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