Description
These two types of fractal are included together in the same module as they are
derived from the same equation.
This equation is iterated until it is obvious that the value of Z is approaching
infinity (>2 will do) or until we loose interest. In the examples that
follow we tend to give up if Z2400 is still less than 2. Z and c are
complex numbers, so we really mean |Z2400| < 2. If |Z| becomes
greater than 2, we count the number of iterations that have passed and use this
to choose a colour. If, after we've iterated more times than we care to, we
still haven't passed 2 we assume the point is inside the set and colour it
black. If you try generating your own patterns you will find that the further
you zoom in the larger iteration cut-off you will need to draw decent patterns.
The following two images illustrate the effect of changing the maximum number of
iterations from 16 to 2400. In this case 2400 is probably too many, but 16
clearly isn't enough. If you choose too many the image will take ages to draw
but otherwise will be ok.
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| 16 iterations | 2400 iterations |
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| These pictures also illustrate the advantage of the new colouring algorithm. The locations of the colour bands no longer change depending on the number of iterations, so if you want to create a series of zoomed images you can increase the number of iterations as you go without loosing continuity. They were created with the new "-d8" colour scheme, as was the image on the right, it's 5000x3750 pixels! |
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The number of iterations required before a point is clearly diverging depends on the two starting values (Z0 and c). As both of these are complex numbers we have four independent variables Zx, Zy, Cx and Cy, where x and y represent the real and imaginary parts of the complex number. I use x and y rather than r and i as they relate more obviously to the 2D images we are generating.
If we lived in a five dimensional universe and had four dimensional computer screens to stare at, we could study the effect of varying all four parameters, and would see on our screens an interesting 4 dimensional shape surrounded by waves of colour. Being a 3D being I personally find it rather hard to imagine what this 4D object actually looks like, instead I have to satisfy myself with looking at 2D slices. Generating this 4D shape could take some time too, as a 1000x1000x1000x1000 pixel screen would have a trillion elements and require 4 terabytes to store the image (32 bit uncompressed).
One obvious slice to draw is where Z0 = (0,0). Here we are free to vary Cx and Cy over whatever ranges we like and study the patterns generated. If we use the values of Cx and Cy as coordinates and plot the number of iterations before |Z| > 2 we get the familiar Mandelbrot image. We can zoom into this indefinitely, seeing more and more complexity, until the detail gets too much for the computer screen to render.
Alternatively, we may choose to pick a particular value of 'c' and vary the starting value of Z. There are an infinite number of 'c' values and each one produces a different pattern. These are referred to as the Julia sets. Every point on the Mandelbrot image has a corresponding Julia set. Interestingly, if 'c' is in the black point in the Mandelbrot then the black part of the Julia set will be joined up, whereas if you choose a point from the coloured region the Julia set will be an infinite number of islands. The example below is in this region as can be seen in the 4th image.
