Mandelbrot Sets

The Mandelbrot set is a captivating and intricate mathematical object that has fascinated mathematicians and enthusiasts alike. It’s a set of complex numbers, often denoted by \(M\), that is defined through a relatively simple iterative process. The Mandelbrot set was first pictured by Robert Brooks and J. Peter Matelski in a paper from 1978¹, although there still is debate on who discovered it first (I’d really recommend reading this article).

The set is defined as follows

\[M = \left\{c \in \mathbb{C} \, : \, z_{n+1} = z_n^2 + c \right\}\] Where \(z_{n+1}\) remains bounded for all \(n \geq 0\) and \(z_0 = 0\).

Where:

• \(M\): The Mandelbrot set.
• \(\mathbb{C}\): The set of complex numbers.
• \(c\): A complex number.
• \(z_n\): A sequence of complex numbers, defined by \(z_{n+1} = z_n^2 + c\).
• The condition for \(c\) to be in the Mandelbrot set is that the sequence \(z_n\) remains bounded for all \(n\).

Numerically, we can check for boundedness after a certain number of iterations.

function mandelbrot_set(x, y, n, fun)
    z = c = x + y*im
    for i in 1:n
      if abs(z) > 2
        return i
      else
        z = fun(z,c)
      end
    end 
    return 0; # no success after n iterations
end

The previous function implements a sort of Mandelbrot set for a function fun(z,c), where we input our complex domain and get a matrix that measures an approximate boundedness, depending on the number of iterations we require.

mandelbrot_set.(X, Y', 100, (z,c) -> sin(z/c))

Here are some examples