lederhosen

Answers to the other day's questions below the cut.

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(Partly, I think, suggested by a passage in Oliver Sacks' Uncle Tungsten in which he discusses music and the periodic table.)

I know several of my artistic friends are tired of hearing lines like "You're so lucky to have that talent!" when a big part of their 'talent' comes not from luck but years of hard work.

For most media, that's the nature of art - people won't see the hard-earned technique unless they actively look for it, but the creative aspect is there for all to perceive. You can look at a Renoir and enjoy it without knowing a thing about Renoir's painting technique; you can listen to Beethoven's Ninth Symphony and feel its beauty and vibrancy without the faintest idea of how to read music, much less how to score for an entire orchestra.

Beethoven, of course, was deaf.

By the time the Ninth was finally performed, he was so utterly deaf that he was still attempting to conduct it several bars after the orchestra had finished, and he had to be turned around to see that the audience were applauding. For him, I presume, technique was the only medium through which he could know what he was writing, and hear it, if not with his ears, then at least in his mind.

Mathematics is a little like that. At first, the technique relates to things that we can perceive directly - we can lay out apples in a square, two by two, and see that two-squared is four; with a little bit of balancing, we can sit another layer of apples on top of the first four and see that two-cubed is eight. This is something comparable, perhaps, to playing a few notes on a scale and hearing what they sound like.

But take it a little further - what about two-to-the-fourth? - and we can't see it any more. We can still build physical analogies of one sort or another (imagine a second apple inside each apple, perhaps?) but they swiftly become so complicated or so bizarre that we can no longer visualise them in a way that lets us grasp the work we're trying to do; we have to build it inside our heads, a strange sort of shape beyond anything our physical senses can capture.

For the last six months, I've been working on a multi-stage sampling project, immersed in the mathematics that involves; I wish I could describe how that feels inside my head. It's a little like a telescope, with four or five stages that fold into one another; it's a little like an orrery, with a cloud of tiny satellites - each of them, viewed from a certain angle, represents a person - orbiting a swarm of invisible moonlets, which in turn orbit moons, each captive to a planet, each planet captive to the invisible sun at the heart of the system. But it's not really any of these things; trying to describe these structures in terms of things we can see is like trying to turn paintings into music.

So the only way I can share that with anybody else if if they've learned the relevant technique - and even then, they'll need to spend some time staring at the dry equations in order to go from 'notes' to 'music'. I suspect this is part of the problem: most people never get enough exposure to the technique that they're able to get past it to see what it represents. It's like staring at a Pointillist painting, and seeing only a sea of coloured dots.

To get very far with mathematics, you need to reach that level of comfort with the underlying technique, internalise it and practice to the point where you can forget it. I can learn how to press a key on a piano and produce a note of whatever pitch and volume I want, but that isn't enough to make me a musician**; knowing the rules of logic is vital as a first step in mathematics, but it doesn't take me very far on the road to doing what I want in the medium. A musician can listen to music they've never heard before, and guess what the next notes will be; there are similar instincts at work when I'm doing my job. Logic tells me what steps I can and can't take, but instinct guides me in choosing which of those countless possibilities will get me closer to where I want to be. It is a creative process in its own fashion, and knowing where that process will begin and end doesn't detract from that creativity, any more than painting loses its creativity when you're looking at a model.

Not, of course, that the distinction between 'technique' and 'creation' is sharply drawn. In mathematics especially, yesterday's creation is tomorrow's technique - and sometimes technique, picked apart and re-evaluated a la Godel etc, becomes creation again.


*Mind you, a good knowledge of technique almost always increases the ability to appreciate art - it's just that you can get a long way even without it.
**Sadly, I had almost ten years of piano lessons without ever really figuring out that there was more to music than pushing the keys in the right way at the right time.

Some days, proving a mathematical result is like building a Rolls-Royce engine, every piece perfectly shaped and meshing precisely with the next.

Some days, it's like patching up a battered Volkswagen with duct tape and chewing gum, in the hope that it can go just a little further before it falls apart.

Guess what sort of day I just had.

(At least I didn't quite end up having to pee in the radiator.)

ambitious_wench posted a link to a scan of part of Russell's proof that 1+1=2, and asked maths geeks to discuss why it was beautiful.

First off, I have to admit a couple of things: I'm not familiar enough with Bertrand's work to follow how the proof works, and I don't know that I'd call it beautiful. There are more than 300 pages of mathematical setup involved before he even gets to that point, and the proof isn't actually completed until eighty-something pages into the next volume of Principia Mathematica. Combined with aforementioned lack of familiarity on my part, that makes it rather too unwieldy to engage my sense of mathematical beauty. But I might be able to give at least some idea of why it's important.

