Educational program - making fractals on the Spectrum.

(C) ALK/Stars of Keladan H.G.             
                                          
MMA> Hey, Shurik,  do article about
  fractals.                            
ALK> Why, that's all  no one is interested anyway
  is processed by them...                      
MMA> Well then, the public, having seen the queue
  the bottom fractal in the demo will say:
  Oh, they pulled the signs off the Amiga again
  and build  on them Ferns and these...
  Mandelbrots...                      
                                          
     ...That's where it struck me  and it hurt me, you can
say for alive. What to do, position
obliges us to finally reveal the whole truth about
fractals.                                






 Let's start, as always, with definitions.    
                                          
Fractus (lat.) - broken, fractional.        
                                          
Fractal, in application to mathematics, there is
some  object described by simple cyclic
logical (or recursive) formulas, and co-
who, most importantly, copies himself
when the scale of calculations changes. But this is a definition. But what about in practice? 
                                          
The point is that  that  fractals do not exist
only “on paper”, they are everywhere. For example
measures, the circulatory system - artery on the pro-
along the entire length it branches and decreases
in  diameter, each new branch is similar to
the previous one also branches  and decreases.
If would not limit size  cells, this
branching  would continue indefinitely
tee. But this is a “simple” fractal. There are complications
her for example, a leaf  of a tree. If you pay attention
detailslook closely at the veins on the leaf,
then  you can notice that each of them
highlights  the main "inflorescence" of the most painful
shikh. And exactly the same can be said by accepting
"secondary inflorescences" for the main ones.          
You can come up with an artificial tail yourself
tal -  for example, draw a square around
draw four more squares of its vertices
with a side 2-3 times smaller than that of a qua-
drata-parent. Repeat this operation once
50 for each generation of squares.:[ ]
                                          
                                          
              Are you tired? ;)               
                                          
                                          
 And now  look  at  this  creation -
beautiful, not is it true? Beauty is in
what is the power of fractals, it fascinates with its
simplicity  and INFINITY! And even more so
armed with a computer  instead of  paper,
you can create such landscapes!              
                                          
                                          
                                          
                                          
   
    
    
    
    
    
    
    
    
    
    
    
    
    
    
                                          
                                          
                                          
          Mandelbrot fractal.           
             Here is its formula:             
Z(i+1)=Z(i)*Z(i)+C, where:                  
Z - complex variable:               
C - complex constant:                
(i),(i+1) - steps of calculations and iterations.    
















Those who  did not  happen to study terrible
the subject "Higher Mathematics" has probably already
"loaded up" and think that this is so
difficult  for them to understand, which is not worth it and
read all this further nonsense.         
                                          
Don't rush! I'll try to explain...    
...not  I undertake  to be absolutely accurate, I will
extremely brief.                         
                                          
A complex number Z can be represented as:
Z=a+j*b, where                              
                                          
a, b - simple ordinary numbers,              
j - an imaginary unit, I won’t bother you
head with this sucks, but surprisingly,
j.  multiplied by  by the same j. equal to MINUS
ALONE.  Likewise. root of -l (insanity.
yes?) will be equalj.                       
                                          
                                          
                                          
If ordinary numbers are taken graphically
to use the number line, then these are
perverted  - on numeric PLANES. These
numbers  have two coordinates, real (a),
and  imaginary  (b), which and are deferred
respectively along the X and Y axes on a plane
tee. The point of intersection of these counts is
This is a COMPLEX NUMBER.                    
            
                     
                     
                     
                     
                     
                     
                     
                     
                     
                     
                     
                     
Do you feel where  I I'm heading? This whole thing
it is possible  (and should!) be displayed on a plane,
namely on the plane of the screen. The question is how!
                                          
To build a Mandelbrot fractal, you need
We need to  enumerate all points on our plane
bones  on  subject of correspondence to the following
condition.                                  
                                          
In the cycle from l to l0000 (no less!) you need
easy to calculate the sequence:     
                                          
A:=a*a-b*b+x                              
B:=2*a*b+y a:=A                                      
b:=B                                      
R:=SQR(a*a+b*b)                           
i:=i+1                                    
                                          
The first  two lines are nothing more than vi-
the formula you gave earlier Z:=z*z+c, but
written in "normal" language. Explanation:a
and b - real  and  imaginary part of the number Z, x
and y  - these are real and imaginary, respectively
components of a complex number C. In our
case - these are the coordinates of the point that
it is necessary to  calculate. During the calculations
for one point x and y do not change.The following  two  lines are most likely only needed
for clarity - after   calculations  A andB
produced assignment of new results
to the original ones for the next cycle.          
                                          
