Larry Riddle, Agnes Scott College

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 first iteration for a hexagon and the limit set.

Function

System

The scale factor **r** for an **n**-gon is

\[r = \frac{1}{{2 \; \left( {1 + \sum\limits_{k = 1}^{\left\lfloor {n/4} \right\rfloor } {\cos \left( {\frac{{2\pi k}}{n}} \right)} } \right)}}\]

where \({\left\lfloor {n/4} \right\rfloor }\)
denotes the greatest integer less than or equal to
**n**/4. The only affine transformations needed for the iterated
function system is scaling by **r** and translations. The
translations can be taken to be

\[\left[ {\begin{array}{*{20}{c}} {(1 - r)w\cos \left( {\frac{{2\pi k}}{n}} \right)} \\ {(1 - r)w\sin \left( {\frac{{2\pi k}}{n}} \right)} \\ \end{array}} \right]\]

for **k** = 1 to **n**, where **w** is the radius of the
initial polygon, that is, the distance from the center of the polygon
to each vertex. Click here for the details.

Rather than use the scaling ratio **r** given above for an **n**-gon, one could continue to use the ratio **r** = 1/2 as with the original Sierpinski gasket. In this case the scaled copies will overlap. This makes them ideal candidates to color using pixel counting. Here are two examples with a pentagon and a hexagon. Click on each for a larger view.