Together of Mandelbrot

the whole of Mandelbrot is a Fractale which is defined like the Ensemble points C of the Plan complex for which the recurring continuation defined by:

Z _{N +1} = Z _N² + C

and the condition Z ₀ = 0

does not tighten towards the infinite one (in module).

If we reformulate that without using the complex numbers, by replacing Z _N by the couple ( X _N, there _N) and C by the couple ( has , B ), then we obtain:

X _{N +1} = X _N² - there _N² + has

and

there _{N +1} = 2 X _N there _N + B .

The whole of Mandelbrot was created by Benoît Mandelbrot and allows indicer the whole of Julia: to each point of the complex plan a whole of different Julia corresponds; these points of the whole of Mandelbrot correspond precisely to the related whole of Julia, and those outwards correspond to the nonrelated whole of Julia.

To draw the unit

It can be shown that as soon as the module of Z _N is strictly larger than 2 ( Z _N being in algebraic form, when X _N² + there _N²> 2²), the continuation diverges towards the infinite one, and thus C is outside the whole of Mandelbrot. That enables us to stop calculation for the points having a module strictly higher than two and which are thus outside the whole of Mandelbrot. For the points of the whole of Mandelbrot, i.e. the complex numbers C for which Z _N does not tend towards the infinite one, calculation will never arrive in the long term, therefore it must be stopped after a certain iteration count determined by the program. It results from it that the posted image is only one approximation of the true unit.

Although that does not have any importance on the mathematical level, the majority of the programs generating of the fractales post the points outside the whole of Mandelbrot in different Couleur S. the color allotted to a point not belonging to the unit depends on the iteration count to the end of which the corresponding continuation is declared divergent towards the infinite one (for example when the module is strictly higher than two). That gives several concentric zones, which surround the unit of Mandelbrot. Most distant consist of points C for which the continuation (z_n) tends “more quickly” towards the infinite one. These various zones delimit in a more or less precise way the unit of Mandelbrot.

Remarkable structures

The principal remarkable structure is the central Cardioïde, of center $A ({1 \ over 4}; 0)$ and of polar equation $\ rho (\ theta) = {1 \ over 2} (1 - \ cos \ theta). \$

One can also quote the Cercle of center $B (- 1; 0)$ and of ray ${1 \ over 4}$ .

Properties

The whole of Mandelbrot is symmetrical compared to the horizontal axis $(O, \ vec U)$ .

That is to say $M$ the whole of the complex numbers belonging to the unit of Mandelbrot.

$f_c is defined: Z \ mapsto z^2+c$ , for all $z \ in \ Complex$ .

The $M$ unit is thus consisted of the whole of the complexes $c$ such as $\ sup_ {N \ to \ infty} \ left| f^n_c (0) \ right| < \ infty$ .

Let us show by recurrence, that the property $P_n: \ forall C \ in \ Complex, f^n_c (0) = \ overline {f^n_ \ bar C (0)}$ is true $\ forall N \ in \ N$ .

Thus, it will have been shown that $c \ in M \ yew \ bar C \ in M$ (because the module of conjugé is equal to the module of a complex) and thus that $M$ is symmetrical compared to the axis of realities in the complex plan.

Initialization: For $n = 1$ , one has $f_c (0) well = C = \ bar \ bar C = \ overline {f_ \ bar C (0)}, \ forall C \ in \ Complex$

Assumption of recurrence: Is $n \ in \ N$ such as $\ forall C \ in \ Complex, f^ {N} _c (0) = \ overline {f^ {N} _ \ bar C (0)}$

From where $\ overline {f^ {n+1} _ {\ bar C} (0)} = \ overline {f^ {N} _ {\ bar C} (0) ^2+ \ bar C} = \ overline {\ overline {f^ {N} _c (0)}^2+ \ bar C} = f^ {N} _c (0) ^2+c = f^ {n+1} _c (0)$

Thus if $P_n$ is true, then $P_ {n+1}$ is true. However $P_1$ is true. Thus the property $P_n$ is true for all $n \ in \ N^*$ .

Thus the $M$ unit is symmetrical compared to $(O, \ vec U)$

Generating software of fractales

Fractint (owner - carried on many platforms)
“Makin' Magic Fractals”
ChaosPro - (owner - for Windows)
XaoS
Gecif
Gnofract 4D - (powerful and fast - for Linux, FreeBSD or Mac OS X)

Algorithms of generation

To draw a whole of Mandelbrot in C#.NET

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