The penultimate line is the most important. This is not
nothing more than  CONDITION under which it is necessary
stop the cycle. It is logical to assume that
here  the total "size" of the number is calculated
Z,  its value. This is where it starts
interesting. It turns out that for different
points  on  plane size R in a loop
calculations will “behave” differently. For
some  points there will be some kind of
exact value,  for other values R
will  rapidly “fly away” to +/- infinity
frailty.  Evaluating  with what speed
the point “flies away” to infinity, you can gra-
physically display it in corresponding colors
volume on a plane. For computer calculation
that  here  introduced  last  that string
cycle i:=i+1. By the number of steps taken -
with this program you can judge the speed
"flying away"  points - the fewer cycles, the
faster (brighter, but not a fact) point on a plane
bones  flies to infinity.Before per-
the first cycle, of course, valuesa,b,iob-
are zeroed.  But  here  occurs  problem -
computer  "does not know" what infinite
ity, or rather knows, but then it’s too late -
message  type Number too big will not slow down
appear.  Here  it is necessary to accept the following
the following  assumption: for "infinity"
Well, that's a very large number. Yes, more about
limits.  To calculate all points on a plane
bones  you still need to select, PDE calculate
poke.  And you need to calculate within the limits:
from -2.S to +l.S for x-,                 
from -2 to +2 for  y-component of the number C,
i.e. near the origin of coordinates - exactly there
"fractal collisions" occur, mainly
waist area plane does not occur
nothing  interesting. WHERE will you settle
to type and with what STEP to sort through the points on
plane (i.e. with what magnification) is
The question is purely creative.                  
                                          
"All this is wonderful," - you say, "but
than  here  SPECTRUM, only he'll be an idiot
count 10000 cycles for each point
planes measuring 256*192. Not only that
this damn thing  will be  count  two days
BASIC, it is still unknown how to display
press the dots with color!" Don't  despair,  I with full responsibility
I honestly declare: Mandelbrot on SPECTRUM -
reality. No need to wait two days - min-
ta from strength! And color - we have an ATTRIBU-
YOU  colors!  Naturally, no BASIC.
Code, and only code.  On  disk  you  you will find
original text  "mandelbr.C"  with comment-
riami  in  ASCII encoding  and  similar
"mandelbr.XAS" (no comments) for XAS-
assembler I don't pretend to be swearing.
the procedures there are the mostкрутые",  но зато они
МОИ !                                     

































Are you looking for a girl, what do you think?
Then go to the "EXIT" box. I'm not sure what to do with it.
те эти сопли. вас ждёт ВТОРАЯ серия!"     
                               © Mr.SECOND
 Small Girl (Julia).          
                 Size:                 
Z(i+1)=Z(i)*Z(i)+C                        



















                                          
Как вы, наверное, уже заметили, эта форму-
ла  точь-в-точь повторяет формулу фрактала
Мандельброта.  И  это неудивительно - ведь
Мандельброт  и Жюлиа - близнецы-братья. мы
говорим   "Мандельброт".  подразумеваем  -
party... ;) ...There was  such a  man, from the company IBM, he
came up with  and began to study using a computer
Yutera  (can you guess which one twice?): Well
and  I don’t know anything about Julia,
Probably it was a friend of Mandelbrot :), and he
I first looked at  on  "brainchild" a little with
another point of view:                      
                                          
If we take as the initial condition of the iteration
Z(0)=0,  and change C-constant from point to
point, then we get a Mandelbrot fractal,
about  what  I tried to tell you unsuccessfully :)
above: 
On the contrary,  if we accept C for
a constant that does not change from
points  to point, and change
Z(0),  equal to the point coordinate 
ki, then we get a set
Julia... 
  Of course,  because  that that
constants C exists infinitely-
fine  quantity, view
fractal Julia _essentially_
depends on the constant C.  
  The lion's share  of these values
"generates" ugly figures
-  something  like spots, faint
distinguishable  against the main background.
According to  my own experience
I  can say that the values
C should be somewhere in the range-
lah +/- 5,and both composition-
lying  this number C (not for-
were?)  in most cases
have  opposite signs.
            
