Understanding the Mandelbrot Set

The Mandelbrot set is a fractal generated on the plane of complex numbers. To understand how elements of the set are computed, we first need to understand what complex numbers are.

What Are Complex Numbers?

Complex numbers arise when we ask: What number, when multiplied by itself, gives -1?

In the real numbers, this isn’t possible. Multiplying a positive or negative number by itself always results in a positive. So, no real number squared equals -1.

To solve this, we introduce a new number called i, defined by:

i × i = i² = -1

We treat i like any other number in algebra. For example:

a·i + b·i = (a + b)·i

Combining a real number a with a multiple of i gives the general form of a complex number:

a + b·i

Algebra with Complex Numbers

Complex numbers follow the usual rules of algebra, including distributivity and associativity. This allows us to expand products like:

(a + b·i)(c + d·i) = (a·c - b·d) + (a·d + b·c)·i

If a = c and b = d, the expression becomes a square:

(a + b·i)² = (a² - b²) + 2·a·b·i

Defining the Mandelbrot Set

The Mandelbrot set is defined using the iteration formula:

zₙ₊₁ = zₙ² + c

We start with a complex number:

c = a + b·i

You may wonder: how do we apply zₙ₊₁ = zₙ² + c if we don’t have a starting value for z? In the Mandelbrot set, we always begin with:

z₀ = 0

Example: c = i

Let’s apply the rule starting from z₀ = 0 and c = i:


            z₁ = z₀² + i     = 0² + i     = i  
            z₂ = z₁² + i     = i² + i     = -1 + i  
            z₃ = z₂² + i     = (-1 + i)² + i  
                            = (1 - 2·i + i²) + i  
                            = (1 - 2·i - 1) + i = -i  
            z₄ = z₃² + i     = (-i)² + i = -1 + i  
            z₅ = z₄² + i     = (-1 + i)² + i = -i

As you can see, the sequence cycles through a set of values and does not diverge to infinity. This means c = i is in the Mandelbrot set.

Example: c = -1


            z₀ = 0  
            z₁ = z₀² + (-1) = -1  
            z₂ = z₁² + (-1) = 1 - 1 = 0  
            z₃ = z₂² + (-1) = 0 - 1 = -1

The values oscillate between 0 and -1 without growing unbounded, so c = -1 is in the Mandelbrot set.

Example: c = 2


            z₀ = 0  
            z₁ = 0² + 2 = 2  
            z₂ = 2² + 2 = 6  
            z₃ = 6² + 2 = 38

Here, the values increase rapidly. Since z diverges to infinity, c = 2 is not in the Mandelbrot set.

Visualizing the Mandelbrot Set

To visualize the Mandelbrot set, we use the complex plane. Think of it as a 2D grid:

Horizontal axis = Real part of c
Vertical axis = Imaginary part of c

Each pixel in an image corresponds to a complex number a + b·i. For each pixel:

Assign a complex value c.
Start with z₀ = 0.
Apply zₙ₊₁ = zₙ² + c for n iterations.
If the value of z stays bounded (commonly checked if |z| < 2), mark the point as in the set.

Points inside the Mandelbrot set are colored black. Points outside are colored based on how quickly z escapes to infinity.

Exploring Beyond z²

You can explore variations of the Mandelbrot set by changing the power in the formula. For example:

zₙ₊₁ = zₙ³ + c

These higher-power Mandelbrot sets generate entirely new fractal patterns and open the door to endless mathematical exploration.

Conclusion

The Mandelbrot set shows how complex behavior can emerge from a simple rule. By extending our number system with complex numbers and applying a straightforward iteration, we uncover one of the most iconic and beautiful objects in mathematics. Whether you’re exploring with z² + c or experimenting with other powers, there's always more to discover. If you are interested in exploring, check out the fractal viewer below to generate your own beatiful fractals.