"Gravity can not be held responsible for people falling in
CHAOS THEORY, FRACTALS AND THE UNATTACHED PROGRAMMER
an Introduction by Ringpull
Chaos theory, eh? What's it all about then? In this article
(and possibly others) I hope to explain and experiment with
strange and currently fashionable branch of mathematics. A
spin-off here is that it also gives me somewhere to vent my views
on the world now that Pervert's Monthly is no more, so there may
be numerous detours on the way!
OK, Chaos theory: What is it, Why is it and Where is it. The
simplest description of a chaotic system is one where a minor
variation on the input produces a huge variation in the output.
The classic example of this is the famous 'butterfly effect',
where the beating of a butterfly's wings in South America causes
a hurricane over Europe six months later. This is because the
atmosphere is a very complex and dynamic system which is very
sensitive to its initial conditions.
Imagine you had two identical planet Earths. On one of these
Earths a butterfly flaps it's wings creating a minor airflow and
on the other Earth it doesn't. As time goes on, this minor air
turbulence starts to show differences between the two Earths. At
first these are very small but the further into the future we go,
the greater the difference until six months later when it gets
very windy and a lot of people have a very miserable holiday.
This has resulted in a strong case for the extermination of all
South American butterflies.
My own variation on this theory is that an accident on the M25
this morning resulted in me meeting another car while driving
along a narrow street in a small town in nothern Scotland this
afternoon. Failing to give way to this vehicle resulted in me
failing my driving test and feeling quite miffed at having wasted
£26 on the test fee.
If you have seen (or read) "Jurrasic Park" you may recall the
mathematician (played by Jeff Goldblum) who keeps saying that the
dinosaurs will inevitably escape because of chaos theory. This
is somewhat misleading as it is not chaos theory but the much
more common Murphy's Law (What can go wrong will go wrong). It
would appear that Crichton could have done some more research,
but fortunately he is now concentrating on the much more
worthwhile task of upsetting women with his new book about
The reason that chaotic systems are predicable in the short
term but become more unpredictable as time goes on is that they
are dynamic. This means that the output is fed back into the
system as the input. For example; today's weather is the
starting point for tomorrow's weather so if today's weather were
different then obviously tomorrow's weather would be different
Chaotic systems aren't just limited to abstract things like the
weather but can also be represented as mathematical formulae.
As soon as something becomes mathematical we can turn it in to a
bunch of instructions and shove it into our computer to be
mangled at our own will, and this is where the fun begins.
What does a graph of a chaotic system look like? You have
probably seen one in the form of the famous Mandelbrot set. This
is in fact a graph of the function z = z² + c where z and c are
complex numbers. Eh? All numbers are complex to me if I don't
have a calculator!
A complex number has two parts to it; a real part and an
imaginary part. Each of these parts can have any value, giving a
number of the form z = x + yi (where i = ?-1).
We plot a Mandelbrot set by picking a point on the screen and
putting the x and y coordinates into the x and y values of c.
Both parts of z start as zero and we keep on squaring z and
adding c. One of two things will happen as we continue this
process. The value of z will either level off or continue on
until infinity. If z levels off then it is said to be in the
set, if it shoots off to infinity then it is outside the set.
On your typical picture of the Mandelbrot set the black area in
the middle is where z levels off and the coloured parts are
where it doesn't. The coloured bands often found on such images
depend on the number of iterations of the formula it takes
before the routine decides that z is not going to level off. A
true Mandlebrot set is monochrome and hence rather boring.
