lederhosen: (Default)
For your entertainment, inspired by a protracted debate with our SM1 lecturer. The basic idea of these questions isn't original to me, but I've modified them somewhat to make the phrasing less ambiguous.*

1. A friend mentions that they have two kids. With no other information, what are the odds that they have (a) two boys, (b) one boy and one girl, or (c) two girls? (For the sake of these problems, assume that half of children are boys, half are girls, and people don't have a natural tendency to children of one sex or the other.)

2. Looking at one of their many bookshelves, you spot a Saddle Club book, which (applying gender stereotypes) you may take as indication that at least one of their children is a girl. Based on this information, what are the odds that they have (a) one boy and one girl, or (b) two girls?

3. You mention the Saddle Club book to your friend. He replies "Yeah, that's Mary's." What are the odds that both their kids are girls?

4. As above, but the reply is "Yeah, that's Mary's, she's my eldest."

5. Different friend, same dilemma - two kids, you don't recall their sexes. Being sneaky, you ask "Would your oldest like to come to my kid's birthday party?" and the response is "Yes, she'd love to." What are the odds that both their kids are girls?

6. As above, but the response is "Mary? Yes, she'd love to."

7. You find a Saddle Club book lying on the ground, inscribed 'To Mary'. You ask around the local schools and in a flagrant breach of privacy they give you the addresses of several dozen Saddle-Club-age 'Mary's in the neighbourhood. You go to the first house on the list and, by examining shoes again, deduce that the family has two children. What are the odds that both are girls?

*Which was surprisingly difficult to do - it's actually very hard to convey no more and no less information than one means to when writing these things.
lederhosen: (Default)
(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.
lederhosen: (Default)
Also, why does MathType not have an 'approximately less than' symbol? It's a perfectly legitimatish concept, dammit.
lederhosen: (Default)
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.)
lederhosen: (Default)
So, have been working long hours and scratching my head a lot the last three days over a tricksy little problem.* After several false starts I finally figured out the right way to approach it, did something sneaky with inventing people who don't exist and then ignoring them**, and got a mucking big ugly-looking expression.

Hey, I can cancel those bits...
...and those bits...
...and those bits too...
...and everything else, leaving zero.

I'd never realised it was possible to be pleased and disgruntled at the same time.

*Tricksy to me, anyway. I'm fairly sure there's a standard result, but I'm a reinvent-the-wheel sort of guy.
**This actually made some sort of sense at the time.

1+1=2

Sep. 16th, 2007 03:14 pm
lederhosen: (Default)
[livejournal.com profile] 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. )
lederhosen: (Default)
Puzzle for physics geeks: Take one nonconducting icosahedron (that's a d20 for all you gamers). Replace its edges with 1-ohm resistors, connected at the vertices, so you now have a web of thirty resistors.

What is the resistance between opposite corners of the icosahedron?

(There are two ways I know of to do this one. The hard way involves a lot of simultaneous equations; the easy way takes about five seconds of mental calculation, once you know the trick.)

Gridding

Mar. 24th, 2007 05:17 pm
lederhosen: (Default)
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... )
lederhosen: (Default)
Via [livejournal.com profile] mathsex, Tupper's Self-Referential Formula. It's not a deep result - all it really comes down to is that an algorithm for encoding small images can be used to encode an image of itself - but it's cute, and an illustration in how good a complex-looking formula can be at obfuscating a very simple rule.
lederhosen: (Default)
First off, a touching Valentine's Day video for medical geeks. (Contains squicky bits.)

And the burning question of the era:

[Poll #922269]

Bonus points if you can name all those options.
lederhosen: (Default)
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."
lederhosen: (Default)
Today I convolved some data. For the uninitiated, this is pretty much the signals-processing equivalent of sex, only uncomplicated by other people.

Then later on, I deconvolved some other data. I'm not sure what this is the signals-processing equivalent of.

And yesterday I finished painting a figure I've spent ages on (Sir Stephen Swift, to those following the Aramia game; he makes an appearance in [livejournal.com profile] edward_dujean's imminent summary). I'm very pleased with how he came out. The face and eyes (always fiddly) turned out well, with very little effort on my part; the blacklining, shading & highlighting on his armour came up very nicely, and the basing isn't too shabby. I'm tempted to enter this fellow next time there's a novice contest; I can see one or two spots where he'll lose marks (I didn't entirely succeed in getting all the casting lines off, and my freehand on his shield is a bit wobbly), but I'd be interested in commentary on the rest of it. One of these days I will get a camera with a decent macro mode so I can bore you all with my miniatures in better focus.

Incidentally, it is entirely a coincidence that one of the major ethnic groups in my setting just happens to have the same rather dark skin that I've found easiest to paint ;-)
lederhosen: (Default)
Some of you will have heard of Paul Erdős, the famous Hungarian mathematician. Erdős was famous for a number of things besides his own mathematical results. He wandered around the world with a suitcase full of mathematics and amphetamines; he would show up unexpectedly on a colleague's doorstep, stay for a few days of mathematical work, and then wander on to another collaboration somewhere else. He talked about people who'd stopped doing mathematics as having 'died', and he spoke of 'the Book', in which God (or, as Erdős called him, the 'Supreme Fascist') kept all the best and most elegant proofs of mathematical theorems.

