FRACTALS

           The Perfect Order

The fractal geometry is mathematically
describing many forms from science and
nature.  It was discovered a long time
ago, and was often used in arhitecture
and  in  art.   But 1980s are the real
beginning.
  For  a  real fractal calculation you
need  speed.   And  that was a problem
until  computers  were fast enough for
such   calculations.    E.g.   imagine
drawing  a Mandelbrot (pic.1) 320X256.
For  every pixel you have to take it's
coords  as  a complex number, get it's
square,  add the coords again, get the
square...   until  it reaches infinity
or  almost  zero.  It would take hours
for a pixel.  Perhaps you could finish
it in few years.  The computers can do
the  same in few secs!  And the pic is
great.   That's  what  I  call  ART in
MATH!   And  with small changes it can
get different forms (Julia, pic2).
  One  of  the definitions of fractals
says  they  are  mathematical  objects
with  infinitely  fine structures.  It
can  be  a  real adventure diving in a
fractal! If you want to draw a fractal
there  are  few examples at the end of
this text.
  There  are  two  kinds  of fractals,
linear  and unlinear ones.  The linear
ones  have  the  same local and global
structure   (Sierpinski-Drieck,  Koch,
etc) All the others are unlinear.
  For  all  the people that see no use
in  fractals,  do  you like a leopard?
It's  leather,  well  it's  a fractal!
Just  think  how  many hears a leopard
has.   It  would  take few millions of
gene-bytes just for the leather.  This
way  it  takes  only few hundred.  The
nature  uses fracs hell of a lot!  Now
it is time that we start using them!


   CALCULATIONS

f(x,y)=(a11x+a12y+b1,a21x+a22y+b2)

This  is  a  function  for getting the
next  step  in making linear fractals.
You  have  to  use  this  function for
every  combination  of  koef.  Use two
screens.   Fill the whole first screen
in  one or  few  colors (don't use 0).
Calculate  the  new positions of every
point  and  draw them on second screen
in   same  colour.   Clear  the  first
screen and do it all over again, until
the fractal looks fine enough.

  Sierpinski-Dreick
comb.   1.     2.     3.
a11     0.5    0.5    0.5
a12     0      0      0
a21     0.5    0.5    0.5
a22     0      0      0
b1      0      0.5    0.25
b2      0      0      0.433


  Koch's curve
comb.   1.     2.     3.      4.
a11     0.3333 0.3333 0.1667 -0.1667
a12     0      0     -0.2887  0.2887
a21     0      0      0.2887  0.2887
a22     0.3333 0.3333 0.1667  0.1667
b1      0      0.6666 0.3333  0.6666
b2      0      0      0       0
  Leaf
comb.   1.     2.      3.      4.
a11     0      0.8496 -0.1554  0.1554
a12     0      0.0255  0.235  -0.235
a21     0     -0.0255  0.1958  0.1958
a22     0.17   0.8496  0.1865  0.1865
b1      0      0       0       0
b2      0      3.0     1.2     3.0


  Canadian
comb.   1.     2.      3.      4.
a11     0.6499 0.6495  0.3182 -0.3182
a12    -0.013 -0.026  -0.3182  0.3182
a21     0.013  0.026   0.3182  0.3182
a22     0.6499 0.6495  0.3182  0.3182
b1      0.175  0.165   0.2     0.8
b2      0      0.325   0       0


 You  could also try to make the fracs
of your own!  Just put your own values
in  the  function.   That  can be real
fun,  'cause  you  never know what you
shall get!
 If  you  get  interested, or you have
used the fractals already, and want to
exchange   ideas   or  fractals,  just
contact  me.   Anyway,  I have quite a
lot  of stuff about fracs, so I'll try
to  write something for the next issue
too.


          XAVER/PLS


PS.   Elf,  thanks  for the pics, and,
btw, where is my candy?