Jamaica 10 ch14 Introduction to Simple Fractals

introduction to simple fractals....tchad
----------------------------------------

  yes, here is a small summary of the
theories and the practical details about
how to make a special form of fractals
known as julia-sets and the mandelbrot-
set.

             COMPLEX NUMBERS

  at first, you have to be familiar to
so called complex numbers...
  complex numbers are based upon the
factor sqr(-1), which is stated as I!
an arbitrary complex number is stated as

                 A + BI

where a and b can be any real
coefficient.

EX.   2 + 3i
      7 + 0.5i
     -1 - 2i    or    0.65734 - 0.48786i

  A is called the REAL part of the
complex number while B is called the
IMAGINARY part.

       HOW TO DRAW COMPLEX GRAPHS

  to visualize complex numbers in an
easy way, matematicians have invented a
so called complex plane...  that means
that you use the a-coefficient as the
x-coordinate and the b-coefficient as
the y-coordinate and then draw a point
at the corresponding place on the
complex plane. the x-axis turns into
real-axis and the y-axis turns into an
imaginary-axis.

EX.
                    i
                    i
                    i
                    i x (2 + 3i)
     (-5 + 2i) x    i
                    i
          ----------i----------
                    i     x (6 - i)
                    i
       (-3 - 3i) x  i
                    i
                    i
                    i


     THE THEORY BEHIND THE FRACTALS

  in the beginning of this century, some
french matematicians called GASTON JULIA
& PIERRE FATOU experimented with this
kind of complex numbers.
  their special interest was caught by a
simple iteration-formula like this :

               z = z^2 + c

where both z and c can be complex
numbers.
  they came to the conclusion that the
iteration developed in different ways
depending on the constant c. sometimes,
z seemed to approach infinity and some-
times it seemed to vary in a chaotic way
and never exceed a special value!
  they started to make maps in the
complex plane over the points whose
complex value resulted in a chaotic
behaviour of the iteration. the process
was executed like this :

     1. set c to any complex number.

     2. examine each point in the
        complex plane and set z to its
        value in the beginning of the
        iteration.

     3. repeat the iteration

        z = z^2 + c

        until the absolute value of z
        exceeds the value 2.
        if it never exceeds 2, then the
        point is within this actual
        julia-set and must be marked
        in the complex plane.

     4. repeat this (from *2) for each
        of the points in the plane.

        THE ITERATION IN DETAIL &
        THE ABSOLUTE VALUE OF Z!

  to fully understand the iteration, you
have to divide it into smaller pieces
and represent the variables with complex
expressions.

               z = z^2 + c
like this:
         x+yi = (x+yi)^2 + a+bi

         x+yi = x^2+2xyi-y^2 + a+bi

identify the real and the imaginary part
of z :

            x = x^2-y^2+a
            y = 2xy+b

  this is the general formula, used
while programming computers.

but what is the absolute value of z??

well, you can say it's the distance from
origo to z's place in the plane.
this can be expressed as SQR (X_<9e>2 + Y_<9e>2)
(according to old pythagoras)

hence, each point that never shows any
sign of developing z to exceed 2 is a
part of the actual julia-set.

DIFFERENT JULIA-SETS & THEIR CONNECTIONS

  there is a unique julia-set for each
c-value and there are only some small
connections between them. the simpliest
julia-set is the one that has the
c-constant = 0 + 0i. this julia-set is
shaped like a perfect circle. even some
small changes in the c-constant, changes
the julia-set radically. it slowly meta-
morphosizes into more and more
complicated shapes. for instance, the
julia-set for c = 0 + i looks like a
crack in the ground or whatever your
fantasy can come up with.
  at a certain limit, the julia sets
changes from being fully connected to
disconnected "islands". this limit is
different, depending on the angle from
origo to z. in fact, this limit is
extremely complicated.

           THE MANDELBROT-SET

  ofcourse, this limit can be visualized
in the complex plane. then it's more
known under "the mandelbrot-set"
  the main-process for the
mandelbrot-set is almost the same as for
the julia-sets.

               z = z^2 + c

the difference is that you don't keep c
for constant anymore. instead you follow
the complex plane with c, in the same
way you did with z when generating
julia-sets. the start-value for z is now
equal to 0. some time, i might include a
small routine for generating both
mandelbrot- and julia-sets.

             VISUAL EFFECTS

  you don't have to mark just the points
that belongs to the real set. in fact,
this is very unusual. to receive the
best effect, you usually choose to
colour the point according to how fast
they are attracted by infinity. for
example, you can let the colour of a
point depend on which iteration it
showed the first sign of being attracted
by infinity. if the point never does
this, the iteration stops at a pre-
defined amount (known as
"max.iterations") and it's coloured
black. this kind of technique is what
you usually see the result of in demos
and such.




           ... to be continued