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$Bulletin(24)$H{COMPUTER SCIENCE$B0AT OPEN DAY}$B2
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$X0$Y2"
$Bulletin(32)$H{THE MANDELBROT SET$B^1
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$H{
$Bulletin(12)
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.

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

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$bulletin(8)-{2} = -1

that 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 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$bulletin(8)-{2} + c, where z, c and the result f(z) are all complex.

Start with z=0.
$b0 Now for any arbitrary value c, apply f(z) repeatedly to itself, i.e.

f$bulletin(8)+{1}(z) = z$bulletin(8)-{2} + c

f$bulletin(8)+{2}(z) = f(f(z)) = (z$bulletin(8)-{2} + c)$bulletin(8)-{2} + (z$bulletin(8)-{2} + c)

f$bulletin(8)+{3}(z) = f(f(f(z))) ... and so on.

For certain values of c, the magnitude of f$bulletin(8)+{n}(z) will become infinite. For others it
will remain finite.  These latter points comprise the Mandelbrot set.  The interesting
parts (for us!) are the borders of the Mandelbrot set, where the magnitude
of f$bulletin(8)+{n}(z) changes suddenly from being finite to being infinite. Small
variations of c in this region produce chaotic behaviour in the number of iterations required
to make the iteration infinite.

We plot the numbers of iterations required to make f$bulletin(8)+{n}(z) infinite versus c and
get the plots you see on the screens.  Each colour represents a different
number of iterations (modulo 256).

"Infinite" in the calculation is actually represented by sqr(2) as if the
polynomial reaches this value, it will certainly become infinite.  It is also
necessary to limit the number of iterations one performs before deciding the
polynomial has stabilised.  The higher this number the greater the detail
(provided one can display it) but the longer the computation time.

The Set itself is usually represented by black "Seas" and "lakes". It is not
generally well known that all these seas and lakes are actually linked, sometimes
by tenuous threads.  Other characteristics of the Set worth noting are how
small "clones" of the set appear at different magnifications, each slightly
different.  The "slide-show" on the displays are simply different parts of the
edge of the Set at different magnifications.

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

$l0
 10 REM Mandelbrot Set in BBC BASIC Courtesy J. Greening (C)
 30 MODE1: 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
100 a=r*r:   b=i*i:   n=a-b+x:   m=2*r*i+y:   C%=C%+1
140 s=a+b:   r=n:     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%<13  GCOL0,1:GOTO 330
300 IF C%<26  GCOL0,2:GOTO 330
310 IF C%<101 GCOL0,3:GOTO 330
320 GOTO360
330 PLOT69,400*x,400*y
360 NEXT:NEXT
370 *SAVE"MANDEL" 3000 8000
380 END

If you want to learn more, try..

Exploring the Mandelbrot Set - Scientific American August 1985

Schondheit im Chaos ("Frontiers of Chaos") - Peitgen and Richter

The Fractal Geometry of Nature - B. Mandelbrot  [Freeman 1983]

"Acorn User" May 1986
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