1. Taking e to a complex power (using θi) walks you through the complex plane at a given angle θ. (That is, f(x) "draws" a continuous curve, where the "pen" has a constant rotational torque of θi.)

2. When θ = pi, f(x) is periodic (the angle is such that f(x) intersects itself or "loops.")

3. The set of complex numbers is isomorphic to a 2D Euclidean plane. Therefore, a walk at angle i*pi through the set of complex numbers is isomorphic to a circle.