( And I will do it in verse. Or, failing that, a parable. )

Responding to problem given here.

The problem (all measurements in pixels): Given a 'canvas' size c, and an integer k (in this particular example, k= 3, 4, or 12) we want to find suitable values for the following:

t (constraint)
u (grid unit width, not counting gutters)
g (gutter width)

so that both the canvas and constraint can be divided into some multiple of k units, each of width u, separated by gutters of width g. We want to exactly cover the constraint, but we don't need to completely cover the canvas.

That translates to:
t = k*m(u+g)-g
c' = k*n(u+g)-g

for some integers m, n (greater than m), and where c' is an integer close to (and not greater than) c. There will be various constraints on t, u, and g - for instance, we'd like t to be between 360 and 500, u somewhere around 50-90, and g small.

( An approach. )

EDIT: Pseudocode below. This is a bit less efficient in some parts than the implementation described above, but should be easier to follow and avoids coding in a GCD function.

( Read more... )

EDIT the second: Misunderstood one aspect of the problem - the constraint doesn't have to divide by k, only the canvas. With that in mind, revised algorithm:

( Read more... )

I was entertained to see that Amazon now has 45 reviews for the Rand Corporation's A Million Random Digits With 100,000 Normal Deviates.

"However, we should all be concerned about the fact that these particular number sequences are now all "copyrighted material". Rand has taken all the really excellent random sequences for themselves!"

"I am still trying to figure out what to do with these Gaussian deviates. I may try to sell them on e-bay."

"First spongebob, now this - the sinister homo-erotic subtexts will do nothing but corrupt the youth."

"To summarize my thoughts on this book, a quote from my six year old daughter fits best: "Daddy, I didn't like it. The results in this [book] do not tell us anything about transposed digits or other self-canceling errors." She was nearly in tears."

"The book is a promising reference concept, but the execution is somewhat sloppy. Whatever algorithm they used was not fully tested. The bulk of each page seems random enough. However at the lower left and lower right of alternate pages, the number is found to increment directly."

"...these random digits are just too old fashioned. I get a feel for the 40s and what life might have been like, but I felt it lacked that "fundamental truth" that would allow this book to span generations to come. In todays world of global communications, econmic uncertainty, terrorism and preemptive wars, I think we all could have used a few negative numbers to really drive the point home. I mean even a few more zeros would have helped."

"If you liked Finnegan's Wake, you'll love this."

Thanks to sclerotic_rings: Fermat's spirals!

Arthur asked me to give him a hand with a mathematical problem the other day. Good news: on investigation, there's a single-line Matlab command that will do what he wants. Bad news: he needs to do it in Excel. (Eventually solved, but rather more work that way.)

Via usekh, this hurts my brain. Still trying to figure out whether it's intended tongue-in-cheek; I suspect so, but it looks as if some of the subjects are taking it at face value. Reminds me of Kim Newman's Pitbull Brittan, a stirring tale of a Thatcherite superhero who goes around walloping striking miners and combating evil unionists.

CBC story, by way of james_nicoll: "...a spokesperson with the Greater Toronto Airports Authority said lightning was causing technical problems with the airport's lightning-detection system".

Also, lovely octopus icons! And I have yet another Girl Genius icon, too.

Also, goddamn earworms! (See current music, below.) At least there's this site to 'help'.

Also also, I am disturbed to read this passage on Wikipedia: Badger Badger Badger, not for its content but for its implications: "The audio doesn't match up exactly with the visuals. This becomes plainly discernible after 30 minutes of continuous playing. At this point, the audio is at the twelfth "badger" in the first line while the visuals show the mushroom. Over the next few hours, the visuals continue to play faster than the audio, achieving maximum separation at 4 hours 12 minutes. Thereafter the gap begins to close, and becomes synchronized again at about 8 hours 30 minutes. This process continues cyclically."

Also also also, somebody at the Onion feels the same way I do about the age-old "how do I write female characters?" issue.

So, cerebrate mentioned that 'mb' is the abbreviation for 'millibit', and the question came up: what possible use is there for such a unit? How can you have a fraction of a bit?

This is much like asking "how can you have a shape with a fractional number of dimensions?" At first glance, it makes no sense. But in trying to extend familiar notions to new applications, it becomes a very useful idea indeed, with applications in everything from data transmission to investment theory.

(Before I start, a caveat: It's been ten years since I did this stuff. Some of the details and/or terminology may be a little off. The gist of it should be correct, though.)

( What is information entropy? )