Mandelbrot Set — Definition, Formula & Examples

The Mandelbrot set is the collection of all complex numbers

c

for which the iteration

z_{n+1} = z_n^2 + c

, starting from

z_0 = 0

, does not escape to infinity. When plotted on the complex plane, it produces one of the most famous fractal shapes in mathematics.

The Mandelbrot set $M$ is defined as the set of complex numbers $c \in \mathbb{C}$ such that the sequence $z_0 = 0,\; z_{n+1} = z_n^2 + c$ remains bounded, meaning $|z_n|$ does not tend to infinity as $n \to \infty$ . A practical test: if $|z_n| > 2$ for any $n$ , the sequence will diverge and $c \notin M$ .

Key Formula

z_{n+1} = z_n^2 + c, \quad z_0 = 0

Where:

$z_n$ = The current value in the iteration sequence
$c$ = The complex number being tested for membership in the set
$z_0$ = The starting value, always 0

How It Works

Pick any complex number

c

. Start with

z_0 = 0

and repeatedly compute

z_{n+1} = z_n^2 + c

. If the values stay small (bounded) no matter how many times you iterate, then

c

belongs to the Mandelbrot set. If

|z_n|

ever exceeds 2, the sequence will spiral off to infinity, and

c

is not in the set. To generate the iconic image, you test millions of points

c

across the complex plane and color each one based on whether it stays bounded or how quickly it escapes.

Worked Example

Problem: Determine whether c = 1 belongs to the Mandelbrot set.

Step 1: Start with

z_0 = 0

and compute the first few iterations using

z_{n+1} = z_n^2 + c

with

c = 1

z_1 = 0^2 + 1 = 1

Step 2: Continue iterating.

z_2 = 1^2 + 1 = 2, \quad z_3 = 2^2 + 1 = 5, \quad z_4 = 5^2 + 1 = 26

Step 3: Since

|z_2| = 2

already reaches the escape threshold and subsequent values grow rapidly, the sequence diverges.

Answer:

c = 1

is NOT in the Mandelbrot set because the iteration escapes to infinity.

Why It Matters

The Mandelbrot set demonstrates how strikingly complex patterns can emerge from a simple quadratic formula, making it a gateway to understanding chaos theory and dynamical systems. It appears in courses on complex analysis and computer science, and generating Mandelbrot images is a classic programming exercise that teaches iteration, complex arithmetic, and visualization.

Common Mistakes

Mistake: Assuming all points inside the main cardioid shape behave the same way.

Correction: Points in the Mandelbrot set can have very different orbit behaviors — some converge to a fixed point, others cycle between values. The set contains infinitely many smaller copies of itself at different scales.