
Frothy Basins

formulas: 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 + ai,  and  a = 1.02871376822...

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 three attractors for this system are made up of
line segments which overlap to form an equilateral triangle. This
attractor triangle can be seen by pressing the 'o' key while the fractal
is being generated to turn on the "show orbits" option.

An interesting variation on this fractal can be generated by applying the
above mapping twice per each iteration. The result is that each of the
three attractors is split into two parts, giving the system six
attractors.

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 xmfract, the actual shade of color used depends on how many
colors are available in the display 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 xmfract
just colors the basins without any shading.

