This construction is similar to the construction of the Sierpinski
gasket by scaling a triangle and translating three copies. In this
case we want to scale a pentagon $P_0$ and translate five
copies. The five smaller pentagons (shown in red) are placed so that
they fit inside the larger pentagon (outlined in black) as
illustrated in the following figure.
Now repeat these steps on the new set $P_1$. Here are the
first three iterations.
The Sierpinski pentagon is the limiting set for this construction.
The scale factor for each function is
\(r = \frac{{3 - \sqrt 5 }}{2} = 0.381966\).
The following IFS consists of a scaling by $r$ and an appropriate translation for each of the five functions [
IFS Details].
\[\begin{align}
{f_1}({\bf{x}}) &= \left[ {\begin{array}{cc}
{0.382} & 0 \\
0 & {0.382} \\
\end{array}} \right]{\bf{x}} \\
{f_2}({\bf{x}}) &= \left[ {\begin{array}{cc}
{0.382} & 0 \\
0 & {0.382} \\
\end{array}} \right]{\bf{x}} + \left[ {\begin{array}{c}
{0.618} \\
0 \\
\end{array}} \right] \\
{f_3}({\bf{x}}) &= \left[ {\begin{array}{cc}
{0.382} & 0 \\
0 & {0.382} \\
\end{array}} \right]{\bf{x}} + \left[ {\begin{array}{c}
{0.809} \\
{0.588} \\
\end{array}} \right] \\
{f_4}({\bf{x}}) &= \left[ {\begin{array}{cc}
{0.382} & 0 \\
0 & {0.382} \\
\end{array}} \right]{\bf{x}} + \left[ {\begin{array}{c}
{0.309} \\
{0.951} \\
\end{array}} \right] \\
{f_5}({\bf{x}}) &= \left[ {\begin{array}{cc}
{0.382} & 0 \\
0 & {0.382} \\
\end{array}} \right]{\bf{x}} + \left[ {\begin{array}{r}
{ - 0.191} \\
{0.588} \\
\end{array}} \right] \\
\end{align}\]
The Sierpinski pentagon consists of five self-similar pieces corresponding to the five functions in the iterated function system.
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The Sierpinski pentagon is self-similar with 5 non-overlapping copies of itself, each scaled by the factor
$r \lt 1$. Therefore the similarity dimension, $d$,
of the attractor of the IFS is the solution to
\[\sum\limits_{k = 1}^5 {{r^d}} = 1 \quad \Rightarrow \quad d = \frac{{\log (1/5)}}{{\log (r)}} = \frac{{\log (5)}}{{\log \left(2/(3 - \sqrt 5 )\right)}} = 1.67228\]
Sierpinski n-gons
By changing the initial polygon and scale factor, it is possible
to generate other types of fractals. The idea is to start with a
regular $n$-sided polygon, then scale the polygon by a factor
$r$ so that $n$ copies of the scaled polygon exactly fit
inside the original polygon. The Sierpinski pentagon is such an
example with $n = 5$ and $r = 0.381966$. The figure below
shows the limit set for a hexagon as the initial polygon.
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Durer's pentagons
In 1525, the great Renaissance artist Albrecht Durer published
The Painter's Manual in which he illustrated various ways for
drawing geometric figures. One section is on "Tile Patterns Formed by
Pentagons". Durer's description forms the basis for the attractor of
an iterated function system.
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