The Mandelbrot Set

by Dominic Ford
Colour scheme
Coordinates
Left:
Right:
Top:
Bottom:
Rendering time: sec
Speed:

Instructions

To navigate around the fractal, click and drag it with the left mouse button. To zoom into or out of the fractal, use the scroll wheel on your mouse.

Each point within the Mandelbrot set is associated with a unique Julia set. To view the Julia set associated with any chosen point, double click.

More information

The Mandelbrot set is one of the best known examples of a fractal. It is a structure with an infinite amount of fine detail: you can zoom in on the edge of the fractal forever, and it will continue to reveal ever-smaller details.

This is an example of chaos theory: an arbitrarily small change in position within the Mandelbrot set can cause an large change in the outcome of the calculations used to work out the colour of that location.

The Mandelbrot set was first discovered in the late 1970s, and was studied by Benoit Mandelbrot in 1980.

Equations

The Mandelbrot set is calculated by iterating the equation \[ z_{n+1} = z_{n}^2 + c. \]

The starting conditions are \[ z_0 = 0 \] and \[ c = x+\mathrm{i}y, \] where \(i=\sqrt{-1}\) and \(x\) and \(y\) are the horizontal and vertical position of the location within the fractal whose colour you wish to calculate.

The calculation is repeated until \(|z_n|>2\), and colours are assigned to each location depending on the number of iterations required until this condition is met.

Above, you can choose various different colour schemes, each of which map the range of values of \(n\) onto a range of colours. If you tick the box labelled smoothing, then smooth colour gradients are drawn between the integer steps that \(n\) takes. This is not strictly mathematically interesting, but does make the fractal look prettier.

A maximum number of iterations needs to be specified, because in some parts of the fractal, the iteration sequence above will never end. This is true in the 'lake' area in the centre of the fractal.