The Orbit of i

At a glance

For 𝑐 =𝑖, the orbit is

The values −1 +𝑖 and −𝑖 repeat in a two-point cycle. Their distances from 0 are √2 and 1, both at most 2, so the orbit stays bounded and 𝑐 =𝑖 belongs to the Mandelbrot set.

Key idea

Once an iteration returns to a value it has already reached, the deterministic next-step rule forces the entire future to repeat. A short, exact cycle can therefore certify infinitely many later steps.

Worked solution, part by part

1. Calculate and plot the orbit

Starting from 𝑧0 =0:

The distinct plotted points are (0,0), (0,1), ( −1,1), and (0, −1).

2. Identify the cycle

The value 𝑧4 = −1 +𝑖 repeats 𝑧2. Since the same rule is applied each time, the value after it must again be −𝑖, followed by −1 +𝑖. Therefore

3. Check the distances

For the two cycle values,

Both are at most 2. The earlier points 0 and 𝑖 also have distance at most 2. Because every later point is one of the two cycle values, no point can escape; by the rule supplied on the worksheet, 𝑐 =𝑖 is in the Mandelbrot set.

Proof of boundedness

The table checks every value before the first repeat. The equal values 𝑧4 =𝑧2 force the following values to repeat in the same order, so the two checked cycle magnitudes account for every later iterate rather than only a finite sample.

Check

Substituting either cycle value confirms the arrows both ways: ( −1 +𝑖)2 +𝑖 = −𝑖 and ( −𝑖)2 +𝑖 = −1 +𝑖. This makes the boundedness argument depend on an exact repeat, not on a picture that merely looks stable.