Sunday, July 31, 2011

How I Get Animated

A few months back, Miss Polly and her gigantic brain thought of a fun participatory event for, and thus asked readers and groupies to write about how they get animated. Incapable of refusing the opportunity to compile a Top Ten list (the 4/4 pop single of information age discourse) and blather on about cartoons (I ONLY WATCH CARTOONS), I wrote a few (thousand) words about my favorite television 'toons and sent them her way.

And that's why today's update is about cartoons instead of stars, books, bugs, or any of the other nerd stuff I'm into. Yeehaw!


9. Downtown

In the 1980s, MTV aired music videos. In the 2000s, it aired asinine reality shows. Something had to fill the gap during the 1990s, and that something was original animated programming. MTV aired some killer cartoons between 1991 and 2003. While you could commend the network for funding and airing high-quality stuff like The Maxx, Oddities, and Clone High, it's pretty hard to admire the way it treated most of its cartoons -- underpromoting them, changing their time slots without warning every other week, and then cancelling them after they failed to attract as big an audience as Beavis & Butthead and Celebrity Deathmatch.

Downtown's thirteen episodes aired in 1999, during the twilight years of MTV Animation. The premise is simple: each episode follows a group of about seven teens and twenty-somethings through a day or a week or so of Manhattan life. It's both a slice-of-life sitcom and a period piece -- a time capsule containing a vision of New York at the turn of the century, before 9/11, Web 2.0, Bloomberg, and obscene living costs. Although pop culture, subculture, and NYC in-jokes are heavily featured, Downtown never really glorifies urban life. Its New York is simultaneously beautiful and disgusting; resplendent with possibility and lonely as hell.

Downtown doesn't just focus on dialogue; it makes conversation into its very subject. It's safe to guess that anyone with even a mild creative spark has occasionally found himself bullshitting with friends, listening to effortless exchange of stories and jokes, and wishing he could bottle it up and record it somehow. Downtown' creators do just this: collect the anecdotes and exaggerated stories from their friends around town and make fantastic cartoons about them. (Most of the voice actors are these friends; you won't see them credited for voice work anywhere else.) It must be emphasized that the writers/actors never glorify or congratulate themselves and their friends for being such hip cats. Pretty much every member of the main cast bears some glaring, self-destructive flaw, but we like and relate to them anyway, the same way we forgive our own friends for their quirks and shortcomings.

Downtown's dialogue follows curving tangents, launches into flights of fancy, and lapses into arguments; people stumble over their words and tell different versions of the same story. When the talking takes off, the animation follows suit, contorting and coloring itself to match the tone of the conversation. It takes the same jokes and stories we tell our friends and makes them as fascinating and funny as they are to us before the telling as is only possible with animation.

HIGHLIGHTS: Episode 6, "Graffiti." Chaka, Fruity, and Matt sneak into the subway tunnels during a service outage so Matt can do some tagging. Adventure ensues. I remember my friend's brother telling us stories about his days as a young scofflaw, when he and his buddies used to visit the subway "graveyards" in abandoned tunnels. This episode holds a certain vicarious nostalgia for me.

10. Megas XLR

I actually rate Downtown higher than Megas, but it helps to talk about the one before bringing up the other.

After MTV flushed their baby down the toilet (it never even bothered airing the final episode), Downtown's main creator, story editor, and lead animator regrouped and put together a pilot for Cartoon Network called "Lowbrow." It earned the most viewer votes during a 2002 contest, and was subsequently greenlit for a thirteen-episode run (followed by a thirteen-episode second season), redubbed Megas XLR.

The backstory is a bit complicated, but here we go anyway: Coop, a basement-dwelling twentysomething gearhead from Jersey City, discovers a busted mecha in the local junkyard. He takes it home, repairs it, pimps it out with sexy flaming 8-ball decals, and replaces the missing head with a 1970 Plymouth Hemi Cuda convertible. Unbeknownst to him, his new toy actually comes from a distant future in which humanity is mired in a losing battle against a ferocious alien empire. "Megas" was a prototype weapon built by the aliens and stolen by the human resistance movement, who sent it back in time for safekeeping. Its human pilot, Kiva, returns to the 21st century to retrieve it, but finds Coop unwilling to give it up. What's more, she can no longer operate it herself: with the cockpit missing, Coop jerry-rigged its control interface into the Hemi Cuda's dashboard with a complicated array of switchboards and video game controllers. Discovering that a lifetime frittered away playing video games in his mom's basement has turned Coop into a first-class mecha pilot, Kiva decides to remain in the 21st century and train Coop to be humanity's savior in the future war against the alien invaders.

The "monster of the week" format is much simpler. Coop does something stupid with Megas and invites some horrible alien terror into Jersey City. Kiva yells at him. Coop fights off the aliens and levels Hudson County in the process. Rinse, repeat, delight.

Like its ill-fated older brother, Megas XLR exhibits a keen fascination with subculture, local color, and the accessories of geek life -- science fiction, anime, video games, etc. Downtown's reality-grounded setting restricted it to only referencing the latter in conversations, but the science-fiction/comedy format of Megas allows for direct parody -- Coop and his pals actually live out the stuff Alex and his friends at the comic book shop can only fantasize about. Everything from Mobile Suit Gundam to Sailor Moon to Transformers to Endless Odyssey gets lampooned, and frequent nods toward video games, comic books, and American "guy" culture are the norm for each episode.

One thing that doesn't get much love from Megas is MTV. In almost every episode, some fictional representation of the network gets the shit blown out of it. Clearly the crew was still smarting from MTV's treatment of Downtown, and it's really hard to blame them for bearing such a grudge.

