TPUG - Toronto PET Users Group/TPUG Users Group CD/TPUG Users Group CD.iso/CRS/crs19.d81/simman.arc/MANDEL1.TXT

Text File | 2009-10-10 | 14KB | 331 lines

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    INTRODUCTION   Mandelbrot set system
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                   by John M. Albergo
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         The Mandelbrot Set is a mathematical construct of infinite
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    complexity.  Its fascinating images can be viewed by the use of a
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    computer with graphics capabilities.  The programs in this kit function
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    together to make the C64 a useful tool in the exploration of this
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    infinite mathematical territory.  If you are unfamiliar with the
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    concept of the Mandelbrot Set, it may be helpful to do some reading on
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    the subject.  The book, CHAOS: MAKING A NEW SCIENCE, by James Gleick is
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    widely available, straightforward and a good place to start.
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         The Mandelbrot Set is a collection of numbers, or points on a
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    plane, that share a common property.  A mathematical operation is
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    performed on each point, repeatedly.  After a certain number of
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    repetitions have occured, and the number has not run off towards
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    infinity, it is considered a part of the set.  In actuality a point is
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    truly a member of the set if it can undergo an INFINITE number of
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    repetitions without growing infinitely large.  Since computing time is
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    limited, we must make our decisions a bit prematurely.  The points
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    outside the set run off toward infinity at different rates.  The varied
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    properties of these numbers contribute to the visual topography of the
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    images.
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    COMPUTING TIME
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         The length of time required to generate images of the set depends
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    on several factors.  Some images will compute in an hour or two.  Some
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    can take several DAYS.  It depends upon how deeply you are magnifying
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    the set, which part of the set you are looking at and how much
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    resolution you want.  The programs are compiled and optimized for
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    speed.  But the infinite nature of the set means that eventually even a
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    supercomputer will be overtaxed.  The images obtainable with this kit
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    compare favorably to some produced by much more powerful machines.
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         The calculating program can be stopped and restarted to allow you
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    access to your computer whenever you want.
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    REQUIREMENTS
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         To use this kit you need:
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    C64 with 2 drives
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    SIMON'S BASIC (a commercial software pkg.)
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    Patience.
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    THE KIT
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         The kit consists of the following:
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    (compiled programs)
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         MANDELCRUNCH - performs the number crunching that is at the heart











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    of any mandelbrot set program.
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         MANDELPAINT - lets you develop and adjust the graphics images to
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    your taste.
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         MAGNIBROT - allows you to "zoom" in on selected regions of the set
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    by adjusting the magnification "window".
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    (basic programs)
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         AUTO-SEQ - displays multiple images in sequence to help visualize
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    the magnification process.
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         COLOURS - lets you adjust the colours in the images.  Also
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    "rotates" the colours as a "special effect".
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    THE FILES
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         The disk in drive 9 holds the files pertaining to the raw data
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    generated by MANDELCRUNCH.  Drive 8 should hold all
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    your program files.  Drive 8 will also
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    hold the "picture" files that are refined from the raw data.
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    RAW DATA FILES
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         MANDELCRUNCH deals with 3 files, which are kept on drive 9.  The
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    structure of these filenames is:
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    (root).(ext)
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         The root name is supplied by you.
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    The extensions are program-generated and will be
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    dat - data file
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    prm - parameter file
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    dst - distribution file
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         The data file is large, since it consists of two bytes for each of
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    the 32000 points in the mandelbrot grid.  Therefore, only 2 mandelbrot
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    runs will fit on a data disk.  Each run also uses a parameter file to
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    keep track of restarts.  When the program is complete, a distribution
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    file is generated which is used by MANDELPAINT to fine-tune the
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    graphics images.
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    PICTURE FILES
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         The fine-tuned images generated by mandelpaint are saved as
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    picture files of 32 blocks each.  The format for picture files is:
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    (root).(ext).(mandelbrot indicator)











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         The root name will match the root name on the MANDELCRUNCH files.
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    The extension may be program generated or user generated.  More on that
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    later.  The mandelbrot indicator is simply a y or an n to indicate of
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    any of the points fall within the mandelbrot set.
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         Magni-Brot also generates picture files.  For these files the
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    format is free-form.
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    RUNNING THE SYSTEM
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         Your kit comes with the program files, the document files, some
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    sample picture files, and a sample parameter file.
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         Put all the program files and the picture file on drive 8.  Put
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    the parameter file on drive 9.  All of the program files use SIMON'S
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    BASIC except for MANDELCRUNCH.
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         Try running the COLOURS program.  Load SIMON'S BASIC into the
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    system, then load and run COLOURS.  When prompted for a filename, enter
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    the complete name of one of the picture files.  Try "onw3>onw4.y".
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    Once the picture is displayed, vary the colors with the "1" "2" "3"
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    keys.  Rotate the colors by pressing "r".  While the colors are
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    rotating, press "+" or "-" keys to control the rotation speed.  To stop
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    rotation, press "r" again.
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         Load and run AUTO-SEQ.  Each press of a key will load successively
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    higher magnifications of a section of the set.  These are all picture
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    files created by MANDELPAINT and MAGNIBROT.  The use of a fast-load
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    cartridge may speed up the loading of images, as these are stored as
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    program files.
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    CREATING YOUR OWN IMAGES
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         As mentioned earlier, MANDELCRUNCH is the heart of the system.
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    Each image depends upon the millions of calculations performed by
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    MANDELCRUNCH.  MANDELCRUNCH generates the data for an area of the
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    mandelbrot set that you specify.  The area may be large enough to
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    contain the entire set, or microscopically small.
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         MANDELCRUNCH needs to know the area to be viewed.  It needs to
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    know the maximum number of "iterations", or how many times each point
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    will be calculated before it is accepted as a part of the set.  It also
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    needs to keep track of where it is in the run, so that it may be
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    interrrupted and restarted.  The parameter file contains these three
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    pieces of information.
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         Parameter files are created 2 different ways.  The easiest way to
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    create parameters is to use MAGNIBROT.  You may also create the
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    parameters manually when you first run MANDELCRUNCH.
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         Load and run MANDELCRUNCH.  SIMON'S BASIC is not needed for this
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    program.  It will ask you for the root name of the area to be
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    calculated.  If you have used MAGNIBROT, the program will load the











