Molding Fractals to Suit      -  john beale 2/12/95

Assume a fractal type, say a mandelbrot set, which has as a parameter
the initial value z0 for the iteration z(n+1) = z(n)^2 + c. This
parameter is usually held fixed for any given fractal image generated.
However, it is possible to vary the parameter spatially across the
image to influence the final geometry, generating an interesting new
class of images. This procedure may be applied to any fractal type
having one or more parameters.

For example, let us say we found a region which was very "dense" in
geometry across the field of view at parameter value A0 and quite
"sparse" in the same region at parameter A1. We generate a grayscale
bitmap with some interesting geometric figure which has gradual
shading from white to black (many sucessive applications of "blur" or
"smooth" operators to a B/W image perhaps).  We now proceed to
generate our new "variable" fractal by varying the parameter as a
function of the grayscale image, with white => parameter = A0 and
black => parameter = A1.

The resulting image is a fractal pattern, dense where the greymap has
a high value, and sparse where the greymap is close to zero. Assuming
the chosen fractal type changes continuously as a function of the
parameter, this may produce a striking design which has a fractal
nature and yet also displays the essential geometry of the chosen
image. Obviously this can be any bitmap; eg letter fonts, a portrait
or even another fractal. The interest of the final image depends
strongly on the choice of a fractal region and parameter range with
this transition behavior.

One could imagine extensions such as varying two or more parameters
simultaneously, perhaps keyed to the intensity and hue or saturation value
of a color image, but this is so complicated I can't imagine what the
result might be like.

If you have any comments or thoughts on this idea, feel free to send
me email at the address below.

-John Beale <beale@jumpjibe.stanford.edu>