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Extraction of strings
The distribution of digits here differs from the previous case.
Before filtering, each
has probability
of being followed by
the string
, probability
of being followed by
the string
, and in general
of being
followed by
s and
s. So, the number of
s is
reduced by:
A symmetrical argument applies to the
s, so after filtering there
are
as many
s and
s as in the unfiltered sequence. The
area of sub-square
equals that of sub-squares
and
, and the
area of sub-square
is
that of sub-square
, so we'd like
, and
. Since the
sum to
, you have
Plugging these values into the Fractal Sequence applet gives a fairly
uniform density of length
addresses (although sub-square
seems
slightly sparse -- a bug perhaps?).
For the same reason as the truncation method, it is not possible to
``tweak'' the
so as to have a uniform density of points in each
of the addresses of length
. Indeed, this fact can be used to
prove that the resulting object is not a ``really nice'' fractal.
Next: Filter foibles
Up: Filtered chaos
Previous: Truncation of strings
Danny Heap
2001-05-18