Exploring the Mandelbrot Set

The Mandelbrot Set is a mathematical abstraction of quite dazzling beauty and intricacy.  From a simple computer program (with sufficient time!) it is possible to generate patterns which embody some of the subtler aspects of natural forms and offer some complex puzzles in pure Mathematics.

What is it?

To understand the Mandelbrot Set one first requires an elementary knowledge of complex numbers, principally 

1) that a complex number z may have a real part x and an imaginary part y such that z = x + iy, where i is the square root of -1 i.e. i*i = -1.

that

2) Any complex number may be plotted on an Argand diagram - a set of axes with the real component running from left to right and the imaginary component running up and down

and that

3) Any complex number has a Magnitude equal to the square root of the sum of the squares of the real and imaginary components (it equals the length of the line from the origin to the point on an Argand diagram).


Given that, we may evaluate functions of z, and in this case we are interested in the function f(z) = z**2 + c, where z, c and the result f(z) are all complex.

Now for any arbitrary value c, apply f(z) repeatedly to itself, i.e.

f1(z) = z**z + c

f2(z) = f(f(z)) = (z**z + c)**2 + (z**z + c)

f3(z) = f(f(f(z))) ... and so on.

For certain values of c, the value fn(z) will become infinite. For others it will stabilise 


Appendix 1: A simple BBC-B program to generate portions of the Set

   10 REM Mandelbrot Set in BBC BASIC
   20 REM Courtesy Prof. J. Greening (C)
   30 MODE1
   40 VDU29,800;512;

   50 FOR x=-2 TO .56 STEP .01
   60 FOR y=1.28 TO -1.28 STEP -.01

   70 r=0:i=0:C%=0

   80 REPEAT
   90 a=r*r
  100 b=i*i
  110 n=a-b+x
  120 m=2*r*i+y
  130 C%=C%+1
  140 s=a+b
  150 r=n
  160 i=m
  170 UNTIL s>4 OR C%>100

  240 IF C%<3 GCOL0,2:GOTO 330
  250 IF C%<4 GCOL0,3:GOTO 330
  260 IF C%<5 GCOL0,1:GOTO 330
  270 IF C%<6 GCOL0,2:GOTO 330
  280 IF C%<9 GCOL0,3:GOTO 330
  290 IF C%<13GCOL0,1:GOTO 330
  300 IF C%<26GCOL0,2:GOTO 330
  310 IF C%<101GCOL0,3:GOTO 330
  320 GOTO360
  330 PLOT69,400*x,400*y

  360 NEXT:NEXT

  370 *SAVE"MANDEL" 3000 8000
  380 END


References:


[1]  Computer Recreations: Exploring the Mandelbrot Set
     Scientific American August 1985 p8

[2] ?Scoenheit im Chaos "Frontiers of Chaos")

[3] The Fractal Geometry of Nature
    Benoit B. Mandelbrot  [Freeman & Co. 1983]

[4] "Acorn User" May 1986
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