Educational Software Cooperative 4/Educational Software Cooperative 4.iso/mand56a/mandel.hlp

Text File | 1996-05-01 | 33KB | 836 lines

                   Mandelbrot/Julia
                     Set Generator
            Operating and Reference Manual
                      Version 5.6

                     Installation

    The Mandelbrot/Julia Set Generator program
requires an IBM compatible computer with at least 512K
of memory, a VGA display and a Microsoft compatible
mouse. The installation process is quite easy. (Users
obtaining .ZIP files from the Internet or a BBS can
skip this installation section.)

    First, make a backup copy of your Mandelbrot/Julia
Set Generator program disk. If necessary, consult your
PC-DOS/MS-DOS manual for a description of the Diskcopy
command. Save the original program disk in a safe
place and use the copy as the working program disk.

    Second, while it is possible to use the
Mandelbrot/Julia Set Generator on a floppy disk
system, a hard disk system is a necessity if you wish
to store a number of image files. To install the
Mandelbrot/Julia Set Generator on a hard disk use the
following steps:

    1. Insert the floppy disk in your computer in
       drive A. (or B if necessary)
    2. Type A: (or B:)
    3. Type INSTALL A C

    Any hard drive letters can be used, for example
INSTALL B D will install the program from floppy drive
B to hard drive D. The installation will create a
directory called MAND5 on your hard drive and you will
need to enter a CD\MAND5 to change to the
Mandelbrot/Julia Set Generator directory before
running the program. Once installed just type MAN to
start the program. All the files for Mandelbrot/Julia
Set Generator need to be in the same directory for the
program to operate successfully.

          Quick Start for Impatient New Users

    Type MAN to start the program. After the mouse
cursor appears click it on the Load Image button at
the left. When the window appears with the list of
image file names, simply clicking on one of them will
display the image using the current color mask. If the
image file contains a specific color mask filename it
will be automatically loaded prior to displaying the
image. Most commands can be interrupted by a simple
mouse click.

    The zoom window feature is started by double
clicking on the displayed image. Once the zoom window
appears, with its crossed center lines, it can be
moved by holding down the left mouse button, while the
cursor is inside the zoom window, and positioning the
window. The zoom window size can be increased or
reduced by holding down the left mouse button and
moving the mouse cursor horizontally while it is
outside the zoom window. Once correctly positioned the
mouse cursor should be clicked on the right gray
panel, which will store the changed dimensions. The
zoom window can be abandoned by clicking the mouse
cursor on the left gray panel. Be careful not to drag
the mouse cursor onto the gray panels while resizing.
The Set Values button should be clicked on next, and
the image file name changed. If this is not done the
original image file will be erased. Clicking on the
Make Image button will start the generation of the new
zoomed image.

        The Command Buttons and Their Function

    Set Values

    The Set Values command allows the user to set the
initial parameters that will be used by the
Mandelbrot/Julia Set Generator to begin generating a
new image. These values are also available for
inspection when an image has been displayed. The
values and their range are:

    Item                             Range
-----------------------------------------------------
    X center value                   -10 to 10
    Y center value                   -10 to 10
    Magnification                    >0
    A value (if a Julia image)       -10 to 10
    B value (if a Julia image)       -10 to 10
    Dwell                            1 to 8191
    Image width in pixels            10 to 4800
    Image type [M J]                 M or J
    Full/Partial image [F P]         F or P
    Default color mask file          xxxxxxxx.MSK
    Display type [0 1]               For future use
    256 color palette number         For future use
    Image file name                  xxxxxxxx.MAN
                                     or xxxxxxxx.MAR

    To change a value simply click inside the
rectangle where the value is displayed and then key in
a new value or file name. File name extensions must be
.MSK for color masks, .MAN for regular images and .MAR
for those that are recursive.

    The A and B values are only displayed with Julia
images. If the recursive image generator is used the
image width must be a member of the 2^n set, ie. 16,
32, 64, 128 etc. The program maintains the
Full/Partial image status and these values cannot be
changed by the user.

    Color Masks

    When the Color Masks command is chosen a popup
window presents the four options:

    Create/Display color mask
    Select color mask
    Select palette
    QUIT

    Clicking on the Create/Display color mask option
allows the user to create, edit and save color masks.

