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.