$n$ -Flake Variations

Year: 2024 Authors: Steven Wilkinson; Blake Settle

Core claim

Sierpinski n-flakes can be extended in two useful ways—by adding a center polygon or by rotating children—and these variations produce distinct geometric and aesthetic behaviors, especially in connectivity and visual patterning.

Topics

Sierpinski n-gons, fractal geometry, polygon recursion, connectivity, mathematical art, pattern generation

Domains

fractal geometry, polygonal geometry, symmetry, iterated constructions, topology/connectivity, mathematical art, generative art, pattern design

Methods

recursive polygon construction, geometric variation analysis, rotation parameterization, visual comparison of iterations, artistic composition with software-generated patterns

Media

regular n-sided polygons, software-generated fractal images, canvas, acrylic paints, printed pattern backgrounds

Paper text

The text below is the locally extracted OCR/Markdown version of the paper. Raw PDF files remain local and are not published here.

Bridges 2024 Conference Proceedings

$n$ -Flake Variations

Steven Wilkinson ${}^1$ and Blake Settle ${}^2$

Dept. of Mathematics and Statistics, Northern Kentucky University, Highland Heights, KY, USA ${}^1$ wilkinson@nku.edu, ${}^2$ blake.settle123@gmail.com

Abstract

Sierpinski $n$ -gons adapt the idea of Sierpinski’s triangle to create fractals from regular $n$ -sided polygons. We develop two variations on this idea. One variation is to fill the center hole with a similar polygon, and the other is to rotate each child polygon a fixed angle from its parent.

Introduction

Sierpinski’s fractals have proven popular both mathematically, as their study provides a good introduction to fractal geometry, as well as artistically since their variations produce several esthetically pleasing patterns. Just a few of the types of Sierpinski variations can be seen in the beautiful works of Aceska and O’Brien [1] and several publications by Taylor [7][8]. We will focus on the variations called Sierpinski $n$ -gons [6], also known as $n$ -flakes.

$n$ -Flakes

To construct an $n$ -flake, begin with a regular $n$ -sided polygon. On the inside of each vertex of this polygon, attach a similar polygon. Furthermore, the new polygons at adjacent vertices of the original should just touch with no overlap. The $n$ -flake is the fractal obtained by repeating this motif on each new polygon as shown in Figure 1.

Figure 1: 5-flake (left) and 6-flake (right): original polygon, one iteration, and several iterations of each.

This construction gives the usual Sierpinski triangle for $n=3$ , but it does not give the standard Sierpinski carpet with $n=4$ . In fact, for the square the result of this process is not even a fractal because there is no hole. See Figure 2.

Figure 2: 3-flake (left) and 4-flake (right): original polygon, one iteration, and several iterations of each.

Wilkinson and Settle

Center $n$ -Flakes

The center 5-flake is a well known construction. See modern day mathematician-artists, such as [3] and [5], who have used the 5-flake iteration idea without removing the center pentagon to produce the pentaflake fractal shown in Figure 3. Both of the above two citations credit sixteenth century artist Albrecht Dürer [4] with the idea of tiling the plane with pentagons and rhombuses. For general regular polygons, the hole shape is a similar polygon only when $n=3$ and $n=5$ . However, one can find a hexaflake with center in many places. Some of the more interesting occurrences are in atennas [2], and in a projection of a hexagonal bipyramid fractal, Hideki Tsuiki’s “H fractal,” into the plane. He made a movie of several projections [9]. The hexaflake appears at time 4:10 in the video. Both the center pentaflake and center hexaflake are shown in Figure 3.

Figure 3: Pentaflake (left) and hexaflake (right): one iteration and several iterations of each.

On the other hand, it is harder to find pictures of these center $n$ -flakes for n > 6 . There are a couple of issues to consider. One is that for pentaflakes and hexaflakes, the center polygon is the same size as the outer polygons, but for n > 6 , the center polygon is larger than the sibling polygons. Also, it is possible to place the center polygon in different ways. Even more, placement is affected by whether $n$ is even or odd. For $n$ odd, as with the pentaflake, the outer polygons have edges in the direction of the center. A simple placement is to align each edge of the center polygon to each inward facing edge of the outer polygons. For $n$ even, just like with the hexaflake, the outer polygons have vertices pointed towards the center. The simplest placement is to match the center polygon vertices to the vertices of the outer polygons. A 7-flake and an 8-flake are shown in Figure 4.

Figure 4: Septaflake (left) and octaflake (right): one iteration and several iterations of each.

