sevens? Similarly, no one knows if a digit stops appearing in pi. This conjecture says that after a certain point in the sequence a digit drops out completely. For example, no more fives appear in pi – something like that. Almost certainly, pi does not do such things, Gregory Chudnovsky thinks, because it would be stupid, and nature isn’t stupid. Nevertheless, no one has ever been able to prove or disprove a certain basic conjecture about pi: that every digit has an equal chance of appearing in pi. This is known as the normality conjecture for pi. The normality conjecture says that, on average, there is no more or less of any digit in pi: for example, there is no excess of sevens in pi. If all digits do appear with the same average frequency in pi, then pi is a “normal” number – “normal” by the narrow mathematical definition of the word. “This is the simplest possible conjecture about pi,” Gregory said. “There is absolutely no doubt t hat pi is a ‘normal’ number. Yet we can’t prove it. We don’t even known how to try to prove it. We know very little about transcendental numbers, and, what is worse, the number of conjectures about them isn’t growing.” No one knows even how to tell the difference between the square root of two and pi merely by looking at long strings of their digits, though the two numbers have completely distinct mathematical properties, one being algebraic and other other transcendental.

Even if the brothers couldn’t prove anything about the digits of pi, they felt that by looking at them through the window of their machine they might at least see something that could lead to an important conjecture about pi or about transcendental numbers as a class. You can learn a lot about all cats by looking closely at one of them. So if you wanted to look closely at pi how much of it could you see with a very large supercomputer? What then? How much pi could you see?

This entry was posted in Books and tagged , , . Bookmark the permalink.

8 Responses to The Books: Life Stories: Profiles from The New Yorker; edited by David Remnick; ‘The Mountains of Pi’, by Richard Preston

  1. mutecypher says:

    Their algorithm for calculating pi to a particular digit is still commonly in use and has yielded many of the recent records. I didn’t realize it was now so high, but pi was calculated to 10 trillion digits in October 2011. Considering all of the work that goes on in speeding up computations, that’s quite an achievement to still have the fastest method after 30 years. And they’ve been a bit of an inspiration, since most of the pi-calculation records over the last decade have been done on (highly modified) personal computers.

    Both brothers are at Brooklyn Poly. There was a documentary on Nova in 2005 about them working on restoring a tapestry – The Hunt For The Unicorn. You can watch it here – very cool.

    http://www.pbs.org/wgbh/nova/physics/chudnovsky-math.html

  2. Kate says:

    I read portions of this in class every year when we study Pi.

  3. Iain says:

    Sheila – thanks for linking to this again. I remember you linking to the New Yorker “Mountains of Pi” piece before, but when I went to look for it online, I found – as you say – that the complete article isn’t available. Now that I know that I can find it in the “Life Stories” collection, I think that a visit to Amazon is in order!

    I can’t wait to read the full profile. Mathematics is a total mystery to me, but this kind of obsessive search for something elusive fascinates me no end.

    • sheila says:

      Yes, I have a similar fascination, Iain – I’m no mathematician, but I love obsessives, no matter what the obsession is. This is a fantastic portrait of obsession!

Leave a Reply

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed.