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Pictures of the CCL

These images are pictures of two dimensional slices of the cubic connectedness locus. The locus (CCL for short) is a four-dimensional analog of the Mandelbrot set. Just like the Mandelbrot set parametrizes the dynamics of the quadratic family z->z^2+c, the CCL describes the cubic family z->z^3+az+b, where a and b are two complex parameters.

The pictures were generated by the program Brot, developed at the Geometry Center by Apprentice Linus Upson and Summer Institute participant Christine Heitsch. Brot provides a friendly user interface and computational tool to explore varying families of two dimensional slices of objects such as the CCL. It was used in the research of Apprentice David Ben-Zvi into the geometry of hyperbolic components, which generalize the discs and cardioids making up the interior of the Mandelbrot set. "Sunrise on Io" and "Black Hole" made it into the October 1993 issue of Scientific American.

Like the Mandelbrot set, the CCL is connected. (The images are typically not connected, since they only represent two dimensional slices which may connect in the other dimensions.) It is NOT however locally connected, as can be seen in "Sea of Mandelbrot", "Comb of Doom" and several others. Another interesting feature is the distinction between "complex" slices, which are reminiscent of the Mandelbrot set and its variants (see for example "Electric Storm", "Frosted Pane" or "Ships on the Sea"), and "real" slices, which look more like images of cantor sets and strange attractors arising in real dynamical systems. (A typical example is "Henon Lookalike".) Most pictures combine real and complex features in often overlapping configurations. (See "Cantor Whirlpool", "Jekyll and Hyde" and "Torrential Storm".)