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    parameters from drive 9 and will begin crunching.
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         If no parameter file exists for the root name, you will be given
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    the option of creating one.  To create the parameters, you must enter
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    rl,rh,il,ih and the iteration limit.  The variables correspond to the
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    low and high limits of the real and imaginary numbers in the area of
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    the complex plane to be examined.  This will make sense to you if you
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    have done some reading on the subject of the mandelbrot set.  Once you
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    enter the information, the program will begin the run.  NOTE:  when
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    entering the limits for the real and imaginary numbers,  make sure that
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    the range of the imaginary numbers equals 80% of the range of the
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    reals.  That is, make sure that  (ih-il) = .8(rh-rl).
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    This is needed to compensate for the nature of C64 graphics.  If you
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    use MAGNIBROT on a previous image, the proper ratio will be
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    automatically generated.
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         See "hints and tips" for information on how to choose your
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    parameters.
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         MANDELCRUNCH will run.  And run and run.  It may keep running for
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    quite a while.  You might start to wonder if it is running at all.  The
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    screen is blanked out while the program runs.  This is a trick to gain
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    some speed.  Since the processor doesn't have to bother with the
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    display, it has more time to crunch.  There are 2 ways to get an idea
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    of how far along the program is.  First, the screen won't stay solid
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    black.  It will change colours during the run.  The c64 graphics are
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    160 collumns and 200 rows.  The screen will change color every ten
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    collumns.  If you are familiar with the c64 color code progression, you
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    will have a good idea where the program is at.  If you don't, don't
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    worry.  The program is interruptible, and when you interrupt it, it
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    will tell you exactly where it is at.
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         To interrupt MANDELCRUNCH, hit the left arrow key at the top left
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    of the keyboard.  This will halt the program, the screen will come to
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    life, and you will see how far along the program is.  If you are
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    computing an image with a high iteration limit, there may be a moment
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    before the program interrupts, as the program finishes the computations
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    for the particular point it is examining.
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         While the program is paused, you have the option of saving it or
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    resuming.  To save press "s" to resume press "r".  Once the parameters
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    have been saved, you can turn off the computer.  To restart
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    MANDELCRUNCH, simply run it and tell it which parameter file to use.
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    It will load the parameters and fast-forward to where it left off.
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         Eventually, MANDELCRUNCH will end. The disk will be busy near the
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    end, as the distribution file is created.  You are now ready to run
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    MANDELPAINT.
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    MANDELPAINT.
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         The graphics images produced by MANDELPAINT are the result of
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    mating the MANDELCRUNCH data to the c64 graphics, and to your visual
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    tastes.
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         The points that withstand the maximum iterations are in the set,
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    and are colored black.  Simple enough.  But what about the rest of the











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    points?  This is where much of the beauty is.
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         As the points get closer to the set, they are pulled to infinity
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    more and more slowly.  The graphics images show the varied topography
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    of these points.  Pinwheels, starfish, curlicues and other shapes
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    become apparent.
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         Unfortunately, the c64 gives us only 4 colors to work with.
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    MANDELPAINT helps to make the most of them.
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         Load and run MANDELPAINT (you must be using SIMON'S BASIC).  The
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    data disk must be in drive 9.  Supply the root name, and the program
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    will read in the distribution file.  The distribution file allows you
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    to see how the data points are distributed within the iteration range.
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    You can examine the distribution either graphically, or as a printout.
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    The distribution graph will show the frequency of each iteration level
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    compared to the highest count.  For example, assume you specified a
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    maximum iteration of 100 in your parameter file.  Look at the graph.
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    The graph is marked at the bottom of the screen.  The big marks are
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    100's , the little ones are 10's.  Assume the highest point of the
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    graph is at 20.  This would mean that more points ran to infinity after
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    20 iterations than any other value.  The rest of the graph is then
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    scaled in proportion to the number of points at 20.  Some graphs will
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    have a pronounced peak, with all other values dropping off rapidly.
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    Others may have many points near the peak.  The graph can help you pick
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    manual cutoff points for colorizing the picture.  The graph can also
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    help you pick iteration levels for your next magnification.  If you
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    used more than 300 iterations, each successive key press will display
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    the graph for another 300 iterations.  After the full range has been
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    displayed, you will be prompted by the menu.
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         After viewing the graph, you can exit back to the main program and
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    choose your colorization method, either Auto or Manual.  Both methods
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    generate cutoff points that define how the set will be displayed.  The
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    cutoff points are like lines on a topographic map.  The faster the
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    terrain changes, the closer together the lines are on a topographic
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    map.  The same applies to the display.  The faster the iteration levels
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    change, the narrower the color bands will be.  If the cutoff points are
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    too close together, it will be difficult to see patterns.  the display
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    may look like random dots.  If the cutoff points are too large,
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    interesting details may be lost.
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         The Auto method saves a lot of time going over distribution lists
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    and charts, and allows you to vary the cutoff points easily.  The
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    display consists of 32000 points, or pixels.  You tell the program the
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    minimum number of pixels to be devoted to any rage of iteration values.
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    For example, specifying a number of 4000 would ensure a maximum of 8
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    color zones on the display.  A value of 500 would mean 64 color zones.
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    The best value is a matter of taste. Experiment.