Tiling Sierpinski Relative Fractals

Taylor and Rowley have investigated the convex hulls of the Sierpinski relative fractals. The convex hull of a shape is the smallest convex set that contains the shape. A set is convex if for any two points in the set, the line segment between the two points is also in the set. While the convex hulls for the Sierpinski relatives can be quite complicated, for the eight symmetric relatives they are simple polygonal figures with at most 8 sides.

Symmetric
Fractals

Below are the eight symmetric Sierpinski relatives. Click on each one to see the convex hull for that fractal, or use the buttons to the left to show/hide all the convex hulls.

111 convex hull 214 convex hull 313 convex hull 412 convex hull

131 convex hull 234 convex hull 333 convex hull 432 convex hull

Tilings

Each of these convex hulls have edges with interior angles of either 45°, 90°, or 135°. Consequently, it is possible to form tiles by combining copies of the convex hull of a particular symmetric Sierpinski relative. For example, the following image shows four copies of the convex hull of relative 412 placed around a common center point. Click on the image to toggle between the four copies of the 412 fractal and the fractals with their convex hulls.

412 convex hull
[Alternative View]
[Cross Stitch Design]

This tile can now be repeated in both the horizontal and vertical directions as shown below. Click on the image to toggle between the image with and without the tile boundaries.

412 convex hull tiling

Notice that in the version without the tiles there are "holes" between the fractals (where the diamonds are located in the image with tiles). These can be filled with copies of the tiles scaled by 1/2 as illustrated in the following image.

412-tiling-level-2-example3

Click the button below to see the effect of filling in all the holes with these 1/2 scaled tiles. But there are still some smaller square holes remaining! And these can be filled with copies of the tiles scaled by 1/4. Click the button to see the effect. And the process can be continued indefinitely to fill ever smaller holes with copies of the tiles scaled by 1/8, 1/16, etc.

master

For another example, the following image shows four copies of the convex hull of relative 214 placed around a common center point. Click on the image to toggle between the four copies of the 214 fractal and the fractals with their convex hulls.

214 convex hull
[Alternative View]
[Cross Stitch Design]

This tile can also be repeated in both the horizontal and vertical directions as shown below (with a slightly different coloring). And as with the relative 412 tiling, the holes between the tiles can be filled with scaled copies of the tiles. Click on the buttons to the left examine these effects.

grid of 214 tiles

Here are examples of tiling with eight copies of Sierpinski relative 111, and four copies of Sierpinski relatives 313 and 333. Click on the tiles to see multiple copies (some scaled by 1/2) tiling the plane. For more examples, see the article by Taylor in the proceedings of the 2018 Bridges Conference.

[Cross Stitch Design]

Finally, you can even fill the holes in some of the Sierpinski relatives with copies of other Sierpinski relatives. Click for a holiday inspired example using relatives 313, 214, and 111.

View some video demonstrations of these various tilings.

Symmetries

If only the design of each Sierpinksi Relative tiling is considered and not the colors used, then the tiles exhibit the eight symmetries of the square. These are the identity (leave it alone), counterclockwise rotations by 90°, 180°, and 270°, horizontal and vertical reflections, and reflections across each diagonal. To understand why, consider how the pattern for a symmetric Sierpinski relative fractal can be described by showing the location of the subsquare that is removed from each square, which we color as white in the images below for the Sierpinski relatives 111, 214, 412, and 313. Next to each basic pattern is what happens when the basic pattern is used to form a tile, and this clearly shows the eight symmetries. Click on each tile to see the corresponding Sierpinski relative.

111 111 tile pattern 214 214 tile pattern

412 412 tile pattern 313 313 tile pattern

Frieze Patterns

Taylor's Bridges 2018 article also gives examples of the convex hulls of symmetric Sierpinski relatives tiled together to form frieze patterns. A frieze pattern is an infinite strip with a repeating pattern that exhibits translational symmetry. There are also four other basic symmetries that can be applied to the pattern along with translation. Click on the name for an example of each symmetry.

translation (T)

Also know as a "hop".
glide reflection (G)

A reflection over a line, followed by a translation in the same direction as the line. Also know as a "step".
horizontal reflection (H)

Symmetric with respect to a horizontal line. Also know as a "jump".
vertical reflection (V)

Symmetric with respect to a vertical line. Also known as a "sidle".
180° rotation (R)

Half-turn rotation. Also know as a "spinning hop"

There are seven possible combinations for these symmetries: T, TG, TV, TR, THG, TRVG, and TRVGH. Some of these are illustrated below with symmetric Sierpinski relatives.

The following images use three different tilings of the convex hulls of Sierpinski relative 214. The first two can be classified as TV, i.e. they display translation and vertical reflection. The third has all five symmetries, so is labeled as TRVGH. Click on each image to toggle between the convex hulls and the frieze pattern.