Because many natural phenomena (rotation, waves, electrical circuits, heat conduction, quantum mechanics) involve periodic behavior, π is needed to express cyclical patterns (circular motion).

And e is the limit of continuous growth and change rate — e is the only constant whose derivative equals itself, so there is no better unit than e to represent change.

What’s worth noting is that π represents cycles, e represents non-cyclical growth — the two are essentially mutually exclusive, contradictory, opposite — yet Euler’s formula allows them to transform into each other!

This is the most astonishing part: two seemingly incompatible modes of existence are unified in the complex world.

It is equivalent to saying:

A seemingly never-cyclical growth process (exponential e), in the complex domain, actually corresponds to circular motion (π, periodicity)!

So Euler’s formula reveals this kind of cosmic order:

All linear growth can be expressed through periodic structures. Or rather, all processes of change, in a higher dimension, are essentially cycles of some structure.

To put it more plainly:

All change is really just “drawing circles” in a higher dimension.