There is a formula that keeps startling me, even after I have traced its derivation many times: eiπ + 1 = 0. Euler's identity, it is called. It involves five numbers that seem to have nothing to do with each other. e, the base of natural logarithms, is the rate at which things grow when they grow continuously — it appears in compound interest, population dynamics, radioactive decay. i, the imaginary unit, is the square root of negative one, a concept that required mathematicians centuries to accept. π is the ratio of a circle's circumference to its diameter, deeply geometric, stubbornly irrational. 1 is the multiplicative identity. 0 is the additive identity. These five constants come from entirely different corners of mathematics. The identity says they are related by a single equation. And not in some approximate, this-is-a-coincidence way — exactly, algebraically, necessarily.
The derivation runs through Euler's formula: eix = cos(x) + i·sin(x). This is itself remarkable. It says that exponentiation by an imaginary number traces a circle in the complex plane. As x increases from 0, eix rotates — it does not grow, it turns. The real part oscillates as a cosine, the imaginary part as a sine. Set x = π and the point has rotated halfway around the unit circle, landing at −1. So eiπ = −1, and adding 1 gives 0. The identity is not a mystery once you know Euler's formula. It is a specific case of a rotation. π radians is a half-turn. A half-turn lands you at negative one. That's all.
And yet. The derivation makes the result less surprising while the result itself keeps feeling more so. The surprising thing is not that the algebra works out — it is that these five numbers are related at all. That the geometry of rotation, via the complex exponential, connects the transcendental constant of exponential growth to the transcendental constant of circular geometry, through the imaginary unit that is itself defined by a kind of self-reference (i² = −1, which is almost a statement about what happens when something turns against itself). The mathematics is doing something deep here: it is revealing that e and π, despite appearing in completely different contexts, are both aspects of a more fundamental structure — complex analysis — that unifies them.
This is a general pattern in mathematics that I find striking: superficially unrelated objects turn out to be instances of a common abstraction. The integers and polynomials, despite their differences, satisfy the same algebraic laws. The symmetries of geometric objects and the symmetries of equations over number fields are secretly the same thing (this is part of what the Langlands program is about). Mathematics keeps discovering that its objects are not isolated islands but outposts of a single connected landmass. Euler's identity is one of the most vivid examples of this hidden connectivity.
Richard Feynman, in a notebook entry from when he was fifteen, called the identity "the most remarkable formula in math." It still gets called that. The remarkable thing is not the formula's complexity — it is its simplicity. Five constants. One addition. One exponent. Zero. The compactness is part of the point: a dense amount of mathematical structure compressed into something you can write in a single line. I keep returning to it not because I expect to find something new but because it is one of those places where mathematics feels less like a human invention and more like a landscape that was always there, waiting to be surveyed.