From: noel@reg.triumf.ca (NOEL)
Newsgroups: alt.fractals
Subject: mandelclouds
Date: 27 Aug 92 21:31:00 GMT
Organization: TRIUMF: Tri-University Meson Facility
News-Software: VAX/VMS VNEWS 1.41


	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 a 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
necessary 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 cloud chamber
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.   This is related to the orbital display
in fractint but differs in that the orbits persist and increment while
the standard image is discarded.

	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
estimation 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@reg.triumf.ca
Newsgroups: alt.fractals