Title: Schottky Group Post by: Ross Hilbert on February 13, 2009, 03:50:20 PM Schottky Group (http://www.fractalforums.com/gallery/0/385_13_02_09_3_50_19.jpg) http://www.fractalforums.com/index.php?action=gallery;sa=view;id=479 A Schottky group is based on a pair of Mobius transformations (called generators) and their inverses. The Schottky group is composed of all possible ordered sets of these transformations (of any length). Each element in the group represents a composite transformation formed by applying the individual transformations in the set in the given order. Each of the generators is associated with a pair of circles called Schottky disks. The disks and their images under all the transformations in the group are rendered to form the fractal. Schottky groups and the algorithms used to display them are described in great detail in the book "Indra's Pearls, The Vision of Felix Klein" by David Mumford, Caroline Series, David Wright. This image was generated by the Fractal Science Kit fractal generator using the built-in Schottky Group Orbit Trap. Note that an additional non-linear transformation was applied to the fractal to form the final image. For additional information about Schottky groups see: - http://klein.math.okstate.edu/IndrasPearls/ For additional information about the Fractal Science Kit fractal generator see: - http://www.fractalsciencekit.com/index.htm |