A formula can feel like a locked door: e to the i pi, plus
one, equals zero. Five symbols, one line, and somehow growth knows about circles.
Start with ordinary exponential growth. Put a point at one,
then let time increase. The point does not add distance;
it multiplies, stretching away from the origin smoothly.
The number e is the natural growth rate: the
one where the function's height and its rate of
change keep matching each other at every instant.
Now make one strange change. Point that growth
direction sideways, into the imaginary axis. Instead of
running outward, the motion begins to rotate.
The rotating point is e to the i t. Its
horizontal shadow is cosine, and its vertical shadow is sine,
synchronized as the point sweeps around the unit circle.
So Euler's formula is not decoration. It says this
exponential is secretly a coordinate pair: cosine t on
the real axis, plus i times sine t upward.
Lift time into a third dimension. The same rotating motion becomes a
helix, a spiral staircase whose shadow is still the unit circle.
From the side, time rises. From above, the height disappears. The shadow
left on the complex plane is exactly e to the i t.
Now replay only the shadow, starting at one and sweeping
through the upper half of the circle. By the time
t reaches pi, the arrow lands at negative one.
At that moment, the sine part has vanished and
the cosine part is negative one. The whole rotating
exponential has become a plain real number.
So e to the i pi equals negative one. Add one
to both sides, and the two opposite points cancel into zero.
That is Euler's identity: not a coincidence, but exponential
growth with its direction turned sideways into rotation.