Even if the Mandelbrot-Set is perhaps the bestknown fractal in the world,
there is not much information on what it is and how to make it. In this
page I will try to make you understand the simple theory behind the
Mandelbrot-Set, and even show you a simple Pascal-program that calculates
the fractal. Hope you enjoy it!
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The Mandelbrot-Set is made by the simple mathematical expression
Z(n) = Z(n-1)^2 + C
The problem is that the constant C and the function Z are complex numbers..
(That is, they have a 'real' part (The numbers we normally use) and an
imaginary part (Numbers multiplied by the squareroot of -1, represented by
an 'i' after it) ). We use to say that C = a + bi
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When calculating the Mandelbrot-Set, you use the x-axis as the 'real'
value, and the y-axis as the 'imaginary' value.
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It is easy to prove that nothing of the Mandelbrot-Set is located outside a
circle with origin (0,0) and radius 2. (Check out fig-1.1)
[Image]fig-1.1
What you do when calculating the Mandelbrot-Set, is to iterate the function
(calculate the function several times) for every pixel on screen, and set
the color of the pixel according to how many times you had to iterate.
The Mandelbrot-Set is actually the big, one-colored thing in the middle of
the figure. When iterating the function there, you can see that the value
increases and decreases chaotic, and never will increase to infinity. To
get out of the calculating-loop, you have to set a maximum number of
iterations. The routine will be something like this:
FOR EACH pixel:
REPEAT
CALCULATE Z(n) = Z(n-1)^2 + C
iterations = iterations + 1
UNTIL value = infinite OR iterations > max_iterations
SET COLOR = iterations
The only problem above should be the calculation.. How do we calculate the
values ? How is this 'Z(n)'-function ???
As I said, Z(n) = Z(n-1) + C. In Mandelbrot-Set, Z(0) = 0. That means:
Z(1) = C
Z(2) = C^2 + C
Z(3) = (C^2 + c)^2 + C and so on..
I also said that C = a + bi. You then get:
Z(0) = 0
Z(1) = a + bi
Z(2) = (a + bi)^2 + a + bi
Z(3) = ((a + bi)^2 + a + bi)^2 + a + bi .....
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So, what is (a + bi)^2 ? Everyone should know that (x+y)^2 = x^2 +2xy + y^2
We get the same thing: (a + bi)^2 = a^2 + 2abi + bi^2
Now the strange thing: I told you i=sqrt(-1), that means i^2= -1, and bi^2
= -b^2 (note: the 'i' disappears, so bi^2 becomes -b^2 (!)
Conclusion:
Z(n).real = Z(n-1).real^2 - Z(n-1).imaginary^2 + C.real
Z(n).imaginary = 2 * Z(n-1).real * Z(n-1).imaginary + C.imaginary
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Now, all we have to do is test when Z(n) is infinite :-)
The value of Z(n) is the length of the vector given by it's real and
imaginary parts. To calculate the length of a vector, we use the formula
Length = Square-root of the value |a|^2 + the value |bi|^2, or simply
Length = SQRT(a^2 + b^2) (Note: We can use the real value 'b' and not the
imaginary value 'bi' here. We are just interested in the value, not if it
is real or imaginary)
So, when is the length infinite ? The bad news: You can't test if it is
infinite..
The good news: Whenever the length exceeds 2, you know that it (some day)
-will- be infinite !
To improve our 'program':
FOR EACH pixel:
REPEAT
CALCULATE Z(n) = Z(n-1)^2 + C
Length = Z(n).real^2 + Z(n).imaginary^2
iterations = iterations + 1
UNTIL Length > 4 OR iterations > max_iterations
SET COLOR = iterations
You might see that I test if the length exceeds 4, not 2, and that I don't
take the squareroot when calculating the length. As you all might know,
calculating the squareroot takes quite a lot of time, and when
SQRT('length') > 2, why not just test if 'length' > 4 ?!?
