Arguably the most famous of all the computer-generated images of fractals, the Mandelbrot set was discovered by Benoit B. Mandelbrot, the god-father of fractals.
The Mandelbrot set is generated by a single equation:
Z = z2 + c
All of these variables are complex numbers, which means that they have two parts: real and imaginary. The real part is an ordinary number, but the imaginary part is called 'imaginary' because it is multiplied by the square root of negative one. Try finding the value of the square root of minus one!
To find the colour of any point in or surrounding the M set, enter its co-ordinates in c and set z initially to zero. Then calculate z2 + c. Input this number back into z, calculate z2 + c, input this value back into z, and so on. This process is known as iteration. Many iterations (a number in the hundreds, perhaps thousands, is required for high magnifications) will reveal whether z escapes towards infinity or never strays far from zero. If the latter is true, that point is deemed to be part of the Mandelbrot set and is coloured black. If it escapes towards infinity, it is coloured by the speed at which it does so. These points are not, by definition, actually part of the set.
Two interesting qualities of the Mandelbrot set are self-similarity and infinite complexity.
Self-similarity is where Mandelbrot sets exhibit certain repetitive characteristics, which become apparent when sets are zoomed in on. One such example is the miniature Mandelbrot sets dotted around the edge of the mother Mandelbrot set. In turn, these Mandelbrot sets have similar off-spring dotted around them in turn, and so on. These sets are often called 'mini-Mandelbrots'. In the Animations section, you will be able to find a few animations of zooming in to mini-Mandelbrots. Note that these smaller sets are never the same as the original; thus self-similarity, not self-repetition.
The Mandelbrot set is infinitely complex because you can zoom in on the original Mandelbrot set as many times as you want, and there will be no loss of detail, no matter how far in you have gone. There are no limits to the beauty of the Mandelbrot set.
Here is a picture of a fractal generated using the Mandelbrot set:
The Julia set is also governed by the equation Z = z2 + c. However, it is slightly different from the Mandelbrot set in that instead of c containing the co-ordinates, z does. c contains numbers that make each Julia set different. These numbers relate to the co-ordinates of the Mandelbrot set . If c refers to co-ordinates outside the border of the Mandelbrot set, the corresponding Julia set is 'disconnected', meaning it explodes into fractal dust. If it is inside the Mandelbrot set, the corresponding Julia set is 'closed', meaning at least part of the Julia set is doomed to never stray far from zero. A few Julia sets are shown below, including the 'Dragon', the 'Rabbit' and a simple Julia set.
Included in the Animations section are animations of the Julia sets with their corresponding points on the Mandelbrot set moving from the 'east' to the 'west', and another one from the 'north' to the 'south'.
Also in the Animations section are the corresponding Julia sets from the mini-Mandelbrots in the Mandelbrot set animations shown earlier, and magnifications of their centres. You should see that these Julia sets are 'closed'. This is as expected, because their corresponding co-ordinates are part of the Mandelbrot set (i.e., they are coloured black).
Here are a few pictures of fractals generated using the Julia set: