Quasi-Fuchsian Groups

!header Quasi-Fuchsian Groups !begin-text-block Popularized in the book Indra’s Pearls, quasi-Fuchsian groups are discrete groups of conformal (i.e. angle-preserving) transformations of the complex numbers — but only ones where the limit set is contained in a Jordan curve. A more down-to-earth explanation is that a quasi-Fuchsian group contains a handful of functions, and applying those functions and their inverses repeatedly can make a given starting point converge somewhere. If we plot all of those somewheres, we often end up with a striking picture — even the simplest case is an Apollonian net, a fractal created by filling a large circle with smaller ones. !wilson !end-text-block !begin-text-boxes resolution 500 Resolution !end-text-boxes

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!text b j A note: high resolutions can lead to somewhat disappointing results, since the limit set is always about a single pixel thick. To produce a brighter image without losing too much quality, apply one or two passes of GIMP’s dialate filter. !begin-text-boxes high-res-resolution 1000 Resolution max-depth 100 Iterations max-pixel-brightness 50 Quality !end-text-boxes !begin-text-buttons download l Generate !end-text-buttons !footer