HIGHLIGHTS: The episode in which Megas gets impounded, forcing Coop to jump through the hoops at the DMV before he can get it back and rescue Kiva from an alien bounty hunter. Another episode has Coop rigging a DDR pad into Megas's control system. And, all throughout, we have more of those delightful flights of fancy we've seen in Downtown.

Oh, and there's also the fact that the only character from Downtown to reappears in Megas as a member of the main cast is Goat -- the last person from the Downtown crew who belongs in Y7 programming. (Goat in Downtown; Goat in Megas XLR.)

8. Neon Genesis Evangelion

I think I love this show. It's hard to be certain. I've only run through it once, and I'm not sure I can ever do it again.

Every family has its own Christmas traditions, and mine is no exception -- but our yuletide ritual, unlike most peoples', involves special psychoactive holiday brownies. (I'm told it goes way back to when my father and mother were dating. They did a surprisingly good job keeping it a secret from my sister and I until we were both in our twenties.)

One Christmas day a few years back, I had just swallowed a brownie and was wondering what I should do with myself when my sister appeared. "I got the Neon Genesis Evangelion box set," she said. "Wanna watch it with me?"

"Evangelion?" I said. "I guess I've heard good things about it before. Yeah, let's check it out."

Marathoning through Neon Genesis Evangelion on a heavy dose of mind-altering drugs was either the best thing I ever did or one of the worst. I can't decide. We watched the whole bloody thing in a thirty-hour span. My mind felt like melted rubber afterwards. It was the most mentally and emotionally draining cartoon I had ever watched in my life.

I don't recall many specifics, but I do remember how at some point -- about halfway through -- it made this surprise turn from "really clever and well-made anime about giant robots" into "harrowing self- portrait of a chemically-imbalanced anime director's nervous breakdown." Evangelion becomes less a show about brilliant and special people defending Earth from mysterious alien invaders than one about brilliant, special, and hopelessly damaged people defending Earth from mysterious alien invaders while doing horrible things to each other and themselves.

I don't think I have the emotional wherewithal to watch it again. I'm also a little afraid that it won't be quite as intense as I remember it, and I'd rather that Evangelion retain its dark mystique for me. Everyone else, however, should watch it immediately if they have not already done so.

HIGHLIGHTS: Rei's reflections from Episode 14. The only part of the show I've watched more than once.

7. Batman: the Brave and the Bold

Batman is everybody's favorite superhero because he's the darkest superhero.

Everybody liked Batman Begins because it was more serious and realistic than other comic book movies. Everybody liked The Dark Knight even better because it was serious and realistic, and also very dark.

The Dark Knight Returns and The Killing Joke are the best Batman stories of all time because they're the darkest, the most grim, and the most serious.

People like Batman: the Animated Series because it is a dark and gritty cartoon show -- doubly in the case of Mask of the Phantasm, its foray onto the big screen. People like Batman Beyond too, since it is dark, gritty, and futurepunk -- doubly in the case of Return of the Joker, its violent direct-to-video movie. They also like Gotham Knight because violent and dark.

People who talk like this don't know Batman -- or, at least, not very well. They seem to be unaware that the grim and tortured urban avenger is only the character's most recent (and therefore most popular) incarnation, and that he spent two decades as a jolly cartoon character who traded fisticuffs with villains like Kite-Man and Crazy Quilt in various museums of giant props. This version of Batman may be out of vogue, but you can't call it a bastardization (as have several YouTube commentators) because it was the and only Batman for many years.

Batman: the Brave and the Bold, which resurrects the lighter, happier Batman, anticipated a fan backlash and actually took a few minutes during one of its first batch of episodes to address these viewers directly.

As you can see, the caped crusader featured on Batman: the Brave and the Bold is definitely not Tim Burton's, Frank Miller's, or Tim Nolan's dark knight detective. This Batman comes out during the day, is almost never seen out of costume, does very little brooding, and regularly travels through time, into outer space, and between dimensions on his crimefighting beat. In the spirit of the Silver Age, The Brave and the Bold happily throws any pretense of plausibility out the window. When the fifteen-year-olds complain about how ridiculous it is for Batman to fly around in a spaceship equipped with an "alien nullifier beam," they're sort of missing the point. Of course it's ridiculous. It's intended to be.

The Brave and the Bold is designed with two target audiences in mind:

1.) Boys between the ages of five and thirteen.
2.) Comic book geekazoids in their twenties and thirties.

Either the writers are Batman fanatics themselves, or they did a lot of homework. As you might have noticed, nerds are always happiest when the television/film take on their favorite comic book makes obscure references only they can pick up on, and The Brave and the Bold is bursting at the seams with DC Comics esoterica. Up until the third season, the show's creators make a concerted point of pairing Batman up with the DC heroes you don't often see outside of the comic book store, like Bronze Tiger, the Metal Men, and Kamandi. When the more familiar superheroes do appear, it's frequently in one of their lesser-known incarnations: Jay Garrick as the Flash, Ryan Choi as the Atom, Guy Gardner as Green Lantern, and the Justice League International instead of the Justice League of America. In the late second season and the third season, the more familiar DC mainstays start popping up. After forty episodes of a superhero all-star cartoon without Superman and Wonder Woman, their belated appearances have the taste of a rare treat.

But this isn't like a superhero version of Family Guy by any means. You needn't understand any of the nods to comic book obscura to enjoy The Brave and the Bold, though it does mean you'll get more of a kick out of it. What makes it such a fun show is its earnestness. Even when it pokes fun at superhero camp, it never treats its material with any smug derision. As silly as they were, the comic books of the 1950s still flew off the newsstands, and not because of their appeal to anyone's jaded sense of irony. They weren't particularly deep, but they certainly weren't unimaginative or boring either. When you listen to Two-Face pun on every possible variation of the number two in a span of ninety seconds, see Batman teaming up with Space Ghost to battle aliens, or watch Aquaman express such irrepressible glee at being such a muscly good guy, you sometimes wonder why superheroes ever got so damned serious to begin with. They can be just as much fun when they're trying to be -- well, fun.

