One of mathematics' most beautiful examples of how an incredibly simple rule can produce seemingly infinite complexity.
The Mandelbrot Set is a collection of points in the complex plane. Each point is tested by repeatedly applying a very simple mathematical rule.
Start with z = 0. Pick a value of c. Then repeatedly calculate the next value of z. If the sequence remains bounded, that value of c belongs to the Mandelbrot Set. If the sequence escapes towards infinity, it does not.
The Mandelbrot Set lives in the complex plane. Every point has two coordinates: a real component and an imaginary component.
The horizontal axis represents the real number a, while the vertical axis represents the imaginary number b.
So every pixel in the image represents a different complex number being tested.
The familiar black region is the Mandelbrot Set itself. Points outside it eventually escape.
The colours indicate approximately how quickly each point escapes. A point that escapes very quickly receives a different colour from one that remains bounded for hundreds of iterations.
There is no complicated formula hidden behind the image. The core calculation is simply:
Yet repeatedly applying this rule creates an enormously intricate boundary. Zoom in and new structures appear: spirals, bulbs, tendrils and miniature copies of the overall structure.
This is one of the great themes of mathematics: complicated behaviour does not necessarily require complicated rules.
How do we decide whether a point escapes?
During the iteration we examine the magnitude of z. Once its magnitude becomes greater than 2, the sequence is guaranteed to escape.
Computers can therefore test enormous numbers of points extremely quickly.
The most remarkable feature of the Mandelbrot Set is its boundary.
At normal scale it looks like a strange collection of bulbs and filaments. Zoom closer and increasingly elaborate structures appear.
The process can continue to extraordinary depths. The mathematics is deterministic, but the visual complexity can feel almost organic.
The Mandelbrot Set is closely connected to another family of fractals known as Julia Sets.
Both use essentially the same iterative equation:
The difference is subtle but important. In the Mandelbrot Set, we vary c and start the iteration at z = 0. For a Julia Set, c is fixed and we vary the starting value of z.
Fractals are mathematical structures in which patterns can repeat across different scales. Nature contains many structures with fractal-like characteristics.
The Mandelbrot Set isn't merely a beautiful computer image. It is an illustration of several important mathematical ideas at once:
Complex numbers give us the mathematical space in which the system operates.
Iteration shows how repeatedly applying a simple rule can generate complicated behaviour.
Dynamical systems allow us to study how systems evolve over time.
Chaos and sensitivity show how small changes in initial conditions can have profound consequences.
And perhaps most importantly, it demonstrates something that appears repeatedly throughout mathematics: simple rules can create extraordinary complexity.