The equation to develop Mandelbrot's set is simple : z = zē + c.
To see if a point is part of the Mandelbrot set, just take a complex number z. Square it, then add the original number c. square the result and add the original number again. Repeat the operation again and again ; if the result keeps on going up to infinity, it is not part of the Mandelbrot set (the black part of the picture). If it stays down to a certain level, it is part of the Mandelbrot set. The Mandelbrot set is the inner black part of the picture, and each different shade of grey represent how far out that particular point is. We can understand the picture imagining that we cut the shape on a metal sheet. We heat it and measure the temperature around the sheet which will have raised. We can give a color to the points which temperature is between a certain range, and an other color the the points in an other range....Basically the temperature gets higher as you get closer to the edges of the metal sheet; this is where is the crucial point : the edge of the Mandelbrot set is extremelly complicated. The picture only gives a slight idea of this complexity ; but we still can see discs of different sizes and we can imagine that there is many more even much smaller. The limited resolution of the image does not enable us to see all of them. The set extends like hairs of imperceptiple lines, which links the main set to mini replicas of this same set. This lines are invible (mathematical lines), but causes the temperature next to them to get higher. We can see this process creating amazing structures. To be able to see that, we need to watch the set's border very closely. We need a microscope. Special application offers this microscope. On the three following pictures, we can have an idea of this complexity :
The third image is a zoom in of the second image which is itself a zoom in of the first image. We can notice that the black shape of the third image looks very much alike the general view of the Mandelbrot set. This is a general property of the Mandelbrot set. However deep we zoom in , we will be able to find a replica of the main view of the set. It is like the signature of the Mandelbrot set.
The following image has been found on the left end of the set, it is highly zoomed in.
Imagining that it is 10 centimetres large, the main set at same scale would be 10E33 (10 to the power of 33), it is to say, 10E15 light years (the milky way is only 100 000 light years of diametre large). This gives an idea of the complexity and infinity of fractals.
