Grimm and grimmer

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A few weeks ago I checked out Coursera, the website that grants access to free online classes in a range of topics led by professors at top universities. Pretty good deal, right? Several of the imminent classes sounded interesting: Introduction to Sustainability, for instance, and Quantum Mechanics and Quantum Computation. However, knowing nothing about either of these subjects I decided to sign up for a class that I have some background in: a literature course called Fantasy and Science Fiction: The Human Mind, Our Modern World. After signing up, though, I realized I didn’t really have an extra 10+ hours a week necessary for all the readings, lectures, and essays.

In fact, the main reason I was interested in the class was the first reading, Grimm’s Fairy Tales, and thus I decided to just read them on my own.

As you might expect, the tales are odd. We know some of the famous ones–Hansel and Grethel, Snow White, Rapunzel, Cinderella (called Aschenputtel in German, a decidedly unattractive name). And while there are several tales of beautiful maidens being rescued from evil stepmothers by kings, there are many more with talking animals and and lies that are not always punished. Most of the stories, really, seem written by children, with a child’s wandering logic and miscellaneous details and rules.

In the woods

Children’s tales are generally fashioned to teach lessons, but Grimm’s lessons are often puzzling. In Hansel and Grethel the poor father is loathe to leave his children in the forest, but agrees with his cruel wife. When his children reappear at the house after Hansel’s flint stones lead them back, the father is overjoyed yet must agree once more with his wife to bring them out into the woods. For, as the story asserts, “he who says A must say B too, and when a man has given in once he has to do it a second time.”

In some stories, young girls who make promises are forced to make good on them (such as in The Frog Prince). Yet, in other instances, cruelty is rewarded. Cat and Mouse in Partnership tells the story of a cat and mouse who save a pot of fat to tide them through the upcoming winter. The cat can’t wait to eat it and steals away three times to eat it by himself. When the winter finally falls and the mouse discovers the cat’s duplicity, the cat eats the mouse. The final sentence of the story simply reads: “And that is the way of the world.” Hardly comforting. Likewise, in The Wolf and Seven Goslings, a wolf asks a baker to cover him in flour so that he might disguise himself and eat the eponymous goslings. The baker at first refuses, not wanting to collude with the evil wolf, but finally agrees. The story gives us this wisdom: “And that just shows what men are.”

Perhaps the strangest two stories I read, though, were The Death of the Hen and The Straw, the Coal, and the Bean. In the first, a hen dies choking and several animals join in to carry her body to a burial site. While crossing a stream, though, they all drown. Her husband remains, buries her, and then buries himself alongside because he’s so filled with grief.

In the second story, a straw, a coal, and a bean escape a woman’s stovetop and set out together on a journey. They, too, come to a stream and the straws lays himself across it so that his new friends might pass over. The coal stops in the middle of the straw out of fear, but the coal’s heat burns the straw and they both fall in the water and…die? Can we say that about a coal and a straw? Meanwhile, the bean finds his friends’ demise so hilarious that she literally bursts with laughter. A kind tailer sews her up with black thread. So, Grimms, what are we to make of it all? Only this: “All beans since then have a black seam.”

Ah yes, an important lesson to teach the children.


Dimension X

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In honor of this biweekly theme I’ve been reading up on string theory, trying to make sense of it all. Finally, after pouring over papers and definitions on scientific websites, then regressing to wikipedia pages, in which every link-text word was just as completely unintelligible, I turned to a better resource: TED talks. You probably know this by now, but here’s the idea: the conference organizers for TED (Technology, Entertainment, Design) invite experts to discuss a topic in 18 minutes or under in a manner that is both highly education and highly understandable. Here’s physicist Brian Greene telling you everything you need to know to start to understand string theory:

My only question: Mr. Greene’s talk is from February 2005 and he affirms that experiments within “the next 3 years, 5 years, 10 years” should show whether string theory is true. Granted, we’re still within the 10-year mark from his original talk, but I wonder what the status of the experiments are. Scientists in Italy announced their findings just last year of particles that moved faster than the speed of light. Does string theory account for that? Does it want to? The Large Hadron Collider in Switzerland that Greene mentions is currently undergoing routine maintenance and will resume colliding particles in March. Stay tuned, I guess…

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:

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?