Agnes Scott College
Larry Riddle, Agnes Scott College

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 first iteration for a hexagon and the limit set.


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.

pentagonScalingHalfSmall   hexagonScalingHalfSmall