### Computing the Set

There is no formula to decide whether or not a complex number **c = a + b i** belongs to the Mandelbrot set (for any

**c**). However, all

**c**such that

**a**, do

^{2}+b^{2}≥ 4**belong to the set (exclusion condition), while all**

*not***c**for which there exists a

**z**where

_{j}= z_{k}**j ≠ k**do belong to the set (inclusion condition). For all other

**c**, one can simply compute the defining sequence (as explained in Mandelbrot Set), until one, eventually, encounters a

**z**where a

_{n}^{2}+b

^{2}≥ 4 (and conclude that

**c**does

**belong to the set). However, for any**

*not***c**which actually

**a member of the set, this will never happen, unless the inclusion condition is met.**

*is*

In reality one simply sets a ceiling ** Max** for

**n**(which can be 256, 512, 100000 or any other number), and decide that c

_{n}belongs to the set whenever c

_{n}=a

^{2}+b

^{2}< 4 for all n ≤

*Max*. This method can be supplemented by an approximation of the inclusion rule, such that membership is assumed if the absolute value of the difference between z

_{n}and z

_{n-1}is less than a predefined constant z

_{small}. By remembering the entire sequence this method can be extended to looking at the difference between any two numbers in the sequence.

### Drawing the Set

The following is looking at at some ways of drawing the set on a medium with a fixed number of points e.g. a computer screen. The medium then consists of x times y points where x is the medium's horizontal resolution while y is the medium's vertical resolution.

The first consideration is the mapping from complex plane coordinates to medium coordinates. Since any area of the complex plan has an infinite number of points, it is necessary to select a set of points which correspond to the medium's resolution, and then map each point to the medium's coordinates. An example will make this clearer:

A drawing medium consists of 2000 x 1000 points, and the part of the complex plan to be drawn is -1.0 to 0.5 along the horizontal axis and -0.25 to 0.5 along the vertical axis. Dividing 1.5 (that is 0.5-(-1.0)) by 2000, gives 0.00075, such that one will consider the points -0.5, -0.49925, -0.4985, ..., 1.0 along the horizontal axis, and similarly -0.25, -0.24925, -0.2485, ..., 0.5 along the vertical axis. These are the 2000000 points from the lower left (-1.0, -0,25) to the upper right (0.5, 0.5).

Additionally one has to decide the ceiling value ** Max** as explained over, and a

**z**if the inclusion rule is to be used.

_{small}

The simplest drawing strategy will then only consider if each point is member of the set or not. Member points are drawn with one particular color, while non-member points are drawn with another.

A more interesting method is to distinguish between how many iterations it takes to decide that a point is ** not** a member of the set. This will be a number between 0
and

**as explained over. A color scale with**

*Max***values will then be used, such that each point is drawn with a color corresponding to the**

*Max***n**where

**a**

^{2}+b^{2}≥ 4