Grandma's recipe
The book Indra's Pearls gives a special "recipe" which- generates particularly beautiful symmetric limit sets,
- contains all examples up to a global Möbius transformation,
- describes all of this using two clear (complex-valued) parameters.
and 
of the matrices describing the original Möbius transformations.
The picture is fully determined by the values of these two numbers.
The following applet shows the limit set corresponding to given values.
The real and imaginary part of these traces can be controlled using the sliders or entered in the text boxes below.
The text boxes can even cope with formulas (like e.g. "2*cos(pi/8)")!
ta.real= ta.imag=
tb.real= tb.imag=
Most of the fractals in Indra's Pearls are encoded using these two parameters. It is quite worthwhile to take the book to hand and reproduce the depicted fractals here. The optical impression can of course diverge from the pictures in Indra's Pearls, as we generate our images only using randomized iterated function systems, in contrast to the more sophisticated methods employed by the book.
Download file:
Yet another degree of freedom
Contents 