    Clicking on the Color mask file name box allows
you to type in a file name. The file name must have
the extension .MSK or you will not be able to select
it later. Ranges of dwell values should be typed into
the squares on the left. Just click on the square,
type in a dwell value and <Enter>. The colors are
selected by clicking on the desired color of the color
wheel in the upper right and then clicking on the odd
and even boxes at the right. The selected color is
displayed between the circular menu and the color
wheel. If the first line of the color mask reads:

    0         9       [blue box]   [white box]

then dwell values from 0 to 9 will be colored blue if
odd and white if even. If a solid color is desired the
color boxes should be filled with the same color. The
end of a color mask should be designated with a
negative value entered into the first dwell box.  The
default color mask on startup is M1.MSK and its values
are:

    Dwell Range     Odd color     Even color
-----------------------------------------------
    0     9         [blue box]    [white box]
    10    19        [red box]     [red box]
    20    510       [yellow box]  [yellow box]
    511   511       [black box]   [black box]
    -1

    In the case of M1.MSK, any dwell values larger
than 511 will be colored black (color 0 in the default
palette).

    The circular menu at upper left has four options.
Clicking on the up or down arrow jumps to the previous
or next 16 color mask entries. A total of 256 entries
can be placed in one color mask. The SAVE option
stores the color mask currently displayed under the
name specified in the Color mask file name box. The
new color mask becomes the currently selected color
mask.

    Clicking on the Select color mask option presents
the user with a large window and the names of the
color masks that have been stored. Clicking on a color
mask name will select and load that color mask. It
then can be viewed by selecting the Create/Display
color mask option.

    Clicking on the Select palette option opens a
window that displays the current VGA color palette of
16 colors. Clicking on the Default box will load the
default VGA color palette. Clicking on the arrows will
select other prestored color palettes, up to number
57.

    The QUIT option returns the user to the main menu.

    Make Image

    Selecting the Make Image command generates a
Mandelbrot or Julia image based upon the parameters
entered in the Set Values command. A warning is issued
before the generation begins to allow the user to
change the file name, as any existing file of this
name will be erased.

    A very simple way to generate images is first to
use Load Image to display a previously generated
image. Double clicking on the image will produce a
zoom window overlaid on the display. Clicking and
holding down the left mouse button allows the zoom
window to be dragged about the image to an interesting
portion of the image. The zoom window can be resized
by dragging the mouse pointer to the left and right
outside the zoom window. Once the zoom window has been
positioned and sized, clicking on the gray panel at
right will automatically store the new zoomed values
into the Set Values area. Be careful not to drag the
mouse cursor onto the gray panels while resizing. The
user will probably wish to enter a new image file name
using the Set Values command (this will prevent the
original image file from being erased), and then
generate a new image of the area defined by the zoom
window with the Make Image command. While the zoom
window is present the procedure can be cancelled by
clicking on the right gray panel around the command
buttons. The zoom window will only work on images that
are 480 pixels wide, or less.

    Load Image

    The Load Image command presents the user with a
list of image file names that have been produced with
the .MAN extension. Clicking on a file name will
display the image with the current color mask if the
selected image has no default color mask file name. A
brief double tone is sounded if there is no default
color mask file name. If a color mask name was
included when the image was generated, this color mask
will be loaded before the image is displayed.
Partially generated images will automatically continue
generation when displayed with this command. Once an
image is displayed double clicking on the image will
produce a zoom window as described under the Make
Image command.

    Make R Image

    The Make R Image command functions similarly to
the Make Image command except a recursive procedure is
used in place of the normal line by line generation.
The image file should be given the .MAR extension so
that it will be properly handled when using the Load R
Image command. In some cases this recursive procedure
will generate images faster that the normal method.
Partially generated images cannot be displayed with
generation automatically continuing as is the case
with the normal Load Image command. Image files are
generally larger with the recursive procedure.

    Load R Image

    The Load R Image command displays a recursive
image previously generated with a .MAR extension in
the file name. A list of such files is presented and
the selected image is clicked on. Partially generated
images will not be automatically continued as with the
Load Image command.

    3-D Image

    The 3-D Image command displays an image generated
with the Make Image command in a pseudo 3-D style. The
display algorithm is a simple one, but very slow. VGA
displays have limitations when displaying 3-D
Mandelbrot images. Best results occur with color masks
that contain multiple colors and have the dwell ranges
broken into many small steps. Large values for the
maximum dwell may result in the top of the image being
lost. Partially generated images will not be
automatically continued as with the Load Image
command.