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To make life easy, we can make a function that has C as input-variable, and
returns the number of iterations the program had to compute for that point
: (In Pascal)
FUNCTION CALC_PIXEL(CA,CBi:REAL):INTEGER; {CA = real value, CBi = imaginary}
CONST MAX_ITERATION=128; {higher value -> better quality}
VAR
OLD_A :REAL; {just a variable to keep 'a' from being destroyed}
ITERATION :INTEGER; {the iteration-counter}
A,Bi :REAL; {function Z divided in real and imaginary parts}
LENGTH_Z :REAL; {length of Z, sqrt(length_z)>2 => Z->infinity}
BEGIN
A:=0; {initialize Z(0) = 0}
Bi:=0;
ITERATION:=0; {initialize iteration}
REPEAT
OLD_A:=A; {saves the 'a' (Will be destroyed in next line}
A:= A*A - B*B + CA;
B:= 2*OLD_A*B + CBi;
length_z:= a*a + b*b; {note: We do not perform the squareroot here}
UNTIL (length_z > 4) OR (iteration > max_iteration);
Calc_Pixel:=iteration;
END;
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To compute the Mandelbrot-Set, you would have a program like:
FOR y:= -1.25 TO 1.25 DO
FOR x:= -2 TO 1.25 DO
color:=CALC_PIXEL(x,y);
This 'program' would calculate the whole set, but you wouldn't be able to
plot the Mandelbrot-set on the screen. That's why we use
screen-coordinates, and calculate the C-value for each pixel: (assumes
640*480 VGA)
FOR y:= 0 TO 480-1 DO
FOR x:= 0 TO 640-1 DO
BEGIN
color:=CALC_PIXEL(real(x),imaginary(y));
PUTPIXEL(x,y,color);
END;
This 'program' would be able to plot the set, but it has to compute the
real and imaginary value for each pixel. How do we do that ???
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First of all, we have to define the area to compute.
To compute the whole set, the area would be (-2, -1.25) - (1.25, 1.25)
When coding a program, it is a good idea to have this values as constants
somewhere in the code. Pascal has all constant values in the top of the
source code:
PROGRAM Mandelbrot;
CONST MinX = -2;
MaxX = 1.25;
MinY = -1.25;
MaxY = 1.25;
[rest of program..]
When the upper left corner is supposed to be (MinX, MinY) , and the lower
right corner is supposed to be (MaxX, MaxY), we get this formula:
real = MinX + x*(MaxX-MinX)/screenwidth;
imaginary = MinY + y*(MaxY-MinY)/screenheight;
Or, to save a lot of divisions (wich takes a lot of time..) :
dx = (MaxX-MinX)/screenwidth; {only needed to be done once !}
dy = (MaxY-MinY)/screenheight; {----------- "" --------------}
real = MinX + x*dx;
imaginary = MinY + y*dy;
Now we should be able to make a 'almost working' program :-)
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PROGRAM Mandelbrot;
CONST MinX = -2;
MaxX = 1.25;
MinY = -1.25;
MaxY = 1.25;
VAR dx, dy:REAL; {Don't confuse with 'real' numbers. This is the
Pascal Datatype REAL}
x, y :INTEGERS;
BEGIN
dx = (MaxX-MinX)/640;
dy = (Maxy-MinY)/480;
FOR y:=0 TO 480-1 DO
FOR x:=0 TO 640-1 DO
BEGIN
color:=CALC_PIXEL(MinX+x*dx, MinY+y*dy);
PUTPIXEL(x,y,color);
END;
END.
Because of the difference in initializing and using graphics on different
platforms, I can give you the complete Pascal source or complete C source
for IBM-Compatible machines with VGA-Cards, and people with UNIX-machines
or those lovely Amiga's have to make their own graphics-routines..
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Now the 'c00l' thing about fractals; You can zoom into them !
Above, we calculated the area (-2, -1.25) - (1.25, 1.25). If you look at
the image below (and I hope you can read those small numbers on the axis!),
you'll see that the whole Mandelbrot-set is withing that area.
BUT, if you calculate the area (0.1, 0.4) - (0.5, 0.6) {as framed in figure
2}, you'll get the image shown in figure 3 !!!
And if you calculate the area (0.37, 0.52) - (0.41, 0.54) {as framed in
figure 3}, you'll get the image shown in figure 4 !!!
(Well, perhaps not. I can't remember the numbers I used to generate the
images, so I just made a good guess above *smile* , but you understand the
idea ?!?)
[Image]The whole Mandelbrot-Set
[Image]Figure 2
[Image]Figure 3
[Image]Figure 4
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Are there anything I have left out ?!? Of the elementary, don't think so,
but if you have any questions or comments, PLEASE MAIL ME, and I'll update
this document to suit your needs !
[Image]
visitors sice 18th November 1995.
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