Phantom math

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“Don’t be too sure,” said the child patiently, “for one of the nicest things about mathematics, to anything else you might care to learn, is that many of the things which can never be, often are. You see,” he went on, “it’s very much like your trying to reach Infinity. You know that it’s there, but you just don’t know where–but just because you can never reach it doesn’t mean it’s not worth looking for.”

The Dodecahedron man in The Phantom Tollbooth

So says the .58 child in Norton Juster’s masterpiece children’s book The Phantom Tollbooth (the child is part of an average family with a mother, father, and 2.58 children–he’s the fractional son). I’ve written about The Phantom Tollbooth before because it’s so full of wonderfully odd explanations of concepts. Milo, the main character, and his retinue make it to Digitoplis where numbers are mined and sent around the world (though they are highly valuable it’s alright when a few numbers drop to the ground because the broken ones can be used for fractions).

They also eat subtraction stew–the more you eat, the hungrier you get. When Milo asks to see the biggest number in the kingdom, he is shown an extremely large 3–it took 4 miners to dig it out. He tries to explain that he wants to see the number of greatest magnitude. The Mathmagician (ruler of the land of Digitopolis) tells him to think of the biggest number possible, and then add one. And then add one again.

“But when can I stop?” pleaded Milo.

“Never,” said the Mathemagician with a little smile, “for the number you want is always at least one more than the number you’ve got…”

Sometimes I wonder if reading The Phantom Tollbooth at a young age gave me a head start on my math career. Thank you, Milo.

Proof positive


The Simpsons--making me smarter yet again

In a Treehouse of Horror episode, Homer gets sucked into a bizarro dimension where he becomes 3D. Take a closer look at that equation behind Homer: 1782^12+1841^12=1922^12. Looks relatively simple, right? If you were to pull out a pocket calculator, it would confirm that this equation is true. And yet, not only would your pocket calculator be wrong, it would belie a mathematical enigma that went unsolved for 358 years.

A "tree" made out of Pythagorean relationships

We are all taught in geometry class about Pythagoras’s discovery of the special nature of right triangles. The relationship of the two legs of the triangle to the hypotenuse is a^2+b^2=c^2. Duh. A-squared plus b-squared equals c-squared has the familiar ring of E equals MC squared…even if you don’t exactly know what it all means.

In 1637, mathematician Pierre de Fermat wondered if such a relationship would be possible, though, when values are raised to a power higher than 2. After some study, he conjectured that in fact no three positive integers a, b, and c can fulfill the equation a^n+b^n=c^n where n is greater than 2.

When I heard about this theory, later termed Fermat’s Last Theorem, it seemed irrational. I thought through the order of cubed numbers (numbers raised to the third power): 1, 8, 27, 64, 125, 216, 343…surely if you went far enough, you would find two cubes that could be added to each other and result in a third cube? But no–it’s impossible. Even so, though, it seemed like the explanation for this should be obvious. It should, it seemed to me, be an elegant solution that matches the elegance of the original conjecture (and I wasn’t the only one who thought this–the fictional Lisbeth Salander from the wildly popular Stieg Larsson series is obsessed with finding her own proof for the theorem).

And yet, while Fermat was correct in his conjecture, it took several hundred years and extremely elaborate proofs to show why.¬†Andrew Wiles, a British mathematician, submitted a proof in 1993 that was based on numerous other proofs that had already gone up against the theorem. His theorem contained a major error, but with another year of diligent work and vetting from other mathematicians, Wiles submitted a finalized proof in 1994, published in 1995. The proof is more than 100 pages long and draws from such esoteric concepts as absolute Galois groups, Hecke eigenforms, and abelian variety. Wiles’s proof has many ramifications in algebraic geometry and number theory, and he was knighted for his work.

I think they just should’ve sent Matt Damon in to solve the thing:

Mathmagic land

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We watched the Disney movie Donald in Mathmagic Land for an 8th grade geography class. Sometimes when I’m out at a bar and challenged to a game of pool, I think of the scene where Donald Duck learns about billiards. Somehow my shots never turn out as clean. Fast forward to 0:15 for the video to start:

If you’ve got a half-hour to see all of Donald’s travels, check out the full-length below:

How to be a mathematician

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What you’ll need: graphing calculator, pocket protector

Songlist: U+Me=Us (Calculus) by 2together

Further reading: Fermat’s Enigma: The Epic Quest to Solve the World’s Greatest Mathematical Problem by Simon Singh

The greatest moment of my young life (besides winning the 4th grade spelling bee with the word “terrarium”), was the Math Masters of Minnesota awards ceremony in 6th grade. Math Masters had three competitions: group problem solving, individual problem solving, and individual fact drill, for which you had to answer as many of the 75 arithmetic problems as you could in 5 minutes. After learning that my team got second place in the team competition and I got third place in the individual problem solving, I anxiously awaited results for the fact drill. I’d answered all 75 questions and felt pretty confident. As the names were called for 10th place, 9th place, 8th, I got increasingly excited at not hearing my name. When 2nd place was announced, I felt like a beauty pageant winner. And then I finally heard my name, 1st place for fact drill.

I knew I was good at math, but it never occurred to me why others might have trouble with the way math is taught. One of my friends explained that her brain works both qualitatively and logarithmically, which research has shown is the intuitive way humans first conceptualize numbers. In her mind, the difference between 1 and 2 is unequal to the difference between 9 and 10, since 2 is twice the size of 1. Also, she was part of our weekly trivia team before moving to the east coast; one night we had a question about how many outfits could be made from 3 hats, 4 shirts, and 5 pairs of pants and she complained that some hats wouldn’t go with more than one or two possible outfits. Sombreros and berets are not¬†interchangeable.

That's a good-looking logarithm

Unfortunately for my friend, we’re taught a linear, quantitative system in school. This is the basic starting point for all higher mathematics (and, more importantly, the counting of money). Yet linearity is not necessarily as integral to the workings of the world as we might think. The land works in logarithmic ways: earthquake magnitude and acidity are common examples. Even the way we perceive time is a logarithmic function: the older you get the faster time years seem to pass because they are a smaller ratio of your overall life.

I was fascinated by these concepts when my friend first told me about the workings of her brain, and I told her she’d make a great addition to a mathematics department, since she would go about solving problems in a much different way. She disagreed, as she would have no common language with the geniuses of the linear/quantitative model.

She’s probably right, but I thought about how I could go into higher studies in math and use what she’d told me to develop some new theory of the universe. I never went beyond high school calculus, though, so I’d have a lot of catching up to do. Sometimes I regret that I didn’t progress further, as I loved pretty much every math class I ever took. Sometimes I think I’m doing that sixth grade version of myself a disfavor. On the other hand, I’m no Will Hunting. I was good at adding and subtracting in record time; who knows if I’d have been good at manifolds or combinatorial topology…


PS. Oh man, I love numbers! Today is 1/2/12. And as of today I have 222 posts, 2 pages, 22 categories, and 522 tags. WHOAWEIRD. Amiright?