Mandelbrot Fractals Explained

Mandelbrot Sets: Created using Javascript, mathematical explanation

The Mandelbrot set is one of the most famous examples of a fractal, an infinitely complex pattern that is often self-similar at different scales. It is a picture generated on the complex plane by repeatedly applying a very simple mathematical rule.

To understand it, you must first be familiar with complex numbers (z = x + yi), which have both a real part (x) and an imaginary part (y), and can be plotted as points on a plane, just like Cartesian coordinates.

The simple rule used to define the Mandelbrot set is the iterative function:

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

where z and c are complex numbers, and n represents the number of times the function has been repeated (the iteration number).

Here's how it works for a given complex number c (this c value represents a single point on the complex plane):

1. Start with the initial value z_0 = 0

2. Plug z_0 into the formula to find the next value, z_1 = z_0^2 + c.

3. Take the result z_1 and plug it back into the formula to find z_2, and so on. This sequence of numbers (z_0, z_1, z_2, . . .) is called the orbit of 0.

The Mandelbrot set, often denoted by $\mathbf{M}$, consists of all the c-values for which this sequence of numbers, or orbit, remains bounded (meaning the numbers do not grow infinitely large or "escape to infinity").

• If the sequence stays within a certain distance (like a circle of radius 2) from the origin, the starting complex number c is colored black (or another solid color) and is considered in the set.

• If the sequence grows larger and larger and tends toward infinity (or "escapes"), the starting complex number c is considered not in the set. These points are typically colored based on how quickly they escape, producing the beautiful, vibrant colors surrounding the black shape.

The resulting image is a "map" that shows how the orbit of 0 behaves for every possible value of c (x - real number, y - imaginary number) on the complex plane.