HIGHLIGHTS: The episodes written by Paul Dini, of Batman: the Animated Series fame. These include the ultra-meta "Day of the Dark Mite," the vintage Scooby Doo team-up in "Batman's Strangest Cases," and "The Chill of the Night," the one "serious" episode of the whole series. And though it's not written by Dini, the Superdickery-inspired "Battle of the Superheroes" episode is mandatory viewing.

6. Æon Flux

When I was in my "tween" years -- which elapsed before the word "tween" entered the lexicon -- late-night MTV cartoons were an indispensable part of my sleepovers with friends. The two-hour Beavis & Butthead blocks were great, sure -- but there was also the other stuff: the arrestingly hip and hypnotically bizarre cartoons that were nothing like any others I'd ever seen. My favorite of these, then and now, is Peter Chung and Howard Baker's Æon Flux.

Aeon Flux defies analogy. Except for maybe the artsy action shorts from The Animatrix and Gotham Knights (and maybe the ultra-wacky Reign: the Conqueror anime), there's not much to which it can be compared. It began as a series of Liquid Television shorts about a silent leather-clad female spy in a strange future world, whose adventures always ended with her own violent death. When the first (and only) season of full-length episodes rolled out, a few things changed: the characters spoke, the plots had a bit more substance, and the heroine (usually) got out of every episode alive. What remained unaltered was the show's disorienting lack of context. Though we're tossed hints, we know next to nothing about the the characters or the world they inhabit. Watching Æon Flux as a twelve-year-old, I stayed glued to the television just because I thought if I watched long enough I would figure out what the hell was going on. I've watched the whole series now -- more than once -- and I'm still not entirely sure. But everyone and everything in Æon Flux is so strange, dangerous, and nasty that, really, we're all probably better off not knowing more. (Speaking of nasty, it's fascinating how ugly these people are. Anime, by contrast, likes to smooth out creases and make everyone sparkle. Peter Chung highlights the wrinkles, bumps, folds, and fluids. Æon Flux's sex scenes are some of the least wankable in adult-oriented animation.)

At the heart of Æon Flux is the yin-yang dynamic between its main characters. The "villain," Trevor Goodchild, is a brilliant but sinister head of state with a genuine desire to improve humanity and its quality of life in the nightmarish metropolitan labyrinth it inhabits. Trevor represents order, intellect, and the desire for control. The titular "heroine," meanwhile, is an agent of desire, freedom, and anarchy who wants to screw up Trevor's big plans -- despite their irresistible mutual attraction to one another.

Most episodes adhere to a basic formula. Trevor hatches a scheme of some sort. Æon moves in to foil it. Chaos ensues. Lots of people die. Twelve-year-olds have hard time sleeping. The experience of marathoning through all ten episodes of Æon Flux leaves you just as exhausted as ten hours of Evangelion. While Evangelion can be quirky, funny, and cute when it's not being mentally and emotionally wracking, Æon Flux is consistently dark, dirty, and confounding. Whenever a story actually gets an unambiguous resolution, it's rarely a happy one. Betrayal, death, dismemberment, mutation, and collapse are the norm.

Nevertheless, Æon Flux is still a whole lot of fun to watch in moderate doses. (I find it's usually best to stop after three episodes to give your psyche time to recompose itself.) "Avant garde" animation doesn't get much better than this, and I wish more American outfits were putting forth this kind of effort today.

HIGHLIGHTS: Trevor's monologues. It's hard to call the guy a villain when he can make such an elegant and compelling case for himself -- and the virtuoso VA performance don't hurt much neither. History will also show that he dropped the "that which does not kill us, makes us stranger" line thirteen years before The Dark Knight's Joker made it popular. Unfortunately, I can't find any quick links to Trevor's better speeches on the You Tube. If you care to look into it, some of his best are in Episode 5, "The Demiurge:"

Light, in the absence of eyes, illuminates nothing. Visible forms are not inherent in the world, but are granted by the act of seeing. Though the world and events do exist independent of mind, they obtain of no meaning in themselves: none that the mind is not guilty of imposing on them. I bid my people follow, and like all good equations, they follow; for full endowment of purpose, they do submit -- in turn, they resign me to a role inhuman, impossible, and unaccountable. But I can no longer stand the sleepless nights...

The height in tension in any game occurs when the rules allow for the influence of human judgement. The advantage in this instance is held by the player who drops the ball to allow the opponent a hollow victory. Half of eternity is still eternity. Infinity and Zero are different degrees of the same value. Pain isn't real beyond the individual who feels it. But the nail is real.

If but the cosmos could possibly have come about innocent of history, and not wholly ignorant of all its potential -- unprejudiced, but aware -- rather than that its only trial be performed at the mercy of all its error.

You can read the rest over at

Thursday, July 21, 2011

Jaggedjaw Jewelwings

Anyone reading this is probably sick of all this talk about insects and would rather I composed an opinion column about the cancellation of Mega Man Legends 3 (seriously, why are you surprised). It is my regret to inform you that since bugs is what I'm into, it's bugs is what I'm blogging.

I usually stop by the pond to visit the damselflies every other day to revitalize myself after the workaday coma. During my last visit, I watched a female land on a leaf about a foot away from my face, and noticed she had something in her mouth.

Given the seemingly dainty, almost ornamental structure of their bodies, the damselfly probably doesn't strike one at first glance as a ravenous carnivore. But these guys are like little airborne sharks: they'll chow down on anything small enough for them to snare in their front limbs and shove into their mouths. (Incidentally, this is would be why I seldom get mosquito bites at this particular pond. The damselflies and dragonflies do an excellent job keeping it free of obnoxious little biters and buzzers.)

