On 8/9/11, Tom Karzes <karzes@sonic.net> wrote:
Me neither. There is, however, a very natural 4-dimensional set that combines both the standard Mandelbrot set and all of its corresponding Julia sets into a single entity.
Let one plane determine c0, and an orthogonal plane determine c1. The product is 4-dimensional, and the iteration is:
z(0) = c0 z(i) = z(i-1)^2 + c1
The planar slice obtained by setting c0 to 0 and varying c1 is the Mandelbrot set. The planar slice obtained by setting c1 to some constant and varying c0 is the Julia set corresponding to c1.
Unfortunately, having a 2-dimensional retina makes it difficult for me to visualize such a set. Being three dimensional can be very frustrating.
Tom
You wouldn't want to be (spatially) 4-D. It messes up the planetary dynamics something rotten, or so I understand. WFL