In Oct. 1999 I prepared some slides for a
general-audience talk (Tsukuba, Japan).
In the process I looked for simple ways to make 3-dimensional renditions
of convex-hull boundaries for Kleinian groups. I settled
a public-domain ray-tracing package, to which I fed lists
of circles from Curt McMullen's
I also used David
kleinian to search through the parameter space
and produce some two-dimensional pictures.
Jeff Brock has made some nice
using these programs.
See also the book Indra's Pearls by
Mumford, Series and Wright, which is full of many really beautiful pictures,
and gives a very good accessible introduction to Kleinian groups without the baggage.
Here are a few of the preliminary pictures I made, before I settled on
a more pleasant color scheme...
limit set of a Kleinian group
near the boundary of Maskit's embedding of the
Teichmuller space of a once-punctured torus
(produced by McMullen's "lim" program),
followed by two 3-D
renditions of part of the convex hull boundary (produced by the
ray-tracing program POV-ray):
The limit set of a quasi-Fuchsian punctured-torus group, followed by
two 3-D renditions of part of the convex hull boundary (it looks more
like a slug than I expected):