By the way,  those
procedures, about
which  
spoke  you-
more,   you can
remake  
for  расчёта
Жюлиа.     А
что касается
пределов  на
комплексной 
плоскости  -
то  они  ос-
таются теми-
же,что и для
Мандельброта
            
  Значение  
  константы 
   Julia:   
C=0.36-j0.37
Множества   Julia   more  suitable
definition of a fractal - no matter how we "increase-
personalized"  scale  images, we will
contemplate  the same picture, except perhaps
the color scheme will shift, but this is not
tell about  Mandelbrot - with increasing
the picture and form still change.        
                                          
If look from a practical point of view
nia, then with the help of Julia-Mandelbrot you can
do the following:                    
                                          
Zooming  - calculate image
"in depth", changing at each step of "brute force"
points. It's better on the Spectrumcalculate
first in advance,  and  then... show how
animation,  believe me, this  will look
it's cool that you could  see  in demos EYE
ACHE II/CBS (Mandelbrot was there) and INSANE
/ЗSC HARDCORE (the same Mandelbrot).       
                                          
                                          
Morphing - for Mandelbrot fractal on
each frame you can change the evaluation criterion
ki  "infinity" and then when viewing
"film" you can see how the fractal smoothly
turns from a perfect circle, bending
returning to its original form.            
...for the Julia fractal on each frame you can
but  smoothly  change the parameter C,as one,
and  other  components. Naturally
with a small  change C fractal shape
will change  insignificantly.  Here  this, I tell you
I will say, not just cool, but very cool! When
viewing  you can see how literally
against an empty background a fractal “blooms”, meandering
weaving and twisting into unimaginable patterns. On
Spectrum  this has not been done (yet :)). Ob-
ownersof the "blue calculator"could do this
see in PC demos.                        
Voxel Mapping - if abstract  from
the colors of the points that make up the fractal, andat-
take the "weight" of a point not by color, but by height,
then you can “show” fractal  landscapes
From the side, moving above
relieffractal, occasionally turning and
changing  height  "flight". Well, what can I say?
how about?  This is also not bad, especially since
Spectrum  Voxel Mapping'and there were all sorts of -
Multicolor  (EYE   ACHE  II), Chunky
(Refresh,Blame),Attribute (Insane). But here
Fractal Voxel  Mapping'and on the Spectrum also
there wasn't, although it's surprising why no one
guessed  instead of  relief map
to create a fractal one?                         
                                          
...Well, so to speak, “for general development”
you can  imagine  Mandelbrot fractal-
Julia, as a formula:                       
Z(i+1)=Z(i)^N+C, where                      
^N  -  this is  construction  in degree N,where N
more  2-x. In general, there will be mathematics here
even more will be calculated slower,
in short, the brakes on Spec will be terrible...
                                          
... but the picture will look beautiful!!!
 Fractal "Fern"           
            Affine set.           




















This  fractal is calculated differently
principle  than  the Mandelbrot family.
If for  Mandelbrot image  is obtained
sequentially, point by point, then for
affine fractals  construction of "sequential
body-coordinate".  ......
 ...... ' .
 Scheme ....  .  .  '   '    .
папоротника ..          '                .
         ..     0     '                .'.
        .  ..       .                   .'
      .'  .  '.                       ' . 
     .   .      . '                  ' .  
    '    .  1    |       2         .'  .  
  .'     '       '                '   .   
 .        '.   .'             . '    .    
 ' 4-┐      '..........   . '       .     
.    │  ...'''.         '.    0    .      
.   .┴''       '.    3    .      .'       
. .'      0      '.........'  ..'         
.'                     ....'''            
''''''''.........''''''                   
Цифрой  0 обозначен большой лист - это на-
ружные контуры папоротника:               
1,2 и 3 - меньшие поthe size of the leaves, which
which are obtained from 0 by rotation, translation and
scaling:                         
4  - petiole,  which is actually also
similar to 0, but strongly compressed in transverse direction
board.                                
                                          
The construction comes down to the following:          
There is a certain mathematical transformation
nie which will be discussed below, with its
using the coordinates of the screen point,
where you want to put the pixel. This
the transformation depends on the coordinates of the previous
current  points and from the so-called probable
numeric coefficient. If you cyclically
call  this is a transformation (about 1000-
4000 times) and after each put it on the screen
point,  then in the end you will get a picture
approximately  reminiscent of the one shown in
the beginning of the section.                           
                                          
