eiπ + 1 = 0

Euler's identity: Math geeks extol its beauty, even finding in it hints of a myste­rious connected­ness in the universe. It's on tank tops and coffee mugs. Aliens, apparently, carve it into crop circles (in 8-bit binary code). It's appeared on The Simpsons. Twice.

What's the deal with Euler's identity? Basically, it's an equation about numbers—specifically, those elusive constants π and e. Both are “transcendental” quanti­ties; in decimal form, their digits unspool into infinity. And both are ubiquitous in scientific laws. But they seem to come from different realms: π (3.14159 …) governs the perfect symmetry and closure of the circle; it's in planetary orbits, the endless up and down of light waves. e (2.71828 …) is the foundation of exponential growth, that accelerating trajectory of escape inherent to compound interest, nuclear fission, Moore's law. It's used to model everything that grows.

Enter Leonhard Euler, the one-eyed Swiss genius whom Frederick the Great lovingly called “our Cyclops.” What Euler showed, in his 1748 book Introduction to Analysis of the Infinite, is that π and e are deeply related, but in a very weird way. They’re connected in a dimension perpendicular to the world of real things—a place measured in units of i, the square root of –1, which of course doesn’t … exist. Mathematicians call it an imaginary number.

Courage, pilgrim! Here’s a visual metaphor. Imagine a graph with real numbers on the horizontal axis and, well, imaginary ones on the vertical. Now, remember the exponential function, f(x) = ex, from high school math? Ordinarily it graphs as an upward swooping curve—the very paradigm of progress. But put i in there, Euler showed, and eix instead traces a circle around the origin—an endless wheel of samsara intercepting reality at –1 and +1. Add another axis for time and it’s a helix winding into the future; viewed from the side, that helix is an oscillating sine wave.

The rest is easy: Take that function f(x) = eix, set x = π, and you get eiπ = –1. Rearrange terms and you have the famous identity: eiπ + 1 = 0.

Onformative

That’s the essence of Euler’s alchemy: By ventur­ing off the real number line into this empyrean dimension, he showed that disruptive, exponential change (the land of e) reduces to infinite repeti­tion (π). You might think you’re getting ahead in the world, but from this astral perch you’re just running laps.

Now, maybe you’ve never thought of math equations as “beautiful,” but look at that result: It combines the five most fundamental numbers in math—0, 1, e, i, and π—in a relation of irreducible simplicity. (Even more astonishing if you slog through the proof, which involves infinite sums, factorials, and fractions nested within fractions within fractions like matryoshka dolls.) And remember, e and π are infinitely long decimals with seemingly nothing in common; they’re the ultimate jigsaw puzzle pieces. Yet they fit together perfectly—not to a few places, or a hundred, or a million, but all the way to forever.

You can take this farther, too. If you write that function above in a more general but still simple form as f(x) = e(zx), where z = (a + bi), what you get is no longer a circle but a logarithmic spiral, combining rotation and growth—now both at the same time! These graceful spirals are also found everywhere in nature, from the whorls in a nautilus shell to the sweep­ing arms of galaxies. And they’re related, in turn, to the golden ratio (yet another infinite deci­mal, 1.61803 …) and the Fibonacci sequence of numbers (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...), which neatly describes the arrange­ment of leaves and petals in plants.

But the weirdest thing about Euler’s formula—given that it relies on imaginary numbers—is that it’s so immensely useful in the real world. By translating one type of motion into another, it lets engineers convert messy trig problems (you know, sines, secants, and so on) into more tractable algebra—like a wormhole between separate branches of math. It’s the secret sauce in Fourier transforms used to digi­tize music, and it tames all manner of wavy things in quantum mechanics, electron­ics, and signal processing; without it, computers might not exist.

The takeaway? Those crop-circle aliens were trying to tell us something.

Lee Simmons (@actual_self) is an editor and writer at WIRED.