This was my first face-to-face look at a damselfly in the middle of a meal. Imagine a person with his hands at his sides and a hoagie sticking lengthwise out of his mouth, and you'll have a decent idea of how this female jewelwing looked as she held this unfortunate fly in her mouth and chewed and chewed and chewed and chewed and chewed and chewed. Little by little, more and more of the fly disappeared. After about ten minutes of chomping, the meal ended -- but only because the rear half of the abdomen snapped off and fell into the dirt. After that the female ground up the last mouthful and flapped off, pretty as you please.

Where organisms are concerned, it is often the case that the appearance of "cuteness" emerges from the viewer's not looking closely enough. Damselflies are no exception. They only seem dainty and ladylike to you because you're not small enough to have the fear of god thrown into you by their hideously jagged jaws. (Note: the name of their order, Odonata, comes from the Greek odonto, meaning "tooth.")

Let's have a look at some photographs taken by some more talented shutterbugs with more powerful cameras:

Nice whiskers!

These actually aren't teeth: the jaw itself is spiky.

Yes, it definitely should remind you of the Aliens movies.

Not really a mouth pic. Just a snap of the damselfly species (ebony jewelwing) with which I'm familiar.

One of the links above leads to a blog (I won't tell you which one; you'll just have to check them all out) discussing observed homosexual behavior exhibited by the blue-tailed damselfly. Toward the beginning it gives one of the most colorful descriptions of the suborder Zygoptera I've had the pleasure of reading:

The damselflies and the dragonflies are known as the odonata; the toothed jaws. Toothed in jaw they are for very good reason, for these are the birds of prey of the insect world… though in the case of the blue tailed damselfly the birds are too often preyed upon… making the chaps more like damsels. Of course the Odonata could be considered birds of prey if raptors were a bit scarier, about one hundred times the size of their snacks, plucking tiny birds out of the sky, like iridescent and clanking clockwork toothy biplanes.

That last line is especially apt in respect to the creatures' antediluvian lineage. Damselflies have not evolved very that much in the last couple hundred million years. Compared to the younger, smaller, and more streamlined insect species, damselflies seem big, awkward, and ungainly. But it's this rococo clunkiness that endows the little buggers with such unparalleled charm. From a distance, they almost seem like little jeweled automatons assembled by some sultan's tinkerer of antiquity to haunt the hanging gardens of the queen. But one look at those nasty jaws reminds you what they are and by what vicious processes the species was allowed to survive and thrive.


Sunday, July 17, 2011

Bugs, roots, words

I'm going to wait until next weekend to post the second half of that last entry. I've had too much else to juggle this past week to produce anything more than 75% of a rough draft, and I'd rather not just rush it out without thinking it over a bit more.

In the meantime, I do have some other stuff for you. First, an update to that earlier piece about those neat little buggers called damselflies.

I had previously thought male ebony jewelwings came in two colors: blue and green. Turns out that they actually come in one color: blue and green.

Here is a photo of a male I snapped earlier today:

And this is a shot of the same bug, taken about two minutes later:

My hunch was correct: whether they appear green or blue depends on the light bouncing off of them. From what I've observed, they're blue when they're closer to water, and green when they're around leafy plants. Whether this is caused by the different amounts of sunlight hitting them (being near the water tends to put them out of the shade, after all) or by the general color of their surroundings, I'm not sure -- but I'm interested in looking further into it. (I suspect the first guess is probably the more accurate.)

Again: this all probably helps explain why I don't have a girlfriend.

Anyway. While visiting Manhattan last night to help a couple of friends lug their amps and synthesizers, I happened to pass a table of books on the sidewalk being sold for a buck each. The broad span of the collection consisted of dry periodicals, hardcover books about statistics, issues of comic books published by Valiant, and lousy fiction paperbacks about love and murder. Only one tome caught my eye: Constance Reid's A Long Way from Euclid, a 1963 hardcover about classical geometry's long and continuous influence upon the development of mathematics.

I spent a lot of time flipping through it between bands and on the ride back home. Just about everything in the later chapters goes over my head, but the first few sections -- which are much, much more elementary and therefore less terrifying to a reader who consistently got Cs and Ds in every math class from algebra I to precalculus -- were great fun to read.

If you read last week's entry, you'll remember my kvetching and complaining about how students are taught the discoveries of science without learning the methods or much of the history. The same is true with mathematics -- maybe even more so, if what I recall from my own primary school years suggests a wider trend.

I, like most students, was shown a right triangle in geometry class, told that a2 + b2 = c2 , and then commenced solving twenty worksheets of right triangle problems. What we didn't learn was the history of the Pythagorean theorem and its tremendous influence upon the course of Western mathematics and thought -- topics any pupil would benefit from knowing. (I learned about these details a few years later during a "history of mathematics" course I took in college as a means to fulfill a math requirement without giving my abysmal computative abilities the chance to destroy my GPA.)

A Long Way from Euclid's first chapter discusses some of the Pythagorean theorem's immediate effects, and I've prepared an excerpt for your reading pleasure. You will either find it really interesting or boring as hell. There is not much likely middle ground. 

