Agnes Scott College
Larry Riddle, Agnes Scott College

Sierpinski n-gons


Description

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.


Iterated
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.

Variations

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