The geometry hidden in Euler's formula

Euler's formula is the standard candidate for the most beautiful equation in mathematics, and it is usually introduced in a way that makes it seem like a coincidence uncovered by manipulating power series.

The series proof is correct and completely unilluminating. It tells you the two sides agree without telling you why anyone would expect them to.

Here is the geometric reading. Multiplication by a complex number is a rotation combined with a scaling. Exponentiation is what you get by repeatedly applying a small change. Put those together and e raised to an imaginary power is asking: what happens if the small change I keep applying is a nudge perpendicular to where I currently am?

A nudge always perpendicular to your position vector, never changing your distance from the origin, is exactly the description of travelling around a circle at constant speed. So the answer must be a point on the unit circle, at angle equal to the total time elapsed. That is the formula, arrived at without a single series.

Set the angle to π and you get the famous identity, which now reads as a rather mundane geographical fact: walk half a turn around the unit circle and you end up at negative one.

The beauty survives the explanation. It just relocates — from mystery to inevitability, which I would argue is the better kind.