Thus, five hundred years before the birth of Christ, mathematics had in hand its famous theorem about the square on the hypotenuse of the right triangle -- a theorem which was destined, in the words of E. T. Bell, to run "like a golden thread" through all of its history. This theorem would serve -- in trigonometry, which is entirely based on it -- as the tool for measurement lying beyond the immediate use of tape measure and ruler. In analytic geometry, it would serve as the basic distance formula for space in any number of dimensions. In its arithmetical generalization (an + bn = cn), it would provide mathematics with is most famous unsolved problem, known as Fermat's Last Theorem. In the most revolutionary mathematical discovery of the nineteenth century, it would be revealed as the equivalent of the distinguishing axiom of Euclidean geometry; and in our own century it would be further generalized so as to be appropriate to and include geometries other than that of Euclid. Twenty-five hundred years after its first general statement and proof, the theorem of Pythagoras would be found, firmly embedded, in Einstein's theory of relativity.

But we are getting ahead of our story. For the moment we are concerned only with the fact that the discovery and proof of the Pythagorean theorem was directly responsible for setting the general direction of Western mathematics.

We have seen how the Pythagoreans lived and discovered their great theorem under the unchallenged assumption that Number Rules the Universe. When they said Number, they meant whole number: 1, 2, 3. . . . Although they were familiar with the sub-units which we call fractions, they did not consider these numbers as such. They managed to transform them into whole numbers by considering them, not as parts, but as ratios between two whole numbers. (This mental gymnastic has led to name rational numbers for fractions and integers, which are fractions with a denominator equal to one.) Fractions disposed of as ratios, all was right with the world and Number (whole number) continued to rule the Universe. The gods were mathematicians -- arithmeticians. But, all the time unsuspected, there was numerical anarchy afoot. That it should reveal itself to the Pythagoreans through their own most famous theorem is one of the great ironies in mathematical history. The golden thread began in a knot.

The Pythagoreans had proved by the laws of logic that the square on the hypotenuse of the right triangle is equal to the sum of the squares on the other two sides. They had also discovered the general method by which they could obtain solutions in whole numbers for all three side of such a triangle. Although these whole number triples (the smallest being 3, 4, 5) still bear the name of "the Pythagorean numbers," the Pythagoreans themselves knew that not all right triangles had whole-number sides. They assumed, however, that the sides and hypotenuse of any right triangle could always be measured in units and sub-units which could then be expressed as the ratio of whole numbers. For, after all, did not Number -- whole number -- rule the Universe?

Imagine then the Pythagoreans' dismay when one of their society, observing the simplest of right triangles, that which is formed by the diagonal of the unit square, came to the conclusion and proved it by the inexorable process of reason, that there could be no whole number or ratio of whole numbers for the length of the hypotenuse of such a triangle:

When we look at any isosceles right triangle -- and remember that the size is unimportant, for the length of one of the equal sides can always be considered the unit of measure -- it is clear that the hypotenuse cannot be measured by a whole number. We know by the theorem of Pythagoras that the hypotenuse must be equal to the square root of the sum of the squares of the other two sides. Since 12 + 12 = 2, the hypotenuse must be equal to √2. Some number multiplied by itself must produce 2. What is this number?

It cannot be a whole number, since 1 x 1 = 1 and 2 x 2 = 4. It must then be a number between 1 and 2. The Pythagoreans had always assumed that it was a rational "number." When we consider that the rational numbers between 1 and 2 are so numerous that between any two of them we can always find an infinite number of other rational numbers, we cannot blame them for assuming unquestioningly that among such infinities upon infinities there must be some rational number which when multiplied by itself would produce 2. Some of them actually pursued √2 deep into the rational numbers, convinced that, somewhere among all those rational numbers, there must be one number -- one ratio, whole number to whole number -- which would satisfy the equation we would write today as

The closest they came to such a number was 17/12, which when multiplied by itself produces 289/144, or, 2 1/144.

But one of the Pythagoreans, a man truly ahead of his time, stopped computing and considered instead another possibility. Perhaps there is no such number.

Merely considering such a possibility must be rated as an achievement. In some respects it was an even greater achievement than the discovery and proof of the famous theorem that produced the dilemma!

Perhaps there is no such number. How does a mathematician go about proving that there isn't a solution to the problem he is called upon to solve? The answer is classic. He simply assumes that what he believes to be false is in actuality true. He then proceeds to show that such an assumption leads to a contradiction, usually with itself, and of necessity cannot be true. This method has been vividly called proof per impossible or, more commonly, reductio ad absurdum. "It is," wrote a much more recent mathematician than the Pythagorean, "a far finer gambit than a chess gambit: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game."

The most recent proof to shake the foundations of mathematical thought was based on a reductio and so, twenty-five hundred years ago, was the first. We shall present this proof, which is a fittingly elegant one for so important an idea, in the notation of modern algebra, although this notation was not available to the man who first formulated the proof.

Let us assume that, although we know we have never been able to find it, there actually is a rational number a/b which when multiplied by itself produces 2. In other words, let us assume there exists an a/b such that

We shall assume (and this is the key point in the proof) that a and b have no common divisors. This is a perfectly legitimate assumption, since if a and b had a common divisor we could always reduce a/b to lowest terms. Now, saying that

is the same as saying that

If we multiply both sides of this equation by b2 (which we can, since b does not equal 0 and since we can do anything to an equation without changing its value as long as we do the same thing to both sides), we shall obtain:

or, by canceling out the common divisor b2 on the left-hand side:

It is obvious, since a2 is divisible by 2, that a2 must be a even number. Since odd numbers have odd squares, a must also be an even number. If a is even, there must be some other whole number c which when multiplied by 2 will produce a; for this is what we mean by a number being "even." In other words,

If we substitute 2c for a in the equation a2 = 2b2, which we obtained above, we find that


Dividing both sides of this equation by 2, we obtain

Therefore, b2 like a2 in our earlier equation, must also be an even number; and it follows that b, like a, must be even.

