Dan Bron wrote: > So what I'm looking for are tips, techniques, or advice on how to diagnose > fractals. That is, given a series whose plot is obviously self-similar, > but whose generating function is not immediately clear, how to discover > that generating function.
Dan: I couldn't quite tell what you are after, but here's a simple example, with plot: the Cantor set. Take the interval [0,1] and remove the middle third. Continue this process on surviving intervals. What you have left is the Cantor set. The function f y below will plot the characteristic function of y iterations of this process. require 'plot' f=:3 : 0 X=.i.3^y Y=.*./"1 ] 0 2 e.~ (y#3)#:X pd 'reset' pd 'type stick' pd X;Y pd 'show' ) What one can tell from the description of the process is that the elements of the Cantor set have ternary expansions with only 2s and 0s. If you halve this expansion, you get all binary fractions, and so the Cantor set has the same number of elements as the real numbers, and in particular is uncountable. Its Lebesgue measure is zero. It is self-similar (take two copies of the Cantor set, shrink by a factor of 2, translate them, and reassemble). Now whether you can tell this from the plot is another question. Perhaps if you give me a hint, I could say more. Best wishes, John ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