Rotated $n$ -Flakes

In another variation we rotate each child polygon a fixed angle from its parent polygon. This new variation can be used with either the original $n$ -flake with no center or the center variation. See examples in Figure 5.

• For a regular polygon of $n$ sides, a rotation about the center of $2\pi /n$ radians takes a vertex to the next one. Therefore, all the possible shapes come from angles in the interval $[-\pi /n,\pi /n]$ .

n-Flake Variations

• Furthermore, the shape that uses the negative of an angle is the mirror image of the shape for the original angle. One can focus on angles in $[0,\pi /n]$ to see all the essential shapes.

Figure 5: Rotated $n$ -flakes: one and many iterations with the original polygon outlined in blue. Left: 4-flake no center, angle $\pi /7$ . Right: 5-flake with center, angle $\pi /5$ .

Connectivity

The non-rotated fractal has a different aesthetic appeal than its rotated cousins. A large part of this difference has to do with the connectivity properties of these fractals. To simplify this discussion, focus on the original $n$ -flakes that have no centers. For the non-rotated fractal, the resulting object is connected. This is clear since each child shares a vertex with its parent as well as a vertex with each adjacent sibling. These particular vertices then remain in all subsequent descendant polygons so that they are points in the fractal. However, rotated fractals are not connected since child polygons no longer share vertices with either their parents or any of their siblings. See the two-iteration pentaflakes in Figure 6. Parent pentagrams are outlined in blue.

Figure 6: The left pentaflake is unrotated, while the right one is rotated $\pi /7$

Artistic Application

The software we used to generate the center variation $n$ -flakes allows for each middle polygon in the last iteration to be colored differently than the outer polygons. See Figure 7. The second author is an artist who

Figure 7: A center hexaflake with centers colored pink.

often begins with a patterned background on which he paints. He started with a two-iteration 3-flake with

Wilkinson and Settle

center that is rotated, where he colored the center white, the background color. He printed multiple copies of this 3-flake, interleaved, on a canvas, and then painted this canvas using watered down acrylic paints so that the background pattern is visible. See the 3-flake and painting in Figure 8.

Figure 8: Left: one rotated 3-flake with white centers. Right: Painting with 3-flake pattern background.

Acknowledgments

This work grew out of a topics course in Mathematics and Art taught during fall 2023 at Northern Kentucky University by the first author with Professor Lisa Holden. The second author was a member of the class. We wish to thank one of the reviewers for letting us know about Hideki Tsuiki’s movie [9] of a 3-D fractal that has a hexaflake with center as a projection.

References

[1] R. Aceska and B. O’Brien. “Artistic Excursions with the Sierpinski Triangle.” Bridges Conference Proceedings, Linz, Austria, July 16-20, 2019, pp. 537-540. https://archive.bridgesmathart.org/2019/bridges2019-537.#gsc.tab=0. [2] S. Choudhury and M. Martin. “Effect of FSS ground plane on second iteration of hexaflake fractal patch antenna.” 7th International Conference on Electrical Computer Engineering, 2012, pp. 694-697. [3] R. Dixon. Mathographics. Dover Recreational Math, 1991, pp. 186-187. [4] A. Dürer. The Painter’s Manual, translated by Walter L. Strauss, Abaris Books, Inc., N.Y., 1977, pp. 161-162. [5] L. Riddle. “Durer’s Pentagons.” https://larryriddle.agnesscott.org/ifs/pentagon/Durer.htm. [6] S. Schlicker and D. Kevin. “Sierpinski $n$ -gons.” Pi Mu Epsilon Journal, 10, 1995, No. 2, pp. 81-89. [7] T. Taylor. “The Beauty of the Symmetric Sierpinski Relatives.” Bridges Conference Proceedings, Stockholm, Sweden, July 25-29, 2018, pp. 163-170. https://archive.bridgesmathart.org/2018/bridges2018-163.#gsc.tab=0. [8] T. Taylor. “Using Triangle Sierpinski Relatives to Visualize Subgroups of the Symmetries of the Square.” Bridges Conference Proceedings, Halifax, Nova Scotia, Canada, July 27-31, 2023, pp. 141-148. https://archive.bridgesmathart.org/2023/bridges2023-141.#gsc.tab=0. [9] H. Tsuiki. “Shadows of fractal imaginary cubes.” Bridges Conference Short Film Festival, Halifax, Nova Scotia, Canada, July 27-31, 2023. https://gallery.bridgesmathart.org/exhibitions/2023-bridges-conference-short-film-festival/hideki-tsuiki.