BUT (and here is the impossibility, the absurdity which clinches the proof) we began by assuming that a/b was reduced to lowest terms. If a and b are both even, they must -- be definition of evenness -- have the common factor 2. Our assumption that there can be a rational number a/b which when multiplied by itself produces 2 must be false; for such an assumption leads us into a contradiction: we begin by assuming a rational number reduced to lowest terms and end by proving that the numerator and denominator are both divisible by 2!

We can only imagine with what consternation this result was received by the other Pythagoreans. Mysticism and mathematics were met on a battleground from which there could be no retreat and no compromise. If the Universe was indeed ruled by Number, there must be a rational number a/b equal to √2. But by impeccable mathematical proof one of their members had shown that there could be no such number!

The Pythagoreans had to recognize that the diagonal of so simple a figure as the unit square was incommensurable with the unit itself. It is no wonder they called √2 irrational! It was not a rational number, and it was contrary to all they had believed rational, or reasonable. The worst of the matter was that √2 was not by any means the only irrational number. They went on to prove individually that the square roots of 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15 and 17 were also irrational. Although they worked out a very ingenious method of approximating such irrational values by way of ratios...they had to face the fact that there was not just one, there were many (in fact, infinitely many) lengths for which they could find no accurate numerical representation in a Universe that was supposedly ruled by Number.

Tradition tells us that they tried to solve their dilemma by persuading the discoverer of the unpleasant truth about √2 to drown himself. But the truth cannot be drowned so easily; nor would any true mathematician, unconfused by mysticism, wish to drown it. The Pythagoreans and the mathematicians who followed them, from Euclid to Einstein, had to live and work with the irrational.

Here was the golden thread impossibly knotted at its very beginning!

It was at this point that the Pythagoreans, rather than struggling to unravel arithmetically what must have seemed to them a veritable Gordian knot, took the way out that a great soldier was to take in a similar situation. They cut right through the knot. If they could not represent √2 exactly by a number, they could represent it exactly by a line segment. For the diagonal of the unit square is √2.

With the choice of two mathematical roads before them, the Greeks, long before the time of Euclid, chose the geometric one; and
"That has made all the difference."

I'd say the book was worth the buck.

Monday, July 11, 2011

Space, Outer and Inner (Part I)

Since May of last year, after the abyssal horror of the Final Fantasy XIII experience forced me to reevaluate how I spend my time, I've been stargazing and reading up on astronomy as an alternative hobby. Among the most remarkable observations I've made is how little the topics of astronomy and outer space seem to interest people.

Last February I showed a few acquaintances the Orion Nebula through my fancy binoculars. (Around that time I was habitually taking a peek at it nearly every evening. Not that it changes or does anything perceptibly different on a night-to-night basis, but the sight of it is astonishing -- all the more so, I felt, because I could see it with my own eyes.) When I mounted the binoculars on a tripod and invited these folks to see it for themselves, they each look for a few seconds and went back inside. Their reactions ranged from underwhelmed to disappointed.

(Here we find an unfortunate side-effect of the Hubble Space Telescope. People unaccustomed to looking through telescopes become familiar with fantastic images like the Pillars of Creation and the Hubble Deep Field, thereby setting expectations that a regular telescope simply can't match. By the time you finish explaining to them that it's unrealistic to expect a portable optical device to match the quality of images produced by a multimillion-dollar roboscope looking out from above the atmosphere through a lens with a 0.05 arc-second resolving power and enhanced by computer processing, your guests have already gone in to watch Nyan Cat for another half hour.)

I don't expect that everyone should be as fascinated by the heavens as I am, but I would think that they should hold at least some interest to most people. Instead, I've found that the subject usually arouses mild indifference or antipathy. During a car ride when I had a Richard Feynman lecture on gravitation in the CD player, my friend in the passenger seat asked, with some exasperation, why I chose to concern myself with something that made no practical difference in my life.

"You're going to stay home and look at the stars? What, you wouldn't rather come out to a bar? Why don't you go out and meet people? You should be trying to meet girls. You need a girlfriend, Pat. If you had a girlfriend, why, you wouldn't be so lonely that you'd choose to stay at home by yourself and look at the stars instead of coming out to the bar."

That which is most immediately useful about studying space (in any degree of intensity) is also what makes it such a tantalizing subject. It lends a certain perspective to your view of the world that is difficult to attain elsewhere in your day-to-day life. (I use "world" in the subjective sense, as opposed to the objective "Earth.")

Here's something you might do for yourself to see what I mean. Go outside tonight around 11:00 or so. (Actually, it might help to wait a week or two so that the moon isn't so damned bright) and find the Big Dipper. (Hint: look north.) If you follow the stars of its "tail" and keep moving in the same curved line, you'll bump into a bright star. Trust me, you'll know it when you see it.
This image, courtesy of Astrobob, will give you some idea of what to look for.

Do note that it won't be as far west as the picture (taken in mid-September) indicates.

This star is Arcturus, the second-brightest star in the night sky of the northern hemisphere, and the centerpiece of the constellation Boötes, the plowman. (Regrettably, the name isn't properly pronounced as "boots;" try "boh-OH-is," instead. However, I will forgive you for saying "boots," as I am guilty of doing it even though I know better.)

Anyway. Once you find Arcturus, begin moving your gaze eastward. The next bright star you find will be Vega (like the Street Fighter character), of the constellation Lyra. If you move your gaze to the southeast, you'll see a second bright star: Altair (like the Final Fantasy village) of the constellation Aquilla. Returning to Vega and looking toward the northeast, you will find a third bright star forming the "head" of a cross-shape formed by three dimmer stars. This bright star is Deneb, of the constellation Cygnus. (As far as I know, no video games reference Deneb.)

Vega, Deneb, and Altair form the corners of the Summer Triangle. Again, from Astrobob:

You should be able to see these three stars even if you live in an urban area with a lot of light pollution. This next step, however, requires you to look from a suburban or rural area, and be carrying a telescope or pair of binoculars.

