Suppose is a random sequence use to drive the chaos game. What is
the effect of filtering
to remove certain substrings?
The simplest case would be to choose a -digit substring and remove
it. Suppose the
-digit is string is
, then this filtering is
indistinguishable from assigning
0 in generating the
sequence. For example, if
were transformations
corresponding to the full equilateral triangle, then filtering out all
s would generate Sierpinski's gasket.
I've implemented two methods of string filtering that you can
experiment with at
Fractal Sequence Applet
The first method truncates
the forbidden string: each occurrence of
has
removed. The second method extracts the forbidden string: each
occurrence of
has
removed.
For concreteness, I focus on the Full Square IFS with substring ``12''
filtered out. This works well since we have some intuition about
working with rectangular coordinates, so the addresses in the square
have the form: lower-left-sub-square has address ,
upper-left-sub-square has address
, upper-right-sub-square has
address
, lower-right-sub-square has address
. This addressing
pattern is iterated in the obvious way.