    Plot Dwell

    The Plot Dwell command reads all the dwell values
of an image stored with the .MAN extension and sums
them. The sums are then plotted with the current color
mask used for each dwell value plotted. Only dwell
values of 2,400 or less will be plotted. These plots
give an indication of how many points in the image
have the various dwell values and can be useful in
constructing a color mask that will display the image
to best advantage.

    Make PCX

    The Make PCX command allows the user to select an
image file stored with the .MAN extension and create a
PCX image file. A 16 color PCX file using the default
VGA color palette can be chosen or several 256 color
PCX formats are available. Click on one of the small
boxes to select what type of PCX file you desire. The
color sequence of each of the 256 color formats is
displayed. The first example has magenta blending into
red for dwell values from 0 to 64, from red to yellow
for dwells from 64 to 128, etc. The PCX image file
format allows users to import Mandelbrot and Julia
image files into other software such as desktop
publishing programs and paint programs. PCX files can
also be used for Windows wallpaper.

    Print Image

    The Print Image command presents the user with
nine different printer types that are supported, or
the command can be quit.

    Epson 9 pin
    Epson 24 pin
    IBM 9 pin
    IBM 24 pin
    LaserJet
    DeskJet 500 B/W
    Epson DM Color
    DeskJet 500 C
    PaintJet
    QUIT

    The 9 pin printers will output at 120x144 dpi, the
24 pin at 180x180 dpi, the Laserjet, Deskjet 500 B/W
and Color at 150x150 dpi and the Epson DM Color and
Paintjet at 90x90 dpi. Be patient, the print drivers
do take time in exchange for attractive output. Color
is the slowest. Black and white images will be
dithered. QUIT returns the user to the main menu.

    The Print Image command is basically for quick
hardcopy. If you wish to print museum quality prints
try a DeskJet 500 series printer. Create your image
and then make a PCX file using one of the 256 color
formats. Next, load this PCX file into the Paintbrush
program that comes with Windows. This is usually found
in the Accessories window. Next print the image from
Paintbrush. You will need a 256 color display to do
this and the Windows print driver that came with the
DeskJet printer. Most IBM PC's and clones being sold
today come with a 256 color display. The DeskJet will
print your image with a superb color balance at just
under 100 dpi. Try an image width of about 750 pixels
to fill out the 8-1/2 inch page. I've used this method
with a Hewlett-Packard DesignJet 650C and 36 inch wide
paper with images 3300 pixels wide to produce colored
output that is truly magnificent. If your printer is
not supported this method can also be used to print
your images. The only thing you will need is the
Windows print driver that came with your printer.

    Remember, images my be created which are much
wider than your screen. The upper left corner of your
image will be the only area visible. To see the entire
image, create a 256 color PCX file and use any paint
program that can read 256 color images. These can be
very attractive.

    Help File

    The Help File command displays the file you are
currently reading. Clicking on the arrows to the right
displays the next or previous page.

    Quit MAND56

    The Quit MAND56 command returns the user to the
DOS prompt.

                 Image File Structure

    Each image file created by the Mandelbrot/Julia
Set Generator begins with a 150 byte header.

Byte  Item       Size           Description
------------------------------------------------------
0     x           8 byte double  x center point
8     y           8 byte double  y center point
16    mag         8 byte double  magnification
24    a           8 byte double  a for Julia sets
32    b           8 byte double  b for Julia sets
40    maxdwell    unsigned int   maximum dwell
42    width       unsigned int   image width in pixels
44    mj[2]       char           M/J, image type
46    partial[2]  char           F/P, full/partial
48    mask[32]    char           color mask file name
80    display     integer        display (not used)
82    pal         integer        palette (not used)
84    name[50]    char           signature
134   fill[16]    char           filler

    All char strings are terminated with a hexadecimal
00 byte.

    The dwell data follows the header. It should be
noted that this is not a true image file, rather the
dwell values themselves are stored. This allows users
to color the image with a large variety of color
masks. Storing an image file might be simpler but for
every different color mask a new image file would have
to be created.

    The dwell data is stored as a series of two byte
unsigned integers. Each unsigned integer contains the
dwell value and a run length corresponding to a string
of identical dwell values. The number of bits required
to hold the maximum dwell is first obtained. If the
maximum dwell is 511, then 9 bits are required, 1023
would require 10 bits, etc. Using 1023 for the maximum
dwell as an example, the right most 10 bits of the 16
bit integer represents the dwell value and the 6 left
most bits contain the run length. As a run length of
zero is not very useful, this value is always
incremented by one such that a run length of zero
equals 1, 1 is 2, etc. Given a maximum dwell of 1023
the following 16 bit unsigned integer represents a
dwell of 1000 and a run length of 32.