If we take our z = z + c formula and rewrite it a slightly
easier to deal with form, we get:
x = x² - y² + cx
y = 2xy + cy
here cx and cy are the coordinates of the pixel. Starting with x
and y both equal to zero and assuming that the equations will
continue to infinity if x²+y² rises above four, we can construct
the following routine to calculate the color of a pixel:
mandel(x,y) // Routine to find color of point x,y
z_x = 0 // Start at zero
z_y = 0
count = 0 // Start the loop counter at zero
temp_x = x*x + y*y + x // The new x value
z_y = 2*x*y + y // The new y value
z_x = temp_x
count = count + 1 // Increase the counter
IF (x*x + y*y) > 4 // Is z going to infinity
COLOUR count MOD 16 // Give pixel a color
PLOT x,y // on the value of the
RETURN // And that's it
UNTIL count > 100 // Number of iterations to try
COLOUR 0 // Pixel is in the black lake
PLOT x,y // Plot it RETURN
As you can see, it is simply a case of continuing the operation
until z exceeds the trigger value. When it does, we assume it
is outside the set and the point is given a colour according to
the number of iterations it took to exceed the trigger value.
This gives the set its characteristic colour bands. If z does
not exceed the trigger value within a set number of iterations
(in this case 100) then we assume that the point is inside the
lake and we colour it black.
By using a couple of simple loops you can use this routine to
plot a Mandlebrot set on your screen. One important point to
remember is that the set lies within the area between -2 and 2 on
both the x and y axis, hence some form of scaling is going to be
required to convert screen coordinates into Mandlebrot set
coordinates. It is also necessary to use real variables rather
than integers for the x and y values.
Once you have written a program to plot the set you can have
all sorts of fun by experimenting with different colour palettes
and some psychedelic colour-cycling! You can also try changing
the formulae to get different fractals, for instance try cubing x
and y rather than squaring them.
The chances are that your Mandlebrot set plotter will be slower
than some of the other programs you may have seen. This is
because there are a number of tricks which can be exploited to
speed up the process. One is using the fact that the set is
symmetrical and mirroring it in the x-axis. Periodicy checking
can speed up calculation of points within the set as the value of
z tends to cycle. Detecting this cycling means we can avoid
having to calculate all 100 (or whatever) iterations. Guessing
routines can speed up calculation of the large areas of colour
outside the set.
Currently the most fashionable trick to speed up the
calculations is to use integer math rather than floating point
routines, which are rather slow if you don't have a maths co-
processor (And the ST doesn't).
All these tricks are employed by the more sophisticated fractal
programs, such as Fractint on the PC. If you have a PC (Yeuck!
Blugh!) you might want to get hold of this program as it is the
most comprehensive fractal plotter around.
When I get around to adding a user interface to my own
assembler integer Mandlebrot plotter I will let you see it in
another instalment of this column. At the moment you have to
enter the coordinates in the source code and re-assemble it.
Next time we will have a look at fractal landscapes and how to
build these miniature virtual realities on your ST! I leave you
with a little song for which I must thank Jack Schofield at the
Guardian for bringing it to my attention.
Sing to the tune of the Beatles' Let It Be.
"When I find my code in tons of trouble,
Friends and colleagues come to me,
Speaking words of wisdom: ''Write in C.''
As the deadline fast approaches,
And bugs are all that I can see,
Somewhere, someone whispers:
''Write in C.''
Write in C, write in C,
Write in C, oh, write in C.
Lisp is dead and buried,
Write in C.
I used to write a lot of Fortran,
For science it worked flawlessly.
Try using it for graphics!
Write in C.
If you've just spent nearly 30 hours
Debugging some assembly,
Soon you will be glad to Write in C.
Write in C, write in C,
Write in C, yeah, write in C.
Only wimps use Basic.
Write in C."
See you Later!
The text of the articles is identical to the originals like they appeared in old ST NEWS issues. Please take into consideration that the author(s) was (were) a lot younger and less responsible back then. So bad jokes, bad English, youthful arrogance, insults, bravura, over-crediting and tastelessness should be taken with at least a grain of salt. Any contact and/or payment information, as well as deadlines/release dates of any kind should be regarded as outdated. Due to the fact that these pages are not actually contained in an Atari executable here, references to scroll texts, featured demo screens and hidden articles may also be irrelevant.