As you probably know, our galaxy's shape is that of a flat disc. "Flatness," especially in this case, is a matter of proportion. Compared to its 575,205,347,300,000,000-mile (or so) diameter, the Milky Way's 76,694,046,300,000,000-mile (or so) thickness is quite flat.

Since we're looking out at the disc from an object inside the disc (and located toward its outer rim), we perceive the bulk of it a wide band arching across the firmament. In the absence of light pollution, it is dimly visible to the naked eye as a pale streak stretching from one horizon to the next. Not understanding the nature of what they were looking at, the ancients named it the Milky Way and invented a myth to explain it. When we came to realize that the arc represented the largest visible portion of our parent galaxy, we came to identify that galaxy by the name given to the visible arc.

(I apologize if I am repeating something you already know. But from what I recall from my thirteen years in the United States' public school system, I don't believe the curriculum ever addressed anything beyond Pluto in much detail. I can't be certain what anyone else reading this knows or doesn't know.)

The Summer Triangle is superimposed over the Milky Way band. If you're looking from a sufficiently dark region, you don't need me to point this out to you. It can be visible from the suburbs, but you'll need an optical aid. (More powerful is always better, but you can still get a decent glimpse using a pair binoculars designed for, say, watching a football game from the nosebleed section.)
All you have to do is point your scope or your nocks into the center of the triangle and look. If you've never tried it, you might find yourself shocked at how much stuff is out there -- how much light from how many stars are hitting your optic nerves all at once. In a place with moderate light pollution, you might see hundreds. In a place with little or no light pollution, you may see thousands.

If you attended public school in the United States (as I did), you probably received a crash course in astronomy. Chances are that curriculum botched it, as my school's did. The lesson probably went something like this:

"The Moon orbits the Earth. The Earth orbits the Sun. The Sun is very large and made of incandescent gas. The other stars are like the sun, but they are very far away. Are you writing this down? The true/false quiz is on Thursday."

We make a grave mistake by teaching students the facts without informing them as to how we arrived at them. This is like making them memorize the solutions to math problems without teaching them how the problems are solved.

Teaching the facts of science without teaching the methods is only marginally better than pointless.

The purpose of science is to arrive at a more accurate understanding of the physical world though repeated observation, testing, and logical reasoning, with methods that are available for anyone (with the proper know-how and materials) to duplicate themselves. When the answers are given without reference to the process, the student cannot properly appreciate them. For all he knows, the facts in his textbook came from some Neo-Delphic Oracle sealed in a wine casket in some Princeton basement.

On a political note: if science wants to assert itself as the superior alternative to religious dogma where finding out the physical facts is concerned, scientific education must make a better effort toward demonstrating to students that it can do one crucial thing that dogma cannot -- it can show its work.

For instance:

We know that the moon orbits the Earth because of how is path across the celestial sphere differs from those of the sun, the planets, and the stars. We know the Earth orbits the sun for the same reason, and by the same logic we find that the other planets orbit the sun as well. (Mr. Copernicus outlines it very succinctly and eloquently, even though he does fudge a few relatively minor details.)

We deduce that the sun is very large because, for starters, we can use radar to acquire a precise measurement of Venus's distance from Earth in kilometers, and then use that number in conjunction with Kepler's Third Law (p^2 = a^3) to find the a value of a (in this case, 1 A.U., or one "Earth-distance") in kilometers. Knowing exactly how far away Venus is from Earth, and knowing exactly how far away the Sun is from Earth, we observe that the sun appears thirty times larger than Venus in the sky, despite being 2.6 times farther away. The conclusion draws itself.

We understand that the stars are very far away because of observations involving stellar parallax. (Hold your raised index finger six inches from your eyes. Close your left eye. Now open it and close your right eye. Notice how drastically the apparent position of your finger seems to change in relation to object behind it. This is parallax. When we view objects like the Moon or planets from different positions on the Earth's surface,and compare where they appear to lie relative to the stars, we observe a similar displacement.

We know that the stars are similar to the sun in composition because spectroscopic analysis turns up evidence of the same basic materials in the sun and in pretty much every other star we can find.

(I should mention that I am by no means whatsoever an expert or authority. I am only an amateur who wants to share what he has learned.)

Before you take a good look, it helps to know what you're seeing, especially in this case. An object like a star is impossible to touch, approach, turn over, smell, etc. When we tell a student that the star he sees is one particular thing as opposed to numberless other particular things, it is no less important that he gets some understanding of how we came to this conclusion. Otherwise, he has as much a basis for believing that the twinkling dots he sees in the night sky are tremendous fusion-fueled orbs billions of miles away as he does for believing that the weatherman is telling the truth when he smiles, winks, and pulls up the Santa Radar on Christmas Eve. (Do weathermen still do this?)

So what are we seeing when we aim our scope into the middle of the Summer Triangle? The short answer is "stars." Thousands upon thousands of the billions upon billions of stars composing the Milky Way galaxy. Most of these stars are at least similar in size to our Sun, whose diameter is somewhere in the vicinity of 64,950 miles. (Our entire world's diameter amounts to 7,926 miles, remember.) Though they appear close together, the distances between each of these points of light you see are so tremendous that to reckon them in miles would make about as much sense as measuring the length between city blocks in millimeters.

What you're looking at is Eternity. But only a sliver of it.

Thanks to advancements in optics, we now understand that the Milky Way is just one galaxy out of billions and billions.

Here's a fun experiment! This BBC article might be outdated and/or inaccurate, but that's okay. It gives us a number to play around with: 156 billion. Let's say that the universe is 156-billion light-years wide. If we're assuming it's a sphere (the truth is probably a lot more complicated and far beyond my ability to approach), that gives it a radius of 78 billion light years -- or 458,200,000,000,000,000,000,000 miles. Now if we calculate the volume of a sphere with a radius of 458,200,000,000,000,000,000,000, we get...

