This article was written by me for a couple of years ago and is recycled and somewhat modified here for DA. It was originally intended to be a comment of "color filters", but grown up to an illustrated article). The following illustrations in my scraps belong to this journal:


fig 1 [link] Max iterations = 50
fig 2 [link] Max iterations = 100
fig 3 [link] Max iterations = 250
fig 4 [link] Max iterations = 500
fig 5 [link] Max iterations = 1000
fig 6 [link] Max iterations = 50000
fig 7 [link] Max iterations = 50000 The level sets colored


They will also be linked in the proper places in the text below


In a picture of so called escape fractals, for example the Julia sets and the Mandelbrot set, points are either inside or outside the set(s). Now let's take the Mandelbrot set for example. In a picture of the M set (or a part of it) the screen is built up of pixels representing various "c" in the iteration formula z -> z^2 + c. Now mainly two things can happen under the iteration process:

1) The variable "z", starting with z = 0, tends to infinity, in which case the corresponding "c" does not belong to the M set, or,
2) The variable "z", starting with z = 0, has a bounded orbit, forever within a magnitude of 2, in which case the corresponding "c" does belong to the M set.

To read more about the math, see the Articles 1 - 9 in the Chaotic Series [link] . The most basic color method is to color those "c" belonging to the M set black, and those "c" not belonging to the M set white.

To this journal are seven JPEG's uploaded under scraps, all coming from the same area of the M set. The first six are black and white, the seventh colored. Let's have a look at fig 1 [link] . It's really obvious what I'm talking about. The black part is a portion of the M set, the white part not. But stop for a moment! In this picture the program runs the inner loop at most 50 times. That is, if for each c = #pixel, the variable "z" has not reach the escape radius (must be at least 2) after iterating the formula 50 times the program gives up and colors the pixel corresponding to that "c" black. But if we increase the maximal iteration number from 50 to 100? The result is displayed in fig 2 [link] . The part of the screen belonging to the M set has decreased, and the border has become more like a fractal, In fig 3 [link] , fig 4 [link] , fig 5 [link] , and fig 6 [link] , the max iteration number is increased to 250, 500, 1000, and 50000 iterations respective. Increasing the max iteration to more than 50000 makes no difference to this motive. The only parts still remaining black are those that really belong to the M set. That is the infinite hierarchy of bulbs (in fact be regarded as minibrots, see Article 13 "Buds are Minibrots" in the Chaotic Series) and the mini copies that seems to float around as dust in this picture. It was in this manner Mandelbrot first saw his set to appear. A question that now presents itself is: Are the minibrots floating around outside the continent constituted of isolated islands, or are they connected with the continent with thin threads, too thin to be captured by the raster of the screen? It was in the beginning of the eighties the mathematician John Hamal Hubbard in a mathematical way proved that the M set is connected. In fact "the thin threads" are solely built up of an infinite hierarchy of smaller minibrots!!! Now look at the last picture (fig 7 [link] ). Just as in fig 6 [link] , the program *gives up* after 5000 iterations. Note that the black areas of these two images are exactly the same. But outside the set the colors are established according to the number of iterations (up to 4999) required for the orbit of z = 0 to reach the escape radius (in this case 10, the bailout being 10^2 = 100). This is the easiest method to color this type of fractal images. You can observe what I'm talking about if you rather fast switch between fig 1 - 6. The colors in fig 7 [link] are like traces of the banks where the black continent has retired, a history of the iteration process. So according to some discussions that have been on some fractal forums regarding coloring methods and filters, were some fractal artists have called every coloring of fractals for "filters", I think that's wrong. The reason for that is that "filtering" means "take away something". In fact the above method of outside coloring make the fractal border of, in this case the Mandelbrot set, much more obvious than leaving the outside white. But this is my opinion. And then we have inside coloring methods! Take care, and don't get lost in coloring methods and filters

BTW, if you look at all the 6 black and white illustrations at the same time [link] or switch fast between them, you will get an intuitive understanding what I am talking about