                   011111 1111101000
                       7FE8 hex

    When an image is being displayed and the unsigned
integer above is read, a line of 32 pixels will be
drawn using the appropriate color from the active
color mask for dwell value 1000.

    Each line of a display is encoded with no
wraparound. This means that each line will end with
the display of an encoded unsigned integer and no
extra pixels of the same dwell will be added for the
beginning of the next line even if there is room in
the run length.

    It should be noted that the maximum run length
that can be stored varies with the maximum dwell
chosen. Files with a maximum dwell of 1023 will have a
maximum run length of 64, those with maximum dwells of
8191 will only store 16. This does not limit a run
length because if it exceeds the space available in a
single unsigned integer it simply creates additional
ones until the run of dwells has been stored. For this
reason images with high maximum dwell values are often
large in size. This method of file compression strikes
a good balance between file size and speed when
displaying an image.

         The Mathematics of the Mandelbrot Set

    The Mandelbrot set is computed by operating on a
fairly simple equation that contains complex numbers
of the form

              x + yi where  i = sqrt(-1)

The Mandelbrot equation is

                     z <- z^2 + c

where

               z = x + yi and c = a + bi

substituting these values into z^2 + c we have

                  (x + yi)^2 + a + bi

               x^2 + 2xyi - y^2 + a + bi

separating the real and imaginary parts of z gives

                  x <- x^2 - y^2 + a

                     y <- 2xy + b

    To determine whether a point (a,b) in the complex
plane is a member of the Mandelbrot set, the real and
imaginary parts of the equation are iterated. The x
and y values are first initialized to zero. The
constants a and b, the point in the plane, are then
substituted into the equations giving

                   x <- a and y <- b

for the first iteration.

    The two new values for x and y, along with the
constants a and b, are now substituted into the
equations again. This procedure (iteration) continues
until the absolute value of x + yi > 2, ie. sqrt(x^2 +
^y2) > 2. For those cases where this value never
exceeds 2, the maximum number of iterations is preset.
A value of about 500 is usually adequate, although
this value is raised to several thousand when smaller
details at high magnification are examined. The number
of times the equations are iterated before the value
of sqrt(x^2 + y^2) > 2 is called the dwell. Those
initial points (a,b) where the dwell is infinite, or
for more practical purposes attains the preset
maximum, are members of the Mandelbrot set. Another
way to describe this is to say that for points within
the Mandelbrot set, the sequence of points produced by
this iteration procedure is bounded inside a circle of
radius 2, where points outside the set are unbounded
and continue to grow and escape the circle.

    The Mandelbrot set exists entirely within the area
defined by

            -2 <= a <= 2  and  -2 <= b <= 2

in the complex plane. A Mandelbrot image is produced
by taking this area of the complex plane and dividing
it into a array of 1200 x 1200 points. Each one of
these points becomes the constant (a,b). The iteration
procedure previously described is used on each of the
1.44 million points, coloring each point in the
Mandelbrot set black and all others white. The
algorithm is:

    maxcount <- 1000
    for b <- 2 to -2 stepdown 1/300
    for a <- -2 to 2 step 1/300
      x <- 0
      y <- 0
      count <- 0
      while sqrt(x^2 + y^2) < 2 and count < maxcount
        x <- x^2 - y^2 + a
        y <- 2*x*y + b
        count <- count + 1
      end while
      if count = maxcount plot(a,b,BLACK)
        else plot(a,b,WHITE)
    end for a
    end for b

    While the algorithm is not that complex, the
amount of computation is enormous. Depending on
programming language and style, the inner loop has at
least four multiplications and a square root. For a
point in the Mandelbrot set this loop is executed 1000
times and there are over a million points to check! It
is not surprising that the Mandelbrot set was not
discovered until the age of computers.

    In the Mandelbrot/Julia Set Generator program some
additional refinements are made to standardize the
initial parameters used to generate a specific image.
Instead of defining the range of (a,b) values used for
an area, a center point and a magnification are
specified. The center point is simply a chosen (a,b)
value. The length of a side which encloses the area of
interest is defined as

                side = 2/magnification

The following values can now be defined

             a_minimum = a_center - side/2

             b_maximum = b_center + side/2

                   gap = side/width

where width is defined as the number of points that
make up a side (or on a computer screen the number of
pixels), and the gap being the distance between each
point.

