I need to focus on things, not this weblog. My last few posts have felt kind of rushed and forced anyway, not exactly the quality content I would like to provide. And work needs to be done! So I refuse to allow myself to spend any more time on here until … Friday. That seems reasonable. I should be able to get good things done by Friday.
I’ll be about elsewise if anyone needs me ^.^
Still sadly without my trusty copy of the Geometer’s Sketchpad, I went and bought myself a compass today. And as such, I think the perfect way to mark such an event would be to discuss the optics of rainbows. Why rainbows happen, where they happen, and what I’m sure everyone has been wondering – what would happen if it rained diamonds?
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This is the first of at least two posts I feel I must write to explain concepts I want to use in future posts, for future problems and ideas. A fairly basic, but necessary post that will allow bigger and better things to come. Linear Algebra, I’m coming for -you!-
Most people have an inherent sense of numbers. Quantities you can add, subtract, multiply, divide. 2 + 2 is always 4, and so it goes. Some may visualize a line of numbers extending off to infinity, with an inherent order. Some may even think of the set of numbers. However you think of it, most people have some idea of how numbers work.
Interestingly, Mathematics, and many Mathematicians, function in a parallel world, at a glance not too unlike our own, but certainly different. 2 + 2 does not necessarily equal 4, because there may not even be a 4 to equal, let alone 2’s to add. Numbers are no longer quantities, but shapeless objects, capable of interacting with each other, but little else.
But it is those interactions that make them worthwhile.
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Almost since the opening of The Dark Knight (or the Dark Kitty, as I like to call it), there has been a small to-do raging in various sections and substrata of the internets. This ruckus centers on this list, the Top 250 Films on the Internet Movie Database, the compilation of hundreds of thousands of users’ ratings of movies.
The crux of the issue is that, shortly after its release, The Dark Knight has rocketed to #1 on that list, upsetting the longstanding favourites such as The Godfather, The Shawshank Redemption, Casablanca, not to mention 247 other films. Battles of wit and will have raged on message boards, attacking and defending the idea that this newcomer, a comic book movie no less, deserves a place among these classics. Indeed, this sparked further wars, people ranking The Dark Knight a 1 out of 10, just to lower its rating. And now it seems as though Shawshank fans and Godfather fans are duking it out, with Shawshaking recently overtaking the Godfather.
In brief, it’s utter insanity, inanity, the best, the worst, and everything you could possibly hope for from the internets.
But it does bring up some interesting points. What it means to be a good movie, what rankings like this actually mean, flaws in the system and … stuff.
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My good friend Jason Rosenhouse had an interesting discussion a few days ago on whether or not modern mathematics is reliable. The gist is that the deepest of maths are increasingly built on a foundation of theorems and proofs that are so complex that it effectively becomes an act of faith to take them as true and correct. And when further theorems are built on these theorems, how confident can you be in the structure of mathematical knowledge?
It’s an interesting point, and it gets to something I think about from time to time – the limits of mathematical knowledge.
The idea is that humans, when doing math, have an understandable bias towards proofs that are readable and (easily) checkable. The question then is what portion of mathematical ‘truth’ can be expressed in such a way.
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So, I know I’ve been a little quiet lately. This is mainly do to the confluence of two factors – one, I’ve actually been doing some work lately. I’ve been doing some programming (and avoiding doing some programming), and mulling over a series of papers on the four color theorem, written between 1892 and 1947. Interesting stuff – the author basically transforms the map coloring problem into solving a system of simple linear equations. I may talk about that some later.
Secondly, I’ve been wrestling with my own frustrations about a much simpler problem. I’m sort of chasing my tail on this one.
To begin with, imagine a function f(x) that satisfies the following equation.
The problem is to prove that it is periodic. Further, what other values besides 
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The New York Times has a good piece today, tonight? tomorrow?, with Dennis Overbye answering questions on physics. Many good topics are covered, like time travel, the big bang, dark energy, the LHC, and the nature of the universe. Interesting stuff and a fairly good overview of a lot of areas of interest, I think.
I think the most amazing thing I have covered lately is dark energy. The galaxies are falling up. How amazing is that? But I have already gone on about that above. In general, I am continually amazed at the passion, resourcefulness and stubbornness of humans, both individually and in groups, and how they manage to pick away at finding out things you would have thought were impossible to find out. There is a robot baking dirt on Mars as I write this. Cosmologists are finding traces of physics from when the universe was less than a trillionth of a second old and hotter than any particle accelerator will ever recreate in a fuzz of microwaves in the sky left over from the Big Bang. They fixed the Hubble Space Telescope, again and again and again.
