Real Powers of Complex Numbers

It is easiest to think of real powers of a complex number using polar coordinates. If we let $z=Re^{i\theta},$ then

z^t = R^t e^{i\theta t}.

As long as $\theta \neq 0$ , the radius will grow exponentially while the argument grows at a constant rate. This means real powers of complex numbers are spirals.

Powers of $i \varphi$

Every time $t$ increments by 1, the point rotates by $90^\circ$ and grows by a factor of $\varphi$ .

The Fibonacci Spiral

A Fibonacci number is placed on one of the axes, then we rotate 90 degrees and go to the next Fibonacci number. It takes a few terms for the adjacent Fibonacci terms to have a ratio that gets close to the golden ratio. You can offset for this by turning the Fibonacci spiral with the slider above.

Fibonacci Spiral Offset

Offset the spiral (Clockwise)0°

0°180°360°

Golden Spiral

(i \varphi)^t

Fibonacci Spiral

Fibonacci Points

Powers of iφi \varphiiφ

The Fibonacci Spiral

Fibonacci Spiral Offset

Powers of $i \varphi$