(C)opyright 1987, Otto J. A. Smith.  (206) 385-1956

                 Making Wallpaper on the Amiga Part I.

     Charles Dewdney popularized the term "Wallpaper for the Mind" to
refer to certain kinds of automatic picture generating programs.  It is
fairly easy to write programs that automatically generate pictures in
any of the languages available on the Amiga, including Basic, although
the easiest method to generate any of the types of pictures mentioned
here is with the program Doug's Math Aquarium, which is ideally suited 
to this kind of endeavor.  Since we are dealing with the Amiga computer
we will assume we have 32 color registers available for 32 simultaneous
colors on the screen.

      By far the easiest kind of picture to create is a moire pattern.
This can be done in Basic using two simple loops that calculate a color
for every point on the screen.  There are two separate simple problems
involved in this.  The first is to determine a value for x and y at each
point on the screen.  These values need not be the same as the pixel
coordinates.  The second problem is deciding on a color for each pixel
based on the values of x and y.

      Here is an example of a formula for generating a color from x
and y:

      COLOR = (X*X + Y*Y) MOD 32

      This formula generates a moire pattern of overlapping and
interwoven circles.  Any formula can be used, provided it is scaled to
values that can be represented as color registers.  If the numbers are
big enough we can take them modulo 32 to assign each number at each pixel
to a color register.  Here is an outline for a very specific simplistic
program that will generate only one picture:

      1.  Allocate a low-res screen with 5 bit-planes.
      2.  For X = 0 to 319 and
          For Y = 0 to 199 do the following:
              Calculate the value of COLOR = (X*X +  Y*Y) MOD 32
              Assign the pixel at X,Y to the color register COLOR.
              Repeat until loops are exhausted.
      3.  Assign colors or whatever to wrap up the program and exit.

       This may not be prove to be the most exciting picture you could
hope to generate. To change this picture you can do several things.
You could change the color scheme.  You may be suprised at how different 
this picture will look with different color schemes.  The apparent sizes
of the overlapping circles can be made to change by changing the color
scheme.  To change the picture you could also change the formula for the
color.  (keep the MOD 32)  For instance COLOR = ((X+Y)/(X-Y)) MOD 32,
or COLOR = (X+X/Y+Y) MOD 32.

        It would also be nice to be able to change the values
of and x and y in the two loops to some other set of numbers rather
than 0 to 319 and 0 to 199.  One way of doing this would be to insert
two lines of code in the inner loop that calculate a new value for x
and y.  For instance:

        FOR X = 0 to 319
        FOR Y = 0 to 199
            NEWX = X/7.15
            NEWY = Y/7.15
            COLOR = (NEWX * NEWX + NEWY * NEWY) MOD 32
            REM   PLOT THE PIXEL X,Y WITH COLOR REGISTER COLOR HERE.
        NEXT Y
        NEXT X

      The values of X and Y bear a relationship to points in a
plane.  The above solution would be more attractive if it preserved
this relationship.  To make this work, we need numbers that define
where we are in the real plane.  Let us define values for XMAX, XMIN,
YMAX and YMIN.  These values define the corners of a rectangle that
looks like the illustration below.

              XMIN,YMIN     XMAX,YMIN
                  _______________
                  .             .
                  .             .
                  .             .
                  .             .
                  .             .
                  .             .
                  ---------------
              XMIN,YMAX     XMAX,YMAX


      We would like the CRT screen to correspond to this rectangle.  To
make this the case, we can calculate NEWX and NEWY as follows:
    
      NEWX = XMIN + ((XMAX-XMIN)/319) * X 
      NEWY = YMIN + ((YMAX-YMIN)/199) * Y

      And we can proceed as above.  By building a program that prompts
for the values of XMAX, XMIN, YMAX and YMIN we can move around on
the infinite plane defined by the formulas in our program.  There are
many ways to make this procedure faster.  In particular NEWX and NEWY
calculations can be factored out of the two inner loops.  This method of
plotting functions of two variables is very general.  The Mandelbrot
set is an example of a plot done this way in which the formula is not a
simple function of x and y, but is a recursive function of x and y.

      The program Doug's Math Aquarium is designed to make the production of
these "Wallpaper for the mind"-type plots much easier.  It allows the entry
of a formula which it then compiles, in order to make the actual plotting 
step faster.  It allows the user to zoom in or out of the picture using the
mouse. It includes tools for scaling, setting the colors, color cycling,
saving and recovering pictures in IFF format and saving and recovering
formulas. Recursive formulas are allowed, so the Mandelbrot set is only one
of a large number of different kinds of plots it is capable of producing.

      Although Doug's Math Aquarium is fast, it is possible to
write special purpose programs for particular formulas, such as the
Mandelbrot set, that are faster.  In order to make the plot faster
the programmer can use scaled integer arithmatic, he can write in
machine language and he can use special features of the particular
formula he is writing for, such as range checking.  Some but not all
of these techniques can be included in the formulas in Doug's Math
Aquarium, because Doug's Math Aquarium does all calculations in Fast
Floating Point Arithmatic.