* Mandelbrot Set * 

* Mandelbrot Sequence is an infinite sequence of numbers z_{0}, z_{1}, z_{2}, z_{3}, .....z_{n}, z_{n+1}, ..... of the form Z_{n+1} = [Z_{n}]^{2} + Z_{0} where the first number is z_{0} (called the seed )& remaining numbers are defined recursively by the said formula.  
* Taking z_{0}= 1, we get z_{1} =2 , z_{2} =5 , z_{3} =26 and so on. The sequence generated is called an escaping sequence as the terms get bigger and bigger without bound. z_{0}= 1 is called a nonblack point.  
* Taking z_{0}= 1, we get z_{1} =0 , z_{2} =1 , z_{3} =0 and so on. The sequence generated is called an nonescaping sequence . It is a periodic sequence as the terms repeat themselves. z_{0}= 1 is called a black point.  
* Taking z_{0}= 0.5, we get z_{1} =0.25 , z_{2} =0.4375 , z_{3} =0.3086 and so on. As n value increases, the numbers get closer to 0.36 though they do not reach that value. The sequence generated is called an attenuated periodic sequence attracted to 0.36 value . It is called an attracted sequence & z_{0}= 0.5 is called a black point.  
* Mandelbrot Set is a set of all complex numbers c or z_{0} that are black points i.e. they generate either periodic sequence or attracted sequence. Points that escape are the only points not in Mandelbrot Set. In the above examples, real numbers were used as seeds. But one can use complex numbers. For example, Taking z_{0}= i, we get z_{1} =1+i , z_{2} =i , z_{3} =1+i and so on. The sequence generated is called an nonescaping sequence . It is a periodic sequence as the terms oscillate between i & 1+i. z_{0}= i is called a black point.  
* Orbit is the path followed by the values of Z at each n.  
IInd Step  
* Let a complex number z > as a pair of (x,y) real numbers and complex number c > as a pair of real numbers (a,b). Formula Z >z^{2}+c which transforms to x (real part of Z) > x^{2}y^{2}+a and y (imaginary part of Z) >2xy+b  
* If (x_{0},y_{0}) =(0,0) and (a,b)=(1/2,1/2), then (x_{1},y_{1}) =(1/2,1/2) ; (x_{2},y_{2}) =(1/2,1) ; (x_{3},y_{3}) =(1/4,3/2) and so on.  
*
The analog of
the Mandelbrot set can be defined for any 



In mathematics, the Mandelbrot set, named after Benoît Mandelbrot, is a set of points in the complex plane, the boundary of which forms a fractal. Mathematically, the Mandelbrot set can be defined as the set of complex values of c for which the orbit of z_{n} under iteration of the complex quadratic polynomial z_{n+1} = [z_{n}]^{2} + c remains bounded. That is, a complex number, c, is in the Mandelbrot set if, when starting with z_{0}=0 and applying the iteration repeatedly, the absolute value of z_{n} never exceeds a certain number (that number depends on c) however large n gets. Why Fibonacci Sequence appears inside Mandelbrot set is not properly understood, letting c = 1 gives the sequence 0, 1, 2, 5, 26… and so on. This sequence goes towards infinity. As this sequence is unbounded, 1 is not an element of the Mandelbrot set.
When computed and graphed on the complex plane, the Mandelbrot Set is seen to have an elaborate boundary, which does not simplify at any given magnification. This qualifies the boundary as a fractal. 

A fractal generated by iterating: z_{n+1} = [z_{n}]^{2} + c
where z_{0}=0 and plotting how fast it diverges to infinity for different
values of the complex number (speed represented as colours). The black
set represents the "prisoner" points that do not diverge: it is
the Mandelbrot set . 

How The Mandelbrot Set Works 
