Figure 2 - a Mandelbrot (Higher-Resolution Image)
Figure 3 - zooming in on the small area near the left hand side lying along the real axis twice reveals self-similar regions
Higher-Resolution Image (Fig 3a), Higher-Resolution Image (Fig 3b)
Fractals are images, made up of a set of points which are produced by mathematical calculation. Fractals are generated by a process called iteration, explained below, and the results are images with infinite complexity. The more you zoom in to a particular area, the more detail that is revealed, rather like the coastline around a country. The fractals also have regions of self-similarity. For example, the Mandelbrot fractal has characteristics nodules, which appear in similar configurations when you zoom in. Fractals can have finite area, but infinite perimeter, so they do not fall in the normal categories of one, two or three dimensions, but have a non-integral number of dimensions.
Iteration is the repetition of the same fairly simple calculation over and over again, each calculation using the result of the previous one.
The terms generated by an iteration are defined as functions of the previous term, and hence the first term must be fixed.
For example, the iteration z(n+1) = z(n) + 2 where z0 = 1 generates the terms: z(0) = 1, z(1) = 3, z(2) = 5 and so on (the odd numbers).
(NB. z(n+1) means z subscript n+1, or the (n+1)th term of z.)
The Mandelbrot set is defined by a simple iteration, involving complex numbers:
z(0) = 0
z(n+1) = z(n)² + c
The initial value of z is fixed at zero, but the value of c is varied.
The Mandelbrot set includes all the points where the above iteration is bounded. This means that after a while the iteration process will settle down to produce a single point, a set of repeating points, or oscillate round a particular point - the points involved are called the orbit. Certain values of c will cause the iteration to tend to infinity, so these initial values are not part of the Mandelbrot set.
An image can be produced by applying the iteration, with the c value depending on the co-ordinates of the pixel being plotted, the x co-ordinate representing the real part of c, and the y co-ordinate representing the imaginary part (like an Argand diagram).
A sample image is shown below in Figure 2. However, this image was produced as a 256-colour image. Points are either in or not in the Mandelbrot set, so how do the colours come about? The iteration is applied repeatedly until |z|² > 4, with the colour being plotted representing how quickly the iteration tends to infinity. The maximum number of iterations is set at 150, so the iteration does not carry on indefinitely, for example with a point which is in the Mandelbrot set, or tends very slowly to infinity. (The points in the Mandelbrot set are actually those in the black region in the centre, and they form a connected 'lake'.) The value 4, known as the bailout value can be changed, to produce different images.
Several images are shown below, produced by zooming in the fractal to show its infinite complexity, and self-similarity.
Figure 2 - a Mandelbrot (Higher-Resolution Image)
Figure 3 - zooming in on the small area near the left hand side lying along the real axis twice reveals self-similar regions
Higher-Resolution Image (Fig 3a), Higher-Resolution Image (Fig 3b)
This uses the same iteration as the Mandelbrot set, but c is fixed for the Julia set, and z(0) is varied depending on which pixel is required. There are therefore an infinite number of Julia sets, each using a different value of c. The Mandelbrot set acts as a catalogue for all the possible Julia sets. The Mandelbrot effectively plots the pixel at the origin for each Julia set, the pixel's co-ordinates defining which Julia set to use.
This link between the Mandelbrot and Julia set means that for every point on a Mandelbrot, a related Julia set can be plotted. If a point in within the Mandelbrot set is chosen, the corresponding Julia set is connected - that is all the points within the Julia set join up with all others, as the points in the Mandelbrot set do. However, if a point which is not in the Mandelbrot set is chosen, the corresponding Julia set is totally disconnected. Hence, at the boundary of the Mandelbrot set, there is the transition in the corresponding Julia sets from connected to disconnected.
The transition from connected to disconnected is illustrated below in Figure 4. The Julia sets for the points around the nodule just to the left of the centre of Figure 2, have been generated.
The values of c used are:
(1) c = -0.75 + 0i Higher-Resolution Image
(2) c = -0.75 + 0.01i Higher-Resolution Image
(3) c = -0.75 + 0.02i Higher-Resolution Image
(4) c = -0.75 + 0.0225i Higher-Resolution Image
(5) c = -0.75 + 0.025i Higher-Resolution Image
(6) c = -0.75 + 0.03i Higher-Resolution Image
(7) c = -0.75 + 0.04i Higher-Resolution Image
(8) c = -0.75 + 0.05i Higher-Resolution Image
(9) c = -0.75 + 0.06i Higher-Resolution Image
Figure 4 - the Julia sets generated corresponding to crossing a boundary between points in the Mandelbrot set, to those outside it, showing the transition from connected to disconnected
