Parameter Spaces of Kleinian Groups
For an excellent overview of Kleinian groups, their limit
sets and parameter spaces, see David
Wright's page. For related sites, try following the links in Curt
McMullens's Gallery.
The Twice Punctured Torus
These pictures were created in an Internet collaboration with John Parker and Ian Redfern.
They correspond to pairs of simple closed curves on a surface called the
"twicepunctured torus", which can be thought of as the surface of a
donut that has (but does not include) two points at infinity.
Click on the pictures to get a detailed PDF version.

One quadrant of a boundary slice of the parameter space.
Conceptually this picture is related to the
Mandelbrot set, although it corresponds to an entirely different parameter
space of dynamics. The
space of all possible limit sets of twicepunctured tori is 4dimensional.
This space has a 3dimensional boundary, and this picture depicts a
projection of a 2dimensional slice of that boundary. Each "ray"
corresponds to a homotopy class of simple loops (equivalently, a simple
closed geodesic) on the surface. As the ray approaches the boundary, the
length of the loop goes to zero and the surface "pinches". At
the boundary, the surface has been pinched into a pair of triply punctured
spheres, and the corresponding limit set is a circle packing. 

A circle packing corresponding to both curves being pulled tight to zero
length on a twice punctured torus (picture by Ian Redfern). 

A near circle packing. One curve has been pulled tight to zero length,
the other curve has a nonzero length (picture by
Ian Redfern). 
The Earle Slice

A short Internet collaboration with Caroline Series and
Yohei Komori resulted in this crude and partial
rendering of one quadrant of the Earle slice, which describes a family of
quasifuchsian groups. The bottom left of the graphic is at the origin of the
complex plane. The description of the
algorithm used makes reference to the terminology of
the paper by Series and Komori.
