The majority of people in my life are uninterested in math, to put it mildly, but that has not prevented them from expressing curiosity about my Euler's Identity tattoo. "What's it mean?" they ask. "Why is it so important to you?" These questions are difficult to answer without invoking so much math as to make the questioners' eyes glaze over (at best; at worst they run screaming from the room). This post attempts to provide a non-mathematical metaphor that, I hope, will convey even to math-phobic readers the same kind of awe and beauty that I experience when contemplating Euler's Identity. I'm going to start with a totally un-mathy topic: literary characterization.
Readers of novels have slightly different ideas about what makes for good characterization than, say, readers of comic books. (I'm deliberately defying the growing pressure to call this benighted artform "graphic novels" by focusing specifically on comic books about superheroes, which is how it all got started, no matter how much the breathless fanboys may inform me that it's grown up since.) Readers of novels define their characters in terms of interiority: they talk about childhood memory, emotional nuance, mannerisms, depth. Readers of superhero comic books define their characters in terms of exteriority: they talk about costumes, superpowers, Achilles' heels, weapons, amulets, etc. It's the difference between viewing a character as a human being versus as a mere bundle of attributes.
Although I have little patience in general with the superhero comic book way of approaching character, it is useful in this context because it provides an interesting approach to numbers. Numbers, too, can be described merely as bundles of attributes. Of any number, there's a list of well-defined characteristics that it may or may not have: odd or even? Real or imaginary? Positive or negative? Whole or fractional? As long as we're willing to view the world through these glasses, then it's not such a stretch to reimagine the set of all numbers as literature's only infinite cast of characters.
Granted, some of these "characters" are a lot more interesting, compelling even, than others. Zero is a good example. We have yet to discover any other number that is as devoted a pacifist and yet simultaneously as rapacious a conqueror as zero. When it comes to addition, zero is a cuddly, harmless teddy bear: you can add anything to zero and that number will just emerge as itself, untouched. But when it comes to multiplication, zero is Genghis Khan, slaughtering all hapless comers, leaving only itself standing. (Don't even get zero started on division.)
In 2005, Roger Ebert popularized the term "hyperlink movie" (coined by Alissa Quart in that same year) in his review of Syriana, where he says it "describes movies in which the characters inhabit separate stories, but we gradually discover how those in one story are connected to those in another." Part of the power of this narrative structure, he explains, is that "the motives of one character may have to be reinterpreted after we meet another one." The TV show Lost, for instance, also has a "hyperlink narrative."
Euler's Identity is the best evidence thus far discovered that all of mathematics is one giant hyperlink narrative. Before Leonhard Euler discovered it in the sixteenth century, mathematics better resembled, say, the work of Stephen King -- a collection of largely unrelated narratives that occasionally overlap in character or location. There was the "number theory" narrative, whose heroes are 0 and 1; the "calculus" narrative, whose hero is e; the "complex analysis" narrative, whose hero is i; and the "trigonometry" narrative, whose hero is pi. Nobody had any idea that the arcs of these characters, which had been entirely separate up to that point, would directly intersect each other.
If math is a hyperlink movie with numbers as its heroes, then Euler's Identity can be thought of as that mindfuck scene at the end that makes the viewer go "No way!" It is the first, and so far only, scene in Math: The Movie in which all five of these compelling number-characters, 0, 1, i, pi, and e, interact directly. (When Heat came out, much was made of the fact that Pacino and DeNiro, two Italian giants of crime noir, had a single six-minute scene together; imagine how much more of a publicity coup it would have been if they were joined by, say, Humphrey Bogart, Denzel Washington, and Laurence Olivier.)
But what are these mathemacting luminaries saying to each other? No one knows. After proving the Identity in a lecture, Benjamin Peirce said, "It is absolutely paradoxical; we cannot understand it, and we don't know what it means, but we have proved it, and therefore we know it must be the truth." Euler's Scene is the mathematical equivalent of something out of a David Lynch film -- arresting, bizarre, inscrutable -- only, this David Lynch film is actually a documentary.