
                                    IFRACS.EXE
                          A FRACTAL DEMONSTRATION PROGRAM

                                  Alfred D'Attore


*****************************************************************************
*****************************************************************************
**       This software is mathematics-intensive.  It requires a very       **
**       fast computer for satisfactory results.  While not strictly       **
**   necessary, a mathematics coprocessor is also desirable.  The screen   **
**     can be normal VGA, 640 by 480 pixels, 16 colors; the most common    **
**                                 monitor extent.                          **
*****************************************************************************
*****************************************************************************


The included programs, written in Borland Turbo C++, comprise representative
examples of the more common fractals.  After initial decompression, command --

FRACTALS 

and you will be presented with a MENU.

Option 1, (one), will lead to fractals selected at random; primarily Iterated
Function Systems, (IFS).  Operation is "automatic," with fractals changing
after a given number of iterations.  Hit the space bar if you get impatient.
In most cases, the program will respond with a new fractal.  If the Tab key,
unshifted, is hit, a PCX type "picture" will be taken of the screen, but only
after the specified number of interations has completed.  Hit the Esc key to
exit.

Options 2 and 3 are fractals of the more othodox type.  In both cases, the
first screen display is optimized for speed.  Both presentations feature a
"Zoom" capability.  That is, pressing the Page up key will bring a rectangle
into view.  Page up decreases its size; Page down increases it.  The cursor
keys can be used to move it in the indicated directions.  Pick any area of the
picture; hit the Enter key, (Return), and then the Space Bar once more.  A
new picture will form exactly as outlined within your rectangle, but -- of
course -- greatly enlarged.  As above, press Esc to exit the programs.

Since both presentations require a long time to form and even more to inves-
tigate, all screens are formed in "thirds;" that is, one vertical line out of
three at a time.  If you permit, a full picture will result; the next two time
periods "filling in" the fractal.  You need not wait, however, to "zoom" in.
You can "zoom" at any time.

Option 4, (four), will lead to a number of fractals based upon the "Hopalong"
function yielding simple but representative and colorful examples.  Hit the
Space Bar to change the picture.  Hit lower case "p" to fit it to the screen.
As in other options, the Esc key exits the program.

All programs provide the capability to store images as PCX files.  At any time,
(preferably when the fractal is fully formed), hitting the TAB key, unshifted,
will start the process.  The bottom blurb will disappear and the question,
"Input filename ?" will appear in its place.  Give the image a name of eight
letters or less, (no extension), and hit RETURN.  A "beep" will announce when
the image is stored.  Afterwhich, the user can proceed exactly as before.

                                 * * * * *

AFRAC.BAS is an automated fractal demonstration.  It is offered for those who
are curious about IFS type fractals and how to program them.  It is written in
QuickBasic.  AFRAC.BAS will run -- albeit slowly -- using the Qbasic
interpreter that accompanies MS-DOS 5.0 or later.  It runs about five to eight
times faster in the regular QuickBasic environment.  Operation is the same as
for option 1 of the FRACTALS.EXE group, except there is no "PCX" capability.

MANDEL.BAS is an example of the Mandelbrot set produced without the use of
complex numbers.  The program is all of a piece, (no subprograms), and most
instructive for that reason.  The only "subtlety" used is a triple FOR-NEXT
loop that attempts to take advantage of symmetry and paint the fractal in
three sweeps instead of just one, to give an illusion of speed, (as above in
FRACTALS.EXE, options 2 and 3).  MANDEL.BAS will run in Qbasic too, but very,
very slowly.

ZAP10.CPP is a particularly simple IFS: z = az + b, where a, b, and z are
complex numbers.  This routine is actually a subprogram of FRACTALS.EXE, 
version 1, except provision has been made here to permit its operation as an
independent program within the Borland Turbo C++ environment.

A varient of the ZAP10.CPP relationship; to wit: z = az + e^b, is also a part
of FRACTALS.EXE, version 1..  Once again, a, b and z are complex numbers.  The
exponent term expands the fractal's scope greatly, producing some very
interesting effects.


                                 * * * * *


IFRACS.EXE is "freeware."  Comments and/or questions are welcome and may be
addressed to me at

                           Compuserve 75222,1254

                                     or

                         1300 South Farm View Drive
                               Apartment B-21
                            Dover, Delaware 19904
                                     USA

