REAL PAVEMENT presents ....... The Fractal Disc
Compiled by Barry Etheridge for PCW World. Released to Public Domain 4/90.
This disc and the program files recorded on it are deemed 'Public Domain' and
may not, under any circumstances, be sold or hired, or offered for sale or for
hire.
TO RUN: Please note that the Basic files on this disc form an integrated and
self-contained program. No file may be run as an independent program apart from
the others, nor without first running "INTRO"
Load CP/M and Basic then type RUN"INTRO
PROGRAM NOTES:
These programs are designed to demonstrate some of the classic features of the
'new' science of chaos, particularly the infinite varieties of 'fractal'
mathematical phenomena, all of which are best appreciated visually! It could not
be the hope of this short introduction to give a crash course in chaos science
or fractal mathematics and the following books are recommended for those whose
interest is stimulated by what they are about to see!
CHAOS: Making A New Science by James Gleick
Does God Play Dice? by Ian Stewart
However, some explanation of the complex arithmetic which is about to go into
action might be appropriate! Iterative functions of complex numbers are what
most of the programs (especially the MAN & CREAT programs) on the disc depend on
for their operation. So some definitions are needed!
A complex number is one which takes the form a+bi where a and b are ordinary
common or garden numbers but i is the so-called imaginary number which is the
square root of -1. A moments thought (1x1=1 -1x-1=1 ?x?=-1) will show that it is
impossible to express this number any other way, but clearly it must exist
(mustn't it?) and it is easy to do maths with it. The square of i is -1, the
cube -1i, the fourth power 1, the fifth i again!
A complex number, then, consists of a real part, and an 'imaginary' part, as
in 0.3 + 1.2i, and many of the constants that you will be asked to input to the
programs are in this form. In response to the question 'Real constant?', you
type 0.3, and to 'Imaginary constant?', you type 1.2, leaving out 'i' because
the program already knows what to expect. An iterative function is one in which
the answer to the equation is fed into the function again to produce a new value
which is then fed into the function again to ..........
For example the Mandelbrot plotting program uses the function z= z x z + c in
which c is a complex number to be tested and z (originally set at 0) is the
complex number which is the product of the equation. So the first few passes
would produce values of:
0 x 0 + c
c x c + c
(c x c + c)(c x c + c) + c etc. ad infinitum
In each case, the answer will produce a new complex number (if we include cases
where a + bi has a value for b of 0)! To prevent this operation looping round
itself for ever, the program will ask for a maximum number of iterations and
will also provide a test value which determines (although not always absolutely)
whether or not the ultimate result of all this strange mathematics on any
particular complex number is a function whose answers get nearer and nearer
infinity. (The 'value' of a complex number is the square root of the sum of the
squares of the two terms).
Now to translate all this into pictures, take a graph with the x-axis
representing the value of the real part of a complex number and the y-axis
representing the value of the imaginary part (so that any complex number can be
plotted; eg. 1+2i is the point (1,2)). You will be asked by the program to
provide parameters (min and max) for these axes which will determine the values
of all the complex numbers which can be plotted within the graph. Now, for each
point (ie. each complex number), the program iterates the function. If the test
value is passed before the maximum number of iterations (referred to as going
out of range), the point is plotted one 'colour' to indicate that the function
(seems to) converge to infinity. If the test value has not been passed after the
maximum number of iterations then the point is plotted another 'colour'.
Think you know what you'll see - then stand by to be amazed!
REAL PAVEMENT HEALTH WARNING! - these programs could seriously mess up your life!
Because of the many thousands of calculations involved in completing a screen,
these programs take several hours to do their stuff, and your computer will be
out of action for the entire time. Moreover, addiction to watching the plots
tracing out complex and endlessly fascinating patterns is not unknown! If you
have FLIPPER, that clever device for running two programs at once, this will
prove an ideal application. Otherwise, run these programs overnight. You have
been warned!
ROLL CREDITS
Mandelbrot programs: modifications by Barry Etheridge on a program by Brian Aird
(published in 8000 Plus) based on an algorithm by A. K. Dewdney published in
Scientific American, 8/85.
Creature programs: written (as a modification of the Mandelbrot programs) by
Barry Etheridge from an algorithm by A. K. Dewdney published in Scientific
American, 7/89
Affine & Chaos programs: by Barry Etheridge
Catalogue & Scrnload program: screen design by Stephen Cook using the PCW screen
design program published by 8000 Plus. List and activate routines by Barry
Etheridge.
All programs employ screen plot, save and load routines by kind permission of
Lawrence Simons. Such routines are copyright and may not be amended, copied or
used in any program (except as an integral part of 'The Real Pavement Fractal
Disc') without the prior written permission of Lawrence Simons.
ENQUIRIES
News (and complaints) of unexpected bugs, crashes or results, suggestions for
amendment or improvement, and programs on similar themes would be welcomed by
Barry Etheridge via:-
PCW World, Cotswold House, 7 Deeley Close, Cradley Heath, Warley, W.Mids.
es would be welcomed by
Barry Etheridge via:-
PCW World, Cotswold House, 7 De