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Troels Jorgensen discovered a family of quasifuchsian fractals (Kleinian limit sets) that cover the entire complex plane. This is  most easily seen by mapping the fractal to a Riemann sphere. In the image below the South Pole of the Riemann sphere (the bottom of the image) is the complex origin (0,0) and the North Pole is infinity.

Wow that's stunning! So the first one is an infinitely large ever-changing fractal? More pics please!  

i highly recommend reading "indra's pearls" as an introduction to making kleinian limit set images - there are many ways to do it, many subleties to consider in its implementation if you want to do it right (incendia does it naively for example).

essentially it's just a set of 4 iterated moebius maps with special properties.

Hi Dave,

Troels Jorgensen is discussed in a couple of the later chapters in Indra's Pearls. I have been plowing through the book a second time, as some of the math can get a bit hairy. My Uf method is based very closely on the Depth First Search algorithm discussed in the book. With my second time through the book I have been making a few revisions to my Kleinian Limit Sets formula, which is still not public as I will be making some changes to better handle the non-free groups. I works very effectively for free groups, but runs much slower, for reasons discussed in the book, with non-free groups. The current method can do a 10,000,000 deep tree search, and is remarkably fast for free groups.

Jorgensen was the first to describe a fully degenerate set with the second trace being parabolic. It is on the Maskit boundary near the 233 610 cusp, but unlike the cusps, which are found as Farey numbers on the Maskit boundary, the degenerate set is positioned at an irrational number. This set is called Troels Point in the book. I will post an image of Troels Point later today. I have to do a bit a Christmas shopping with the wife first.

By the way Dan Goldman has a good algorithm in C for calculating the Maskit boundary and double cusp positions.
 

Thanks Ron.
That's sorted out some of my study reading time, though knowing I have the new volume from the "Wheel of Time" series for Christmas my study reading time will have to wait a while as I have 12 volumes to re-read (incuding New Sping) before reading the new one

Here is another Jorgensen quasifuchsian fractal that maps the entire complex plane. This is the first one mentioned in the Indra's Pearls book. To my eye, it doesn't map as completely as the first example I posted.

Jorgensen has a special formula for generating limit sets which is described in Indra's Pearls. Here is an example using his formula: