Charged Curve

CD6a, an example of mathematical and fractal art, a Sierpiński curve Julia set, the Julia (and Fatou) set of a rational function.

Here is a fine art print of this picture.

A Sierpiński curve Julia set is connected (see Sutherland's book, or these notes, for the definition of a connected set). As described in Peitgen et al, Section 13.5, Adrien Douady and John Hubbard made the connection between the complex dynamics of connected Julia sets and the physics of electrostatics. The idea is to consider a connected Julia set as carrying a negative electrical charge with the Fatou set being the resulting electrostatic field which acts as an attracting force field on any small positively charged particle placed outside the Julia set. For the physics to carry over properly the situation needs to be 3-dimensional and this can be done by extending the Julia set out from the plane perpendicularly, so it becomes an infinite column whose cross-section is the Julia set in the plane. This is the main mathematical idea behind the colouring method used in this picture. Whilst the colouring method is different, this fractal has exactly the same form as this one.