Larry Riddle, Agnes Scott College

The Sierpinski gasket can also be formed by splitting a square into four equal pieces, each scaled by 1/2 and then removing the upper right square. In the figure below, the letter "L" is used to show the orientation of the squares. The same construction is then applied recursively to each of the three remaining pieces to get a Sierpinski (right) triangle. [See a video demonstration of this construction.]

As a variation, however, we can rotate or reflect the scaled squares, thereby changing the orientation. For example, suppose we rotate the upper left square by 90° counterclockwise. This would have the following pattern.

Construction

Animation

Now we can repeat this construction on the three remaining squares. Notice, however, that the "upper right square" must be interpreted with respect to the current orientation of the square that is being subdivided. So the second iteration would look like the following.

Repeat these steps ad infinitum, always taking into account the current orientation. This yields the following fractal image.

The IFS for this fractal is given by the following three functions corresponding to the three squares from top left to bottom right.

\({f_1}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}}
{0} & {-1/2} \\
{1/2} & {0} \\
\end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}}
{1/2} \\
{1/2} \\
\end{array}} \right]\) |
scale by 1/2, rotate by 90° |

\({f_2}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}}
{1/2} & 0 \\
0 & {1/2} \\
\end{array}} \right]{\bf{x}}\) |
scale by 1/2 |

\({f_3}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} {1/2} & 0 \\ 0 & {1/2} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} {1/2} \\ 0 \\ \end{array}} \right]\) | scale by 1/2 |

Symmetry

Group

A square has 8 symmetry transformations that preserve the basic shape. These are the identity (leave it alone), counterclockwise rotations by 90°, 180°, and 270°, horizontal and vertical reflections, and reflections across each diagonal, as shown below. These 8 transformations form a finite group, the symmetry group of the square, also known as the dihedral group of order 8.

Each transformation has been labeled by a number from 1 to 8. We can apply any of these transformations to the squares in the construction above and the three remaining squares will still fit together as before. Only the orientation of each square may be different. Repeating the construction recursively on the remaining three squares will produce a fractal that is a relative of the Sierpinski gasket. We can identify each pattern in the design for the fractal by the three digits associated with the transformations applied to the three squares, with the digits read from the top left to the lower right. So the construction shown above would be labeled as 211 since the transformation used for the upper left square is a rotation by 90° (transformation 2) while no change is made to the other two squares. Click here for details on the dihedral group of order 8 and formulas for the iterated function systems.

214 |
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316 |
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184 |
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748 |
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432 |
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427 |

In these examples, the last one is totally disconnected, the penultimate one is simply connected (no holes), and the first four are connected, but not simply connected, with infinitely many holes just as with the Sierpinski gasket. There is also a fourth type that has infinitely many connected components, such as the fractal 211 shown above. See the papers by T.D. Taylor for more details about the connectivity properties of Sierpinski relatives.

Each Sierpinski relative can be described by a sequence of three digits corresponding to the transformations applied to each of the three squares in the pattern for that fractal. Because there are 8 possible transformations, there are 8x8x8 = 512 possible sequences. But different sequences might produce the same fractal. Notice that example 1 (214) and example 5 (432) above are both symmetric (with respect to a diagonal reflection). There are 8 such symmetric fractals as shown below in two counted cross-stitch designs. The four on the left all have 1 as the middle digit in the sequence corresponding to the transformations [see construction videos], while the four on the right all have 3 as the middle digit [see construction videos].

Larry Riddle, 2009/2010

Sierpinski Theme and Variations

(7 iterations on 25 count per inch fabric, 12.5" x 12.5")

Click on each picture for a larger view.

Each of these 8 symmetric fractals can be formed from 8 different sequences of the transformations on the square. This accounts for a total of 64 sequences. The remaining 448 sequences produce fractals that are non-symmetric. They come in pairs, however, with the corresponding fractals symmetric to each other with respect to a reflection across the diagonal. For example, the picture below shows the fractals for the pair (627) and (745). Since these symmetric pairs are essentially the same fractal, the total number of fractals is 8 + 224, or 232. [Counting Details].

Dimension

Each of these relatives of the Sierpinski gasket is self-similar with 3 non-overlapping copies of itself, each scaled by the factor
**r** = 1/2. Therefore the similarity dimension, **d**,
of each attractor is the same as that of the Sierpinski gasket, i.e. the solution to

$$\sum\limits_{k = 1}^3 {{r^d}} = 1\quad \Rightarrow \quad d = \frac{{\log (1/3)}}{{\log (r)}} = \frac{{\log (1/3)}}{{\log (1/2)}} = \frac{{\log (3)}}{{\log (2)}} = 1.58496$$

Properties

The article by Fillebrown et. al. develops a method for generating rules for determining which points (x,y) are or are not in a particular fractal. These rules make use of the binary representation for the coordinates x and y and use the sequence of bit-pairs of the point. See Binary Description of the Sierpinski Gasket for an example of such a rule.

Taylor and Rowley have investigated the convex hulls of the Sierpinski relative fractals. While these convex hulls can be quite complicated, for the eight symmetric relatives they are simple polygonal figures with at most 8 sides. Consequently, it is possible to form tiles and friezes by combining copies of the convex hull of a particular symmetric Sierpinski relative. For more details and some examples, see Tiling Sierpinski Relative Fractals.

- Heinz-Otto Peitgen, Hartmut Jurgens, and Dietmar Saupe.
*Chaos and Fractals: New Frontiers of Science*, 2nd Edition, Springer-Verlag, 2004 , pp230-237 (pp244-251 in the 1992 first edition). - Michael Frame and Benoit Mandelbrot.
*Fractals, Graphics, and Mathematics Education*, Mathematical Association of America, 2002, pp6-7. [Chapter 1 available online]. - Sandra Fillebrown, Joseph Pizzica, Vincent Russo, and Scott Fillebrown. "Points in Sierpinski-like Fractals," in
*The Beauty of Fractals*, Mathematical Association of America, Notes #76, pp63-74. - Dave Ryan's Fractal World, 18th September 2005. Website is no longer available.
- Taylor, T.D. "Connectivity Properties of Sierpinski Relatives," Fractals, Vol. 19, No. 4 (2011), 481-506.
- Taylor, T.D., C. Hudson, and A. Anderson. "Examples of Using Binary Cantor Sets to Study the Connectivity of Sierpinski Relatives," Fractals, Vol. 20, No. 1 (2012), 61-75.
- Taylor, T.D. "Totally Disconnected Sierpinski Relatives," Slides from talk at the CMS Summer Meeting, Halifax, June 2013.
- Taylor, T.D. "The Beauty of the Symmetric Sierpinski Relatives", Proceedings of Bridges 2018: Mathematics, Art, Music, Architecture, Education, Culture, Tessellations Publishing, 163-170.
- Taylor, T.D. and S. Rowley. "Convex Hulls Of SierpiĆski Relatives", Fractals, Vol. 26, No. 6 (2018).
- For other examples of similar gasket fractals, see Robert Fathauer's website at https://mathartfun.com/fractaldiversions/GasketHome.html.