The Mandelbrot set is a particular set of complex numbers that has a distinctly recognizable shape. Each point on the plane of complex numbers is either in the Mandelbrot set or not. A point's membership is determined by a simple iterative process:
For a given complex number \(c = x + yi\), where \(x\) and \(y\) are real numbers and \(i\) is the imaginary unit, we define the sequence \(\{z_n\}\) by the iterative rule: \(z_{n+1} = {z_n}^2 + c\) with the initial condition \(z_0 = 0\). The number \(c\) is in the Mandelbrot set if and only if the absolute value (magnitude) of \(z_n\) remains bounded as \(n\) approaches infinity. In other words, \(c\) is in the Mandelbrot set if and only if there exists a number \(N\) such that \(|z_n| < N\) for all \(n\).
In practice, when visualizing the Mandelbrot set, we often limit the number of iterations and consider \(c\) to be in the set if \(|z_n| < 2\) after a certain number of iterations (the choice of 2 is arbitrary since any number greater than 2 would give the same set). The number of iterations required before \(|z_n|\) exceeds 2 (or some other escape radius) can be used to determine the color (or in this case brightness) of each point \(c\), creating the familiar fractal images.
Although the definition of the Mandelbrot set is simple, it results in incredibly complex shapes. In this visualization, you can explore this fascinating object and see how it keeps revealing more detail the closer you look, never repeating itself exactly. It is a beautiful demonstration of the deep links between mathematics, nature, and art.