A Mobius transformation is a map from the complex plane to itself (or Riemann sphere if you prefer) of the form f(z) = (az+b)/(cz+d) where a,b,c, and d are nonzero (complex) constants such that ad-bc is nonzero. Under composition, these transformations form a group which is isomorphic to the group SL(2,C). The idea is to look at subgroups generated by a pair of elements of this group. Each element is visually represented by plotting its fixed point. For certain pairs of elements, this generates a very elaborate picture.

I worked on this project with a student in Fall 2007 (it was his senior seminar project). All of this is described in considerable detail in the book Indra's Pearls by David Mumford, Caroline Series, and David Wright. We implemented the Depth First Code as outlined in that book.

Source code (it is not commented): MobiusDFS. If you do take a look at the source, the two key parameters to play around with are ta and tb. Most of the interesting pictures happen when those values are around 2.

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