Euler's Formula: Complex Multiplication Is a Rotation
An interactive live film: e^{iθ} climbs a helix over the complex plane whose shadows are the cosine and sine waves; multiplying z by it turns z by θ and keeps |z|, the same numbers as the 2D rotation matrix R(θ).
Narration
The floor is the complex plane: the real part along one axis, the imaginary part along the other. The point e to the i theta is cos theta plus i sin theta, and it always lies on the unit circle.
Lift the point by its angle. As theta grows, it climbs a spiral, and from above you see only the unit circle, because its length is always one.
The rest of the narration is in the book's Companion.
