Wednesday, April 15, 2009


Riemann's zeta function and the building blocks of reality?

Okay, one more post and then I'm getting to work.

Last night Allison and I watched the episode of BBC's Horizon that aired the other day, "Alan and Marcus Go Forth and Multiply". An extremely well done introduction for the layman to some of the most interesting and mindbending aspects of higher mathematics. But right in the middle there is a discussion of Bernhard Riemann's zeta function, which in my (albeit layman's understanding) is concerned with predicting the distribution of prime numbers. And then it gets, for a moment, really interesting.

Mathematician and Oxford professor Marcus Du Sautoy takes co-presenter Alan Davies to a lab somewhere, where a quartz sphere has been hooked up to an apparatus which converts vibrations in the quartz into electrical impulses. When Davies is instructed to strike the quartz with a ball-bearing, the resulting electrical impulses show up on a meter. And Du Sautoy explains that the distribution of wavelengths on the meter are uncannily similar to the distribution of primes in Riemann's zeta function.


He then calmly explains that the same correspondence has been found between the zeta function graph and the energy states of the electrons inside a uranium atom. Oh, and it shows up in various statistical models as well, like the distribution of buses in London and so on.

By this point, I'm up out of my chair. I'm an avid reader of math and science popularizations, despite the fact that I can't really do complicated math to save my life, and I've never even heard about this before.

A search online last night showed that I wasn't the only one, either. There was even some suggestion that the quartz-zeta function business was a hoax. But then I started turning up stories from years past that mentioned the connection with uranium, which had apparently been first noted by American mathematician Hugh Montgomery and physicist Freeman Dyson back in the 70s.

I'm going to be doing some more digging into this, and hunting down a copy of Du Sautoy's The Music of The Primes, but in the meantime I had to ask all of you nice people. Am I the only one who hasn't heard of this before? (Allison had a vague memory of running into the idea before.) And if I'm not, and this isn't widely known, why isn't it widely know?

I remember something many years ago about Fibonacci (sp?) sequences and prime number sequences showing up in nature, like the zig-zags of diamonds on the rind of pineapples, and even in the stock market.

Neat stuff.
With the distribution of primes being one of the messier and more interesting infinite series, I am strangely pleased to read this. Go primes!
Now that you mention it, I remember reading about the phenomenon way back sometime. And I do have a copy of Du Sautoy's "The Music of The Primes" somewhere around my office waiting to be read, so I guess I'll read it now. But, yes, I have heard about this before, more then once, and it is incredible.

About 10 years back, when I was stationed in Mississippi for the day job, and I began to write SF seriously, I wrote this hard SF story in which the Riemann zeta function was the gateway into a conceptual breakthrough.

I was a lousy writer at the time: lots of inspiration, lots of crazy ideas, but no good way of implementing them in a good story. As somebody wiser than me already said (and I'm probably paraphrasing here): "It's not the ideas you got, but what you *do* with them."

The beginning writer I was back then, used them to some esoteric principles that might have been interesting to layman physicists and geeks, but didn't have any impact storywise.

Way back then, I had a notebook on my nightstand, and wrote down all the ideas I had in the middle of the night. Nowadays, if I can't remember the idea the next day -- nay, the next week or next month -- I just forget about it: only if an idea or concept stays without me nevertheless, won't leave me even if I try, then I use it.

Anyway, I was fascinated by the Riemann zeta function and his hypothesis (and much more: Poincaré Resonances, wave/particle duality, the true nature of time, and more), so I wrote a story exploring it at the quantum level -- through an almost purely mathematical agent -- that eventually explained it all. The budding SF writer conjecturing a theory of everything.

Of course I ditched the story. Because of your post I dug it up from the place where I shouldn't be looking, and reading back things I thought up ten years ago either I was so smart I don't understand my younger self anymore, or I learned to let ideas ripen like, well, cheese, wine or whisky, or both.

Anyway, as far as I can understand my younger self, back then I thought the Riemann hypothesis might explain why time is flowing in both directions at the quantum level, while it's only going in one direction at the macro level ("time symmetry breaking through Poincaré Resonances introducing uncertainty at the macroscopic level". And yes, I don't have a clue what I meant, as well. It sounded good at the time...;-)

There is a lot of esoteric mathematics that might find a practical application sometime, and I agree that the Riemann hypothesis -- still one of the unsolved problems of Hilbert's list of more than a century ago -- is enormously alluring, and might hint at a huge unsolved physical problem. On the other hand, it might only be a tool to explain something relatively minor, like neutrino oscillations (which will then become major...;-).

Fascinating stuff, although someone like Ed Witten will probably shake his head at our naiveté.
Jetse, you are one smart cookie, my friend. Give that story another shot, I think you've got something there.

This is the kind of thing that Greg Egan would go to town with. I wish I had the maths to really do the subject justice.
Not sure if I'm *that* smart, Chris.

However, I have written a story, very recently, that goes at least as deep and as far.

Care to critique it, tell me if it works or sucks? It's gone through one thorough critique and rewrite so far, so it's not unpolished first draft (About 8K). In exchange I'll gladly critique one of your pieces.
Jetse, any other year and I would *jump* at the chance. With the World Fantasy Award judging this year, though, I'm not left with enough time to read anything at all that isn't for WFA consideration. Which considering that I've got a new Kim Newman novella taunting me from my inbox, really and truly sucks.

Hit me up in November if you're still tinkering with it, though. Assuming my brain hasn't rolled out of my ears by then, of course...
Sorry Chris, I knew but still forgot you were a WFC judge (I congratulted you on that in Calgary). I know how much you have to read for that, so my profound apologies for adding to your burden.

Have a good read!
No worries, Jetse!
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