Agnes Scott College
Larry Riddle, Agnes Scott College

Sierpinski Pentagon

Description


Sierpinski
Sierpinski

 

Construction
Animation

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.

Iterated
Function
System

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 [Details].

\[\begin{align} {f_1}({\bf{x}}) &= \left[ {\begin{array}{*{20}{c}} {0.382} & 0 \\ 0 & {0.382} \\ \end{array}} \right]{\bf{x}} \\ {f_2}({\bf{x}}) &= \left[ {\begin{array}{*{20}{c}} {0.382} & 0 \\ 0 & {0.382} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} {0.618} \\ 0 \\ \end{array}} \right] \\ {f_3}({\bf{x}}) &= \left[ {\begin{array}{*{20}{c}} {0.382} & 0 \\ 0 & {0.382} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} {0.809} \\ {0.588} \\ \end{array}} \right] \\ {f_4}({\bf{x}}) &= \left[ {\begin{array}{*{20}{c}} {0.382} & 0 \\ 0 & {0.382} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} {0.309} \\ {0.951} \\ \end{array}} \right] \\ {f_5}({\bf{x}}) &= \left[ {\begin{array}{*{20}{c}} {0.382} & 0 \\ 0 & {0.382} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} { - 0.191} \\ {0.588} \\ \end{array}} \right] \\ \end{align}\]

The Sierpinski pentagon consists of five self-similar pieces corresponding to the five functions in the iterated function system.

sierpinskiSimilarity
[Enlarge]

Similarity
Dimension

The Sierpinski pentagon is self-similar with 5 non-overlapping copies of itself, each scaled by the factor r < 1. Therefore the similarity dimension, d, of the attractor of the IFS is the solution to

\[\sum\limits_{k = 1}^5 {{r^d}} = 1 \quad \Rightarrow \quad d = \frac{{\log (1/5)}}{{\log (r)}} = \frac{{\log (5)}}{{\log \left(2/(3 - \sqrt 5 )\right)}} = 1.67228\]

Variations

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.

hexagon
[Enlarge]

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.


[Enlarge]

References

  1. Colthurst, Thomas. "Fractal Polytopes," preprint, 1992. [Available in postscript format at http://thomaswc.com/resume.html]
  2. Durer, Albrecht. The Painter's Manual, translated by Walter L. Strauss, Abaris Books, Inc., N.Y., 1977.
  3. Jones, Juw. "Fractals Before Mandelbrot-A Selective History," in Fractals and Chaos, Crilly, Earnshaw, and Jones, Editors, Springer-Verlag 1991.
  4. Schlicker, Steven and Kevin Dennis. "Sierpinski n-gons," Pi Mu Epsilon Journal, 10 (1995), No. 2, 81-89. [Available at Pi Mu Epsilon Journal Past Issues]