    The Mandelbrot set is an interesting image, a sort
of cardioid with a spiked head attached at the left.
The boundary of the set sprouts self similar buds of
different sizes. Vastly more interesting images are
forthcoming when we examine the boundary of the
Mandelbrot set under higher magnification. To obtain
higher magnifications we simply divide a smaller area
into our array of points. For example, the area
defined by the center point (-0.77,0.17) and
magnification 20 is located in the upper valley
between the head and the cardioid shaped body.

    If we continue with these magnifications, very
different and interesting images can be produced by
coloring the dwell values in specific ways. Along with
coloring points in the Mandelbrot set black, we can
assign different colors to other points based upon
their dwell value. For example, we might assign yellow
to dwell values in the range 400 to 499, red to 300 to
399, etc. When we do this a great deal more detail
begins to appear in the boundary regions. This region
of interest exists only in a narrow band just outside
the Mandelbrot set. The skill one uses in choosing the
various colors for differing dwell values is very
important when attempting to produce an attractive
image.

    The Mandelbrot/Julia Set Generator uses a file
called a color mask to store the colors used in
painting the various dwell values in an image. This
technique allows many different coloring schemes for a
single image. Consider the following color mask:

    Dwell Range    Odd Color   Even Color
----------------------------------------------
    0     9        blue        white
    10    19       red         red
    20    510      yellow      yellow
    511   511      black       black
    -1

    Dwell values from 0 to 9 will be colored blue if
they are odd numbers and white if they are even.
Values from 10 to 19 will be colored red, 20 to 510
yellow and 511 will be colored black. Choosing the
maximum dwell value to be in the set 2n - 1 maximizes
the file compression method the Mandelbrot/Julia Set
Generator program uses.

    Generating Julia set images is a similar process.
The point (a,b) is chosen from one of the interesting
boundary areas of the Mandelbrot set. This value is
held constant and the (x,y) value is initialized to
the various points in the complex plane defined by

            -2 <= x <= 2  and  -2 <= y <= 2

    This would be a magnification of 0.5, actually the
Julia image can often be enlarged slightly to fill the
screen and magnifications from 0.6 to 0.9 are often
used.

    The algorithm for generating a Julia set is

    maxcount <- 1000
    a <- constant
    b <- constant
    for y <- 2 to -2 stepdown 1/300
    for x <- -2 to 2 step 1/300
      count <- 00
      while sqrt(x^2 + y^2) < 2 and count < maxcount
        x <- x^2 - y^2 + a
        y <- 2*x*y + b
        count <- count + 1
      end while
      if count = maxcount plot(a,b,BLACK)
          else plot(a,b,WHITE)
    end for x
    end for y

                  Selected References

Barnsley, Michael, Fractals Everywhere. San Diego, CA:
    Academic Press, 1988.

Briggs, John and Peat, F. David Turbulent Mirror. New
    York: Harper & Row, 1989.

Devaney, Robert L.Choas, Fractals, and Dynamics. Menlo
    Park, CA:Addison-Wesley, 1990.

Devaney, Robert L. and Keen, Linda, Editors. Chaos and
    Fractals, The Mathematics Behind the Computer
    Graphics: Proceedings of Symposia in Applied
    Mathematics.Providence, RI: American Mathematical
    Society, 1989.

Gleick, James Chaos, Making a New Science. New York:
    Viking Penguin, Inc., 1987.

Mandelbrot, Benoit B. The Fractal Geometry of Nature.
    New York: W. H. Freeman and Co., 1983.

Peitgen, Heinz-Otto and Richter, Peter H. The Beauty
    of Fractals, Images of Complex Dynamical Systems.
    Berlin: Springer-Verlag, 1986.

Peitgen, Heinz-Otto and Saupe, Dietmar, Editors. The
    Science of Fractal Images. New York: Springer-
    Verlag, 1988.

Pietgen, Heinz-Otto, Jurgens, Hartmut and Saupe,
    Dietmar Fractals for the Classroom, (Volumes I &
    II), New York: Springer-Verlag, 1992. (There is a
    single volume work entitled Chaos and Fractals,
    New Frontiers of Science, which is essentially the
    same work as the two volume set above.)

Pickover, Clifford A. Computers Pattern Chaos and
    Beauty: Graphics from an Unseen World.New York:
    St. Martin's Press, 1990.

Pickover, Clifford A. Computers and the Imagination:
    Visual Adventures Beyond the Edge. New York: St.
    Martin's Press, 1991.

