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Cell[CellGroupData[{

Cell["The ShowIFS command", "Section",
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Cell[CellGroupData[{

Cell["The basic algorithm", "Subsection",
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require a different algorithm to get a nice approximation. One such example \
is the Sierpinski pedal triangle. Given an acute triangle, the first step in \
the construction of the Sierpinski pedal triangle is to drop a perpendicular \
from each vertex to the opposite side. The points of intersection of the \
sides of ",
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the interior, perform the same procedure on the darker remaining triangles, \
and iterate. This leads to a fractal figure called the Sierpinski pedal \
triangle. In fact, the three dark triangles are similar to the whole; thus, \
the Sierpinski pedal triangle is a self-similar set. The  package function ",
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three vertices. Here is the pedal IFS for the triangle above."
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Cell[TextData[{
 "Alternatively, the ",
 StyleBox["ShowIFS", "Input"],
 " command may be called with a real second parameter that specifies a \
\[OpenCurlyDoubleQuote]tolerance\[CloseCurlyDoubleQuote]. Essentially, \
triangles will be subdivided as long as their \[OpenCurlyDoubleQuote]size\
\[CloseCurlyDoubleQuote] exceeds the tolerance. This yields a much nicer \
approximation to the set."
}], "Text",
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generalized to arbitrary iterated function systems. Rather than applying \
every function in the IFS to a given initiator and iterating, we randomly \
apply the functions one at a time and iterate. With each application of a \
function from the IFS, the generated point should get closer to the attractor \
and the randomness of the algorithm will (hopefully) generate a somewhat \
uniform distribution of points over the attractor. Some sets are more \
naturally generated using a stochastic algorithm."
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probabilities and one to compute dimension."
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Cell[TextData[{
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points using the random algorithm (assuming the IFS consists of similarities \
and satisfies the open set condition).  Not surprisingly, the IFS for the \
Sierpinski gasket requires a uniform probability distribution for the three \
functions."
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Cell["\<\
The IFS for the Sierpinski pedal triangle example, however, requires \
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While we have an equation that the dimension of our pedal triangle example \
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