Friday, January 28, 2011

Catching Up With Ramanujan: Partition Numbers, Fractals, and Mathematics

"Hidden Fractals Suggest Answer to Ancient Math Problem"
Dave Mosher, Wired Science, Wired (January 28, 2011)

"Researchers have found a fractal pattern underlying everyday math. In the process, they've discovered a way to calculate partition numbers, a challenge that's stymied mathematicians for centuries.

"Partition numbers track the different ways an integer can be divvied up. The number 3, for example, has three unique partitions: 3, 2 + 1, and 1 + 1 + 1. Partition numbers grow so fast that mathematicians have a hard time predicting them.

" 'The number 10 has 42 partitions, but with 100 you have 190,569,292 partitions. They get impossibly huge to add up,' said mathematician Ken Ono of Emory University.

"Since the 18th century, generations of mathematicians have tried to find a way of predicting large partition numbers. Srinivasa Ramanujan, a self-taught prodigy from a remote Indian village, found a way to approximate partition numbers in 1919. Yet before he could expand on the work, and convert it to a clean equation, he died in 1920 at the age of 32. Mathematicians ever since have puzzled over Ramanujan's manuscripts, which tie the primes 5, 7 and 11 to partition numbers...."

First of all, the Lemming takes issue with "impossibly huge to add up." But the Lemming's ranted about the distinction between "impossible," "difficult," and "impractical" before. Recently. (January 10, 2011, January 7, 2011)

Back to Ramanujan, partition numbers, and all that. Dave Mosher tells about how a bit more recently another mathematician, A.O.L. Atkin, studied Ramanujan's work - and even more recently Ken Ono, Amanda Folsom, and Zachary Kent noticed something fractaly about the series of numbers. Except Mr. Mosher didn't write "fractaly." Or maybe that should be fractally. That may not have been a word until the Lemming started writing this paragraph.

Which is an example of how languages develop, and why an official French dictionary is so thin - and that's another topic. Topics.

Ken Ono and another chap, Jan Hendrik Bruinier, published a paper that describes a function - they call it "P" - that produces any integer's partition number.

What happened was that the mathematicians noticed - not the same numbers, but the same patterns of numbers repeated over and over in what they were studying. Sort of like Mandelbrot sets. (October 18, 2010)

An immediately-practical result of the P function involves cybersecurity. Anther fairly new word. And yet another topic.

"...The combined research doesn't quite reveal a mathematical representation of the universe's structure, Ono said, but it does kill partition numbers as a way to encrypt computer data.

" 'Nobody's ever going to do that now, since we now know partition numbers aren’t random,' Ono said. 'They're completely predictable and we should no longer pretend they're mysterious.'..."

Hats off to Dave Mosher, by the way, for not using the words "chaos" or "chaotic" in the article. Not once. Considering that it has to do with fractal mathematics - and a new wrinkle in fractals at that - that's a remarkable accomplishment.

But then, Mr. Mosher is that all-too-rare phenomenon, a "science journalist" who knows a bit about science. And journalism: the sort that involved rooting out facts. He's got a fair number of articles on the Wired website.

Mathematics, Chaos, and Stuff the Lemming Doesn't Understand

The Lemming's noticed the words "chaos" or "chaotic" fairly often in news coverage of major accidents, disasters, and developments in science and mathematics.

For example, there was a serious disconnect between video and narration in coverage of a particularly messy situation in an African country. As the Lemming recalls, a bomb had gone off minutes before. Buildings and bodies had been blown around. Some of the local folks were dead, others injured. Still others had formed impromptu teams and were clearing rubble by hand, in an effort to get at survivors who weren't already being carried to ambulances. This scene of intense, complex, activity was described as - you guessed it - chaos.

The Lemming will grant that the Africans weren't lined up in neat rows and columns, or standing around watching the show. But "chaos?!" Emergency response should only be that 'chaotic' everywhere. Those folks were getting things done, fast. Yet again another topic.

The Lemming suspects - strongly - that journalists use the words "chaos," or "chaotic," when they mean "complicated," "rapidly changing," or "something I don't understand."

There are quite a few things the Lemming doesn't understand. Mathematics from Calculus on up - or out - for one. But the Lemming tries to remember that there's a huge difference between not understanding the underlying patterns of a particular entity or incident: and there being no underlying patterns.

Randomness does exist - in the way dice roll, for example. But even then, statistics make it possible for mathematics to handle true randomness - and tell the difference between that and something that simply isn't obvious or simple. (July 14, 2010)

But assuming that something the Lemming doesn't understand is, therefore, "chaotic" - bereft of order? The Lemming has pretty good self-esteem: but doesn't have that big an ego.

Then there's something an Italian said, about four centuries back now:

"Mathematics is the language in which God has written the universe."
Galileo Galilei, Italian astronomer & physicist (1564 - 1642), via The Quotations Page

Which brings up still another topic, in another of the Lemming's blogs:Related posts:

2 comments:

Brigid said...

Plural agreement: "in the way die roll"

The Friendly Neighborhood Proofreader

Brian, aka Aluwir, aka Norski said...

Brigid,

Got it. 'One die, many dice' - I have trouble with that one for some reason. May have something to do with it being a non-Germanic term.

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