Impulse #04 24 февраля 1999

Программирование

Fractals - Fractals from a theoretical point of view.

              FRACTALS from a theoretical perspective



                                                (C) Adrien 
Douady 

 Julia set and Mandelbrot set


   Julia sets of quadratic maps and mnozhezhestvo Mandelbrot 
appear in a situation that from a mathematical point of view is 
extremely easy - from the sequences of complex numbers, defined 
by induction using the relation: Z (n +1) = Z (n) ¤ + c, where 
c - is a constant. 


   The behavior of the above sequence of numbers depends on the 
parameter c and the starting point Z (0). If we fix c

and change the Z (0) in the field of complex numbers,
then we get the Julia set, and if we fix Z (0) = 0 and change 
the parameter c, then we obtain the Mandelbrot set. If

take Z (0) far from zero, then the sequence will rapidly tend 
to infinity. Of course, this is also true when the point Z (n) 
for some n is far from zero. But there are such values ​​Z (0), 
for which the sequence (Z (n)) never goes away, but always 
remains limited. When given c, these values ​​form the filled 
Julia set Kc for the polynomial Fc: Z-> Z ¤ + c. Present the 
same Julia set consists of boundary points Kc.



   It is only natural that the form of a set
Julia depends on the choice of c, but
surprised at how this relationship
strong. And, by changing c, you can get an incredible variety 
of Julia sets: one of them look like large "thick" clouds

others resemble the sparse bushes of blackberries,
third look as the sparks flying in the sky
during the fireworks. There are two main types of Julia sets: 
some of them are in one piece (we're talking connected), and 
others represent the cloud of points (We call them the Cantor 
set). For mathematics there is a good opportunity to introduce 
a new set - a set values ​​of c, for which the Kc is connected. 
I call it the Mandelbrot set, as Benoit Mandelbrot was the 
first recipient his image with a computer and initiated its 
study. 


   Julia set is among the
the most interesting fractals. Most of them are self-similar. 
Looking at the picture of any set of Kc in the microscope, 
we'll see a picture that, firstly, depend little on whether, in 
what place we look, and secondly does not significantly 
olichaetsya from what we saw and without microscope. At the 
same time, the Mandelbrot set M does not possess the 
self-similarity: Yes, M is really an infinite number of copies 
of itself and, therefore, in some place where we might look at 
the boundary of M in the microscope, we can see some of the 
small copies of M. But those copies woven into a network of 
filaments, which form a very strongly depends on at what point

look. Moreover, if we consider
two copies of a comparable size, the ratio of the distance 
between them to their size will depend greatly not only from 
the point at which we observe, but also on the increase

microscope.


          Invariant sets


   Invariant with respect to any
Conversion is the figure of the complex plane are not affected 
by this transformation. The simplest example is the figures 
that are invariant under quadratic transformations f (x) = x ¤ 
+ b * x + c. 


   The method of constructing such sets will show the example 
of the transformation: 

          f (x) = x ^ 4 +2 * Q * x ¤ + E (*)


   First, choose any specific
values ​​for Q and E, for example,
Q =. 13 + .4 i, E =. 08-.5i

   The process of construction - an iterative, so we define the 
Number of iterations: iteration = 5000


   Initial value: X0 = 0

   Iteration formula:

          X (i +1) = eSQRT (-QeSQRT (Q ¤ + X (i)-E))


   Equation (*) is generally the case 4
root. We must choose for each iteration of a single root. The 
choice can be implement a random manner. e means plus or minus. 
All calculations - over complex numbers. If the plot Xi: the 
axis X - Re Xi, the axis Y - Im Xi for data values, the 
resulting figure will resemble an island. The form of the 
resulting shape depends on the values ​​of Q and E. 

   Similarly we can construct a shape that is invariant under 
any other transformation.


IMPERIO> The text was found and treated me to you
to the smallness of pomozgovat on this subject, here. From a 
practical part of this question I really still do not 
understand, but perhaps in the next issue of IMPULSE is a 
serious discussion on this topic. I still urge all those who 
may be something to bear on the topic of fractals to contact me 
personally - would be very grateful.