But then, I’m a mathematician, what do I know ^^
First, a few words on modular arithmetic. I really wanted to avoid talking about modular arithmetic, but I tried writing this post without it, and it became overly complicated and I used the word remainder, and remainders of remainders, about eleventy trillion times. So here it is. Modular arithmetic is basically shuffling around integer multiples of things. We say that two numbers A and B are ‘congruent mod k’, if A = n*k + B for some integer n. So, for example, 17 and 7 are congruent mod 10 since 17 = 10*1 + 7. 107 is also congruent to 7. -3 is also congruent to 7, since 7 = 10*1 + -3. Notice the link to division and remainders. 107 gives a remainder of 7 when divided by 10 and is congruent to 7 mod 10, because in each case you’re subtracting the largest possible multiple of 10 from 107. When two things differ by an integer multiple of k, they are congruent mod k. We can think of divisibility then as when a number N is congruent to 0 mod k. 100 is congruent to 0, since 100 = 10*10 + 0, 100 gives a remainder of 0, 100 is divisible by 0, it’s all saying the same thing.
There are a small set of rules people can use to determine whether an integer is divisible by numbers like 2, 3, 4, 5, 6, 8, 9, 10, etc. Rarely does anyone go much beyond this, or even address the lonely 7. For the most part, these rules are simply shortcuts for calculating congruences. Congruencies? For example, we know that any number N can be written in the form N = 10*A + B, where B is the last digit of the number, and A is the rest of the digits. We can also say then, N = 5*(2*A) + B. Notice that B Thus, N is congruent to B mod 5. If N is divisible by 5, it is congruent to 0 mod 5, so we want B to be congruent to 0 mod 5. The only values from 0 to 9 (since B is a single digit) that are congruent to 0 mod 5 are 0 and 5. Thus, N is only congruent to 0 (and thus divisible by 5) if it’s last digit is a 0 or a 5.
All the other divisibility rules accomplish the same thing, finding a shorter way to calculate these congruences with the divisor. Dividing by 3, it turns out that N is congruent to, and thus leaves the same remainder as, the sum of the digits of N.
But there’s a way at getting into the problem indirectly, sort of sideways. Calculating whether or not a number is congruent to 0 mod the divisor without actually calculating the remainder.
In brief, start with your number N, (7931, for example). Take the last digit, double it, and subtract it from the remaining digits. 793 – 2*1 = 791. Repeat. 79 – 2*1 = 77. Eventually you will get to a number that is obviously divisible by 7, or obviously not. If you reach a number divisible by 7, the original number is as well. We reached 77, 7*11, so 7931 is divisible by 7. And if you check it is. This is very fast, as you effectively lose a digit with each step ^^
Below, I explain why this works, and extend the same trick to derive simple divisibility tests for 11, 13, 17, and 19.
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As I see it, this is a very graphic demonstration of how irrational numbers relate to rational numbers. It definitely concerns some inherent property of rational numbers. The problem is, I really don’t know what that property is or how to describe it.
We start with simple periodic functions like sine and cosine. Being periodic, they repeat with a certain period : ) But that period can be controlled. Below, Cos(x), Cos(3/2*x), and Cos(2*x) are all plotted on top of each othe, the first in blue, second in red, third in yellow-ish. As x increases, the functions oscillates and eventually repeat.
The Clearly, Cos(2*x) oscillates twice as fast as Cos(x) and has a period half as long.
Basic stuff, but then we start combining things.
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Via slashdot, I found this article on an Amazonian tribe that does not seem words to describe numbers and counting. Rather, they have relative quantifiers that indicate a quantity from 1 to 4, 5 or 6, or ‘many’. In effect, small, medium, or large.
Of course, it’s probably a bit of a stretch, but as counting forms much of the basis of math (though I wouldn’t mind hearing discussions on that), this relates, as I see it, to whether mathematics is something inherent to the universe itself, or rather something more akin to an evolutionarily advantageous artifact of our brains. Some of our brains, anyway.
Evolution almost certainly plays a role here, as the language in question is spoken by about 300 people, and they likely haven’t spent a lot of time doing commerce with other cultures.
There were some interesting thoughts raised in the comments. For instance, one person noted that most people can easily recall about five separate objects at a glance, and the perception of quantity is likely linked to how easy it is to remember the objects. Another noted that this may be a different kind of language artifact – the researchers were having them count foreign objects, but should’ve asked them to count how many children they had, or some other familiar kind of object, the idea being that perhaps different words were used for different types of things. This is interesting, because linking quantity and type descriptors would greatly diversify what could be expressed.
There are far more questions I’d like to ask. For example, the researches only went up to 10 objects. What if they went straight from 10 to 1000? While both are ‘many’, there’s a pretty significant visual difference from the sheer amounts.
Also, I do wonder whether or not, even with the limited range of quantifiers, if the speakers here make use of any kind of computational-esque rules. For example, two ‘medium’ groups combined would always produce a ‘large’ group. Very fuzzy sort of addition. Of course, two ’small’ groups combined don’t necessarily produce anything definite.
It’s all very interesting.