Above all, he was famous for the volume and range of his collaborative work; while Erdős was an impressive mathematician in his own right, his greatest contribution to 20th-century mathematics was the way in which he brought people together. Mathematicians celebrate this with the concept of the Erdős number: Erdős' number was 0, the 500-odd people who co-authored papers with him had number 1, their co-authors have number 2, and so on. Most mathematicians link to Erdős in less than five steps.

Erdős' first two co-authors, back in 1934, were a fellow called Turán (who has no further part in this tale) and another Hungarian Jew by the name of George Szekeres. Not too long after that, George and his wife Esther - also a Hungarian Jew - fled Europe for Australia via Shanghai. They became fixtures of the local mathematical scene; George was Professor of Mathematics at UNSW for many years, and although he retired in 1976 he was the sort of person to whom that just means 'no longer drawing a paycheck'. I met them once or twice, although [livejournal.com profile] reynardo's mother knew them better than me.

Anyway, Rey just called to let me know that George and Esther both died yesterday; they'd been in poor health for a while, and they passed away within about an hour of one another. While trying to remember how to spell 'Szekeres', I came across a rather sweet story... )

Miscellany

Aug. 3rd, 2005 09:59 am
lederhosen: (Default)
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 [livejournal.com profile] 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 [livejournal.com profile] 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.
lederhosen: (Default)
Provoked (and I mean that in the nicest way) by a post from [livejournal.com profile] laochbrann:

What are the rules that govern behaviour selection in a potentially violent confrontation?
- For any given behaviour in such a confrontation, what are the strengths, weaknesses, opportunities and risks associated with that behaviour?
- How can a game be designed that encourages players to select effective behaviours in potentially violent situations?


(FWIW, I have seen a Flash game that touches on this, but it's pretty limited.)

One of the catches in using game theory to evaluate/suggest behaviour is that the context of the game is crucial to answering this sort of question. If you treat a given scenario as an isolated problem, you can end up with a very different answer from the one you get if you view it as one in a long sequence of interactions.

Read more... )

I guess the short answer to those questions is "it's very difficult" ;-)
lederhosen: (Default)
So, [livejournal.com profile] 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? )
lederhosen: (Default)
Prompted by a now-deleted post on a snark* community complaining that 'grok' is a fictional term and is not proper language, regardless of whether it's in the OED...

Since everybody else seems to have strongly-held views on the subject of New Words, I thought I'd have a go at coming up with a compromise that will annoy everyone equally. And just to make sure of that, I'll start by calling on mathematics - specifically, information theory - to dictate a few rules of language. The math-phobic can skip the cut; it's just there to justify some of the principles presented after it.

Read more... )

[livejournal.com profile] lederhosen's principles of vocabulary. Note that some of these conflict, and have to be weighed against one another, except for rule 9 which is non-negotiable.

1. Important (in particular, common) concepts should have compact expressions. The more it's used, the shorter it ought to be.

2. As our world and context changes, so does the importance of various concepts. It's no longer as important to be able to say "bear" in a hurry as it used to be; "computer", OTOH, has become ubiquitous.

3. To keep language effective, it needs to be able to change to reflect these facts. Where there's a need for a new word, or a shortening of an existing word, we should be willing to accept such novelties.

4. If a lot of people adopt a neologism, this is evidence that such a word was needed, and can be taken as grounds for its acceptance.

5. Exception to #4: if there's already a perfectly good & compact word for this purpose, use that one instead. Neologisms should be created due to need, not ignorance and laziness.

6. Exception to #4: Where possible, neologisms should be user-friendly. As far as possible, this means following existing patterns of language. Adapting existing English is great; borrowing from other languages is good. Words derived from Latin etc. are more likely to be readily understood and accepted than words made up from scratch.

7. Exception to #6: Sometimes, insistence on following existing patterns may get in the way of #1 and #3. Latin constructions tend to become fairly long; as such, they're admirably suited for necessary but uncommon pieces of vocabulary - for instance, many academic terms - but less so for things like "blog".

8. User-friendliness also means avoiding ambiguity. English already has more than enough homophones, thank you very much.

9. Numbers are not letters and should not be used phonetically, EVER, with a possible exception for Sinead O'Connor when covering Prince.

I quite like 'grok' because it satisfies almost all of the above principles. It offers a compact and unambiguous word for an important nuance that isn't adequately conveyed by any other short form - 'understand' and 'comprehend' are longer, and as with 'know' they lack the connotations of fully absorbing and coming to terms with the concept. (Indeed, the fact that it's hard to explain 'grok' except by example is a proof that the niche exists.) The only one it doesn't satisfy is relationship to pre-existing language, and I think the others greatly outweigh this.

*Carrollites will no doubt appreciate the irony.

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