(4/3) * (π) * (458,200,000,000,000,000,000,000)^3 = oh jeez.

Uh...I'm getting 40,295,250,862,323,019,768,114,206,530,184,000,000,000,000,000,000,000,000,000,000,000,000,000 cubic miles. This is a very sloppy, simplistic, and definitely incorrect approach (it doesn't account for any relativistic space-stretching voodoo, for one thing), but it gives us some idea of how far existence might go. (Earth's volume is about 259,875,899,200 cubic miles. Subtract that from the "number" up above and contemplate how much stuff that's not us is.)

It is very easy not to think about this in your daily life. Most of us probably don't give it a second thought. Stargazing is a pursuit that rather forces you to confront this reality -- though usually in small doses. (When faced with a number as large as the one above, the size of the Milky Way is no less piddlingly infinitesimal than that of the Earth.)

I don't know of anyone who has not been taught that the Earth revolves around a massive Sun and that the universe outside our neighborhood is bigger than we can possibly imagine. Most of us are even used to the idea, but the fact is so rarely brought to bear on our day-to-day existence that we rarely appreciate the significance of this fact.

You can look at Hubble images and listen to Carl Sagan wax poetic about the cosmos, but neither brings you into direct contact with the subject. This is why you need to sometimes go out and be touched by it -- to see, with your own eyes, a thousand stars stretched out over billions of miles.

When formulating an opinion or viewpoint on any subject, it is important to consider as much of the whole of the situation as possible. Otherwise you risk an incomplete conclusion.

When we evaluate ourselves, our world, and ourselves in the context of our world, we cannot omit the underlying facts of the situation. Though we rarely have any direct contact with it, Eternity is there. When assessing himself and his environment, the average person fails to factor it into his personal equation.

Why should he, though? Why fuss over something that brings no immediate (or even perceptible) weight upon his health, his friends, his career, and his hobbies?

This has already careened far enough along for now, but for now I'll close with a question. Even if a person has not given it any direct thought, he is very likely to have already arrived at his answer without being asked.

"Which is more real: the part(s) of reality existing closest to me, or the parts of reality occupying the greatest percentage of the whole?" 


Edit: Says a commentator on Twitter:

they're both equally real this question is dum

I was afraid of this. Maybe that's not the question I want to ask. Maybe what I'm turning over in my addled brain is a matter of significance.

Are the constituents of reality which exist in closest proximity to my experience more significant than the overwhelmingly larger span of reality (matter, energy, time, space, et al) existing outside of my environs?

Why or why not?

And why should this question strike most as an "empty academic point?"

Tuesday, July 5, 2011

Blah blah blah

This weekend I bit the bullet and spent a very, very long time revising some of my old articles about a certain Japanese video game series. It was an unbelievable pain in the ass, but it's DONE. I didn't even really want to do it, but -- well, OCD is a pretty powerful motivator. The things have my name on them and they were pretty insubstantial compared to the later additions, so I fleshed them out a bit to bring them up to par. This would also be why I don't have a post prepared for this week.

God, I need something else to think about. Something to flush the Espers and dragons out of my brains.

On the agenda for the next few weeks:

  • Give The Zeroes (my unpublished n-v-l; only when it gets published in some form or other will I be comfortable with calling it my novel) one last editorial sweep and get ready to e-bookerize it. I might also look into a pay-per-print service, but we'll have to wait and see about that.

  • Comics! I'm doing a little "miniseries" about a famous roller from Greek myth. One page is complete. Six more will follow. Also, I've been seriously considering setting up a new website for my drawn comics, so I can keep them and the old pixel comics in a separate place. It's about time I find a new host, anyhow...

  • Oh, and there's still that OTHER thing -- n-v-l number two. I've been taking a two-week breather from it, but it's starting to really gnaw at me again. Guess I'll just have to let it. For the time being, making n-v-l number one available to the public and making cute little comics to attract the immediate attention of that public takes priority.

    No rest for the broke and driven. Maybe something good will come of it all.

    My friend Jeff got married two weeks ago. I was the best man, so I had to give a toast. The geek that I am, I thought it would be appropriate to offer the happy couple my blessing by raising a glass to Shakespeare's 116th sonnet (which I imagine gets a whole lot of play in the wedding circuit). For your reading pleasure:

    SONNET 116

    Let me not to the marriage of true minds
    Admit impediments. Love is not love
    Which alters when it alteration finds,
    Or bends with the remover to remove:
    O no! it is an ever-fixed mark
    That looks on tempests and is never shaken;
    It is the star to every wandering bark,
    Whose worth's unknown, although his height be taken.
    Love's not Time's fool, though rosy lips and cheeks
    Within his bending sickle's compass come:
    Love alters not with his brief hours and weeks,
    But bears it out even to the edge of doom.
    If this be error and upon me proved,
    I never writ, nor no man ever loved.

    Note #1: My recitation omitted the "O no!" in line five. I couldn't find a way of working it in without killing the rhythm.

    Note #2: "Bark" is an old term for "ship." You know -- ships were made of wood, wood comes from trees, trees are covered in bark...

    Note #3: When Shakespeare wrote this poem, "proved" rhymed with "loved" and "come" rhymed with "doom." This certainly isn't the only time the evolution of English pronunciation has worked against the bard. I always get blue balls from the very first lines of Macbeth:

    When shall we three meet again
    In thunder, lightning, or in rain?
    When the hurlyburly's done,
    When the battle's lost and won.
    That will be ere the set of sun.
    Where the place?
    Upon the heath.
    There to meet with Macbeth.