2.50 Frothy Basins
(type=frothybasin)
Frothy Basins, or Riddled Basins, were discovered by James C. Alexander
of the University of Maryland. The discussion below is derived from a
two page article entitled "Basins of Froth" in Science News, November
14, 1992 and from correspondence with others, including Dr. Alexander.
The equations that generate this fractal are not very different from
those that generate many other orbit fractals.
Z(0) = pixel;
Z(n+1) = Z(n)^2 - C*conj(Z(n))
where C = 1 + A*i
One of the things that makes this fractal so interesting is the shape of
the dynamical system's attractors. It is not at all uncommon for a
dynamical system to have non-point attractors. Shapes such as circles
are very common. Strange attractors are attractors which are themselves
fractal. What is unusual about this system, however, is that the
attractors intersect. This is the first case in which such a phenomenon
has been observed. The attractors for this system are made up of line
segments which overlap to form an equilateral triangle. This attractor
triangle can be seen by using the "show orbits" option (the 'o' key) or
the "orbits window" option (the ctrl-'o' key).
The number of attractors present is dependant on the value of A, the
imaginary part of C. For values where A <= 1.028713768218725..., there
are three attractors. When A is larger than this critical value, two of
attractors merge into one, leaving only two attractors. An interesting
variation on this fractal can be generated by applying the above mapping
twice per each iteration. The result is that some of the attractors are
split into two parts, giving the system either six or three attractors,
depending on whether A is less than or greater than the critical value.
These are also called "Riddled Basins" because each basin is riddled
with holes. Which attractor a point is eventually pulled into is
extremely sensitive to its initial position. A very slight change in
any direction may cause it to end up on a different attractor. As a
result, the basins are thoroughly intermingled. The effect appears to be
a frothy mixture that has been subjected to lots of stirring and
folding.
Pixel color is determined by which attractor captures the orbit. The
shade of color is determined by the number of iterations required to
capture the orbit. In Fractint, the actual shade of color used depends
on how many colors are available in the video mode being used. If 256
colors are available, the default coloring scheme is determined by the
number of iterations that were required to capture the orbit. An
alternative coloring scheme can be used where the shade is determined by
the iterations required divided by the maximum iterations. This method
is especially useful on deeply zoomed images. If only 16 colors are
available, then only the alternative coloring scheme is used. If fewer
than 16 colors are available, then Fractint just colors the basins
without any shading.