I was inspired by an article on the Mandelbus, Scientific American
February 1989, which describes the orbit of iterations for a point near the
mandelbrot, to explore further this concept of orbits and get a better
visualization of where these iterative calculations are going.  This line of
thought brought out what I like to think of as the Mandelcloud or the cloud of
all orbital points from within a defined region of the mandelbrot set. If you
take the range of real and imaginary numbers that frame the whole mandelbrot
set (-2.0,-2.0) -> (2.0,2.0) and divide this into a two dimensional matrix of a
size appropriate to your graphical output device, you can trace the orbit of
any point and plot its destination as long as it is still bounded by the range
of points within your matrix.  The destination won't often be represented
exactly by any individual element in the matrix, but it is only neccessary to
find the closest one and index that.  If the matrix is initialized to zero, an
orbit that visits a point can increment the associated two dimensional element
and then continue on iterating and incrementing until the orbit escapes towards
infinity or an iteration limit is achieved. What is left in the matrix after
the calculation is finished is the complete orbital history for the starting
point.  The calculation becomes interesting if this is done not just for one
point but for a family of adjacent points or all the points represented by the
matrix. It is possible to view the calculation as it proceeds by displaying the
orbital point as a coloured dot on the screen.  For a family of points some
orbits will return close enough to be binned into the same matrix element.  If
this element is incremented each time it is visited, a different colour can be
associated with the number of orbital returns. It is in effect a two
dimensional relative frequency histogram of orbital returns.  Popular points or
areas will have higher colour intensities.  The initial co-ordinate for each
test point will obviously map the matrix directly, so it is redundant and I
have chosen not to plot it.  Each subsequent iteration will produce a coloured
point on the screen in what appears to be a chaotic fashion.  If the iterative
process proceeds for each point in a line, the adjacent points have very
similar orbits and will produce a family of curves and spirals that are
distinct for that region.  Sometimes it looks like a cloudchamber photograph
and other times like a spirograph.  If the line cuts across the mandelbrot you
will see very distinct changes in orbital paths as the boundary is crossed.  As
orbits from points near the mandelbrot tend to return close to the originating
point, it is possible to map only a selected region of the mandelbrot and plot
the orbits associated with that area, effectively zooming in on the orbital
cloud.  If you select a small area cutting the Mandelbrot, many interesting
images may be can be found.  Many of the points on the orbit will be beyond 
the area bounded by your matrix, but they may be tossed away and the 
calculation continued.  Orbital paths near the mandelbrot tend to return close 
to the origin and will hence fall into the area bounded by the matrix and they 
may be counted and displayed.  The resultant pattern from such a calculation 
is remarkably beautiful.  Very complex multicolour moire patterns emerge from 
the crossing of the orbital paths and numerous spherical or teardrop shapes 
define the areas of orbital return.  The orbital paths of adjacent points 
inside the mandelbrot are dramatically different from those approaching from 
the outside.  The juxtaposition of these two orbital families create the image 
of the mandelcloud. 

	What I find pleasing about these images is that they are not so
obviously digitally produced and have a a grainy pointalist style.  What
is also very beautiful but hard to convey in pictures is watching the
initial pattern develop on the screen. They unfold very gracefully and
the geometry is intrinsically beautiful.  Another facet of these images
that requires a digital image to appreciate is the cycling of the 
colour pallette.  Many different patterns exist within one image and not all
of them can be hilighted at once.  It often takes quite a different colour
scheme before other patterns emerge.  

	Anyway, I have found these images quite strikingly different, while
still recognizably related to the mandelbrot and have spent some time
exploring this avenue.  I think the analysis of the curves produced from
adjacent points may provide an interesting method for determining membership
in the mandelbrot.  Perhaps a look at rate of change of slope would do the
job?  I haven't thought of the best way to do this yet. I believe that there
are already some techniques using second order differentials used in
distance estimatation methods.  I'm not sure if what I have envisioned would
be significantly different.  It might be possible to approach the boundary
from both sides of the set with relatively few iterations by detecting the
differences in orbital path. 

						Noel Giffin