review of the Pythagorean Theorem

Pythagorean Theorem  (a2 + b2 = c2)

named for pythagoras ~500bc, proved by everyone and their grandmother

3 out of 3 sides, 3 out of 3 angles, all squared up, put'cher dukes up

Note: The gods of formatting have denied me the power of exponent-ing text, so please take 'a2' as 'a squared.'  A thousand pardons.

Jargon: For our purposes, we’ll use ‘theorem’, ‘function,’ and ‘equation’ interchangeably: an idea that’s been demonstrated to be true, and consists of the steps for implementing that idea.  A ‘conjecture’ is a theorem that hasn’t been proven yet but seems credible.  Strictly speaking, I’m playing fast and loose with these terms but if the math mafia comes for me I’ll just take their lunch money.  


Oh, what, math proofs don’t deserve reviews?  They’re somehow beneath works of art like, I don’t know, pirates of caribbean?  I’ll concede that art and reviews go together like peanut butter and judgemental chocolate; art has plenty of subjective value we can argue about until the cows come home.  In contrast, math is completely objectified (‘hey, my eyes are up here!’). But I don’t think subjectivity is necessary for artistic value. Even though it’s dry and no-nonsense and goes around in tweed trousers all day, there’s form in math’s functions.

The nerds have been at us for some time now to acknowledge mathematical beauty (newton was an inveterate pest on the subject of ‘elegance’ but luckily the black plague kept everyone too busy to pay much attention).  Even hollywood’s joined the cause with a beautiful mind and the imitation game and all sorts of illfated attempts to make math sexy. And although it can drive russell crowe insane and still defeat the nazis, math makes a poor main character.  A fundamentally abstract idea doesn’t show up well on camera.

But we make art about abstract things all the time: love, struggle, loyalty, etc.  The failure to (ahem) artitize math is because it’s so often worshipped as a sacred tablet, whole unto itself.  The nerds may insist it’s the bedrock of science, the universal language, but it’s something much more human. It’s our description of the relationships among us and this messy reality we live in.  Math isn’t rock, it’s water.

The pythagorean theorem is the wayne gretzky of equations---it’s old, it’s great, and it’s the only one most of us have heard of. The ‘pyg’ is way ahead on public relations, though the golden ratio and quadratic formula do okay for themselves.  As theorems go, it’s a granddad, dating back to ancient mesopotamia, china, and india. The greeks, however, were better at marketing. They realized they had a hit on their hands and didn’t waste a second slapping their brand on it. Good thing pythagoras had such a conveniently rhymey name otherwise we’d be stuck with the shamashian theorem (named for the mesopotamian governor of the universe), which sounds like the square of the length of the side of bacon with your eggs.

The pyg not only has the advantage of antiquity, it happens to be one of the most proved theorems in math.  Sticking with our ancient greek theme, there’s a veritable orgy of proofs, both algebraic and geometric. Just about every notable mathematician would pass the time proving it, from ptolemy to einstein.  The pyg gets around.

It happens to have a notoriously buttoned-up goth anti-twin, fermat’s last theorem.  Fermat scribbled his ‘last’ theorem in the margins of an unrelated text, noting that it was ‘cute, a real shawty’ which he’d proved but couldn’t be bothered to detail.  It then went unproven for nearly 360 years (no one ever found fermat’s proof, we’re pretty sure he was just trolling), amassing a huge number of failed attempts, until andrew wiles of oxford finally slew the dragon in 1995, earning glory and groupies in the math world and a great big ‘huh?’ from the rest of us.  To the cognoscenti, the pyg was the friendly partygoer, someone they saw every weekend, while fermat's theorem was her aloof older sister, far too cool to even think of approaching.

[If you’re curious, f’s theo states that there is no integer n > 2 that satisfies an + bn = cn (a,b,c nonzero integers).  So the pyg is in the clear because it’s at n = 2 but anything larger (n=3,4,etc) won’t work.  It seems so simple, and yet.]

The more approachable sibling, good ol’ a2 + b2 = c2, is admirable for its symmetry, simplicity, and emotive power.  It lives in ‘right’ triangles, in the dependencies among the two legs and the long slide of the hypotenuse, in the two acute angles forever separated by ninety degrees.  A triangle gives off a strength and steadiness more complex shapes lack. A rectangle is only two right triangles hugging, a pentagon is a humdrum barn silhouette, the hexagon’s stop sign is a drag, and anything more is just desperate.  Everyone knows a decagon is a big showoff, the peacock’s tail of geometry. But the triangle is grounded, weightbearing. Capable, in short.

Any function describing such a commendable shape is bound to be good.  But the pyg’s own rhythm is intuitively pleasing, like an old beatles song.  A squared plus b squared equals c squared.  It has a meter.  The exponents drive the beat up, pulling the phrase onwards.  It trills, it blares, it’s the only mathematical formula you can clap along to.

It’s also adaptable.  It works in non- and euclidean space; it’s comfortable hanging out with algebra and geometry both; higher dimensions leave it unfazed, you can apply it to three- and four-dimensional shapes; it can help you draw the ellipse describing a planet’s orbits; it has kids called pythagorean triples (sequential integers that satisfy it, e.g. 3,4,5); it can even navigate the complex relations among moving + still objects in relativity to help you find ‘true’ position and time.  Inspiring, no?

And there’s a friendly reminder in the pyg’s higher powers that simple, linear addition---this plus that, him plus her, you plus me---could be exponentially more than just its parts.   

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