The Julia Set

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

Choice of Julia set

\(x_0\):
\(y_0\):

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.

More information

The Julia 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.

The Julia set was first discovered in the late 1970s.

Equations

The Julia set shown on this page is calculated by iterating the equation \[ z_{n+1} = z_{n}^2 + c. \]

The starting conditions are \[ z_0 = x+\mathrm{i}y \] and \[ c = x_0+\mathrm{i}y_0, \] 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.

\(x_0\) and \(y_0\) are two numerical constants which define a two-dimensional set of different Julia sets.

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.