Image credit: Wolfgang Beyer
The Mandelbrot set is one of the most recognizable fractals. It is generated on a complex plane using a simple equation.
First, pick a complex number (complex in the mathematical sense, not complexity sense) and call it c.
Next, implement the equation
For example, if we start with the point c=1, Then the series drifts off to infinity (1, 2, 5, 26, …). If we start with c=-1, we get a bounded series (-1, 0, -1, 0, …). Series that are bounded belong to the Mandelbrot set, and unbounded series do not. Now, we can graph the Mandelbrot set by using colors. Black denotes that the starting point belongs to the set, and colors denote how quickly the series diverges.
The Mandelbrot set shows that complicated features can be generated using simple rules. We also observe a sensitivity to initial conditions in some regions, where the diverging points and converging points are woven infinitely close together.
The video below zooms into the Mandelbrot set, showing how patterns repeat on all sorts of scales. Complexity emerges from simple equations.