Since there are four “figures”, to construct
it is necessary  to use one  of the four
formulas:                                   
                                          
x(i+1) = a1*x(i) + b1*y(i) + e1  (1)      
y(i+1) = c1*x(i) + d1*y(i) + f1           
                                          
x(i+1) = a2*x(i) + b2*y(i) + e2  (2)      
y(i+1) = c2*x(i) + d2*y(i) + f2           
                                          
x(i+1) = a3*x(i) + b3*y(i) + e3  (3)      
y(i+1) = c3*x(i) + d3*y(i) + f3 x(i+1) = a4*x(i) + b4*y(i) + e4  (4)      
y(i+1) = c4*x(i) + d4*y(i) + f4           
                                          
xy(i+1) and xy(i) - traditionally, the following and
previous coordinate values:           
a,b,c,d,e,f - conversion coefficients,
four groups for each.                
                                          
That is, each transformation is set to 6
coefficients   (a,b,c,d,e,f), a  full
configuration future fern floor
is fully specified by 24 parameters.        
                                          
Now  the question arises, which of the four
transforms to use? Use
you need  all 4. At each step, naturally,
used only one. How to choose -
randomly,  but  not  evenly.  Degree
probability of using  transformation
is proportional to  area  corresponding
figures. The sum of the areas is conditionally equal to 1 or
100%,  respectively  sum probabilities
should also be equal to 1.                 
                                          
                                          
Что же касается реализации этого на Спек- 
труме, то для папоротника имеем:          
                                          
                                          
                                          
┌-┬-----┬-----┬-----┬-----┬-----┬-----┬--┐
│N│  а  │  b  │  c  │  d  │  e  │  f  │р │
├-┼-----┼-----┼-----┼-----┼-----┼-----┼--┤
│1│ 0.00│ 0.00│ 0.00│ 0.16│ 0.00│ 0.00│ 1│
│2│ 0.85│ 0.04│-0.04│ 0.85│ 0.00│ 1.60│84│
│3│ 0.20│-0.26│ 0.23│ 0.22│ 0.00│ 1.60│07│
│4│-0.15│ 0.28│ 0.26│ 0.24│ 0.00│ 0.44│07│
└-┴-----┴-----┴-----┴-----┴-----┴-----┴--┘
Р - вероятность использования, в %%       
Реализоватьprobabilistic  execution of that
or another formula simply:                  
1. Calculate a random number within 0-
255 (one byte, if implemented on ASMA):  
2. If the number is less than 3, then the formula is (1):  
3. If the number is less than 21, then the formula is (3):  
4. If the number is less than 39, then the formula is (4):  
5. Otherwise, the formula remains (2):          
0  3   21  39 255
├---┼--------┼--------┼-----------------┤ 
3/256 18/256 18/256   (256-39)/256     
 1%  7%   7%  84%       
                                          
Where do  come from?  For each
figures  1-4 we set three coordinates each, this
may be the coordinates of the beginning and end of the sheet
plus more  sheet point, most distant
from  the line connecting this beginning and end.
Thus, we have four systems
we equations:                             
                                          
┌ x1 = x1*a1 + y1*b1 + e1                 
│ y1 = x1*c1 + y1*d1 + f1                 
│ x2 = x2*a1 + y2*b1 +e1                 
│ y2 = x2*c1 + y2*d1 + f1                 
│ x3 = x3*a1 + y3*b1 + e1                 
└ y3 = x3*c1 + y3*d1 + f1, where            
                                          
х123,y123 - specified coordinates (known):
a,b,c,d,e,f - unknown quantities.       
Having solved  each system of 6 equations with 6
unknown each,  you can find the one you are looking for
group  parameters a, b, c, d, e, f for each
leaf.                                    
How to decide? I don’t know and don’t ask :)   
                                          
Well...  what  knew - told.  Go for it
fractals on Spectrum - a little-studied thing
naya.                                      
                                          
------══════════════════════════════------
 * * * * *                 
                                          
Р.S. The article was prepared based on materials:   
1. Magazine "Technology-Youth" N9,199X:    
2. Own research and observations:    
3. ProgramsFractal eXtreme Explorer/PC: 
4. DemoRiseDemo/GLOBAL Corp.,NOUMENON/i:
5. FAQ fido-conferences DEMO.DESIGN.      
                                          
R.P.S. I almost forgot, there is this one in the box
what a program, it's called FRACTALS. Burzhuyskaya
she, however. Well, in general, you understand me...