Pickover, Clifford A. Mazes for the Mind. New York:
    St. Martins Press, 1992.

Schroeder, Manfred Fractals, Chaos, Power Laws,
    Minutes from an Infinite Paradise.New York: W.H.
    Freeman and Co., 1991.

Stevens, Roger T. Fractal Programing in C. Redwood
    City, CA: M&T Publishing, Inc., 1989.

Stevens, Roger T. Advanced Fractal Programing in C.
    Redwood City, CA: M&T Publishing, Inc., 1990.

Stewart, Ian Does God Play Dice? The Mathematics of
    Chaos. Oxford: Basil Blackwell, 1989.

Stewart, Ian and Golubitsky, Martin Fearful Symmetry,
    Is God a Geometer? Oxford: Blackwell, 1992.

                     Registration

    You may freely copy and distribute this shareware
Version 5.6 of the Mandelbrot/Julia Set Generator.
Shareware users who find the Mandelbrot/Julia Set
Generator useful should support the author and
register their copy. The form found below should be
used for registration. Registered users will receive a
copy the lastest version of the Mandelbrot/Julia Set
Generator with additional images and a printed manual.
Registered users will also receive support, by letter
mail, e-mail or phone, for six months from the date of
registration.

    The Mandelbrot/Julia Set Generator is a "shareware
program" and is provided at no charge to the user for
evaluation. Vendors who distribute shareware programs
may charge a small fee for an evaluation copy. Feel
free to share this program with your friends, but
please do not give it away altered or as part of
another system. The essence of "user-supported"
software is to provide personal computer users with
quality software without high prices, and yet to
provide incentive for programmers to continue to
develop new products.  If you find this program useful
and find that you are using the Mandelbrot/Julia Set
Generator and continue to use the Mandelbrot/Julia Set
Generator after a reasonable trial period, you must
make a registration payment of $25. plus $2. shipping
to Theron Wierenga. The registration fee will license
one copy for use on any one computer at any one time.
You must treat this registered software just like a
book. An example is that this registered software may
be used by any number of people and may be freely
moved from one computer location to another, so long
as there is no possibility of it being used at one
location while it's being used at another. Just as a
book cannot be read by two different persons at the
same time.

    The registration fee is $25. ($35. outside the
United States.) Please include $2.00 for shipping and
handling. A complete listing of the program, which is
written in Borland C/C++, is also available for an
additional $20.00. All prices are in U.S. dollars.

   Checks should be made out to:

   Theron Wierenga, P.O. Box 595, Muskegon, MI 49443

                  Ombudsman Statement

    This program is produced by a member of the
Association of Shareware Professionals (ASP). The ASP
wants to make sure that the shareware principle works
for you. If you are unable to resolve a shareware-
related problem with an ASP member by contacting the
member directly, ASP may be able to help. The ASP
Ombudsman can help you resolve a dispute or problem
with an ASP member, but does not provide technical
support for members' products. Please write to the ASP
Ombudsman at 545 Grover Road, Muskegon, MI 49442-9427
USA, FAX [phone] or send a CompuServe message via
CompuServe Mail to ASP Ombudsman 70007,3536.

                     User Support

    Registered users will receive support, by letter
mail, e-mail or phone, for six months from the date of
registration on any problems they encounter. Customer
support and order phone [phone]. The author is
available by e-mail on the internet at
[e-mail].

                   Registration Form

      Mandelbrot/Julia Set Generator, Version 5.6

  Name______________________________________________

  Address___________________________________________

  City_________________________State_____Zip________

  Email address (if available)______________________

  Disk size:   5 1/4 in._______     3 1/2 in._______

  Registration fee . . . . . . . . $25.00 __________
  Registration fee (Outside USA) .  35.00 __________
  Borland C/C++ program code . . .  20.00 __________
  Shipping . . . . . . . . . . . .        ___2.00___

  Total enclosed . . . . . . . . .        __________
           (All prices are in U.S. dollars.)

                  Method of payment:
  Check or MO_______ MasterCard________ Visa________

  Account number__________________ Expir. date______

  Signature (necessary)_____________________________

  How did you receive your copy of this program?

  __________________________________________________

  Suggested improvements____________________________

  __________________________________________________

  __________________________________________________

  __________________________________________________

    The Mandelbrot/Julia Set Generator, Version 5.6

               is a software product of

   Theron Wierenga, P.O. Box 595, Muskegon, MI 49443

      Customer support and order phone [phone]
                [e-mail]