This is a continuation of my post on Fibonacci Bands in Aperiodic Patterns. This time we'll look at using these bands to generate patterns with threefold rotational symmetry. The image on the left shows a set of horizontal Fibonacci bands. A second set of bands has been added to the figure on the right, rotated 120 degrees.
And a third set of Fibonacci bands has been added to the image below, rotated another 120 degrees.
In the image above, 48.8% of the pixels are colored yellow. These correspond to places where a single 0-band crosses a pair of 1-bands. Here's a breakdown of the pixels in the image.
0 Bands 1 Bands Color Percent
3 0 blue 7.3%
2 1 green 22.1%
1 2 yellow 48.8%
0 3 red 21.7%
Here is the same image, colored black only where three 0-bands meet (the blues in the original image).
Notice the triangles are all the same size, and they appear in two different orientations. Together, they form an isometric grid without any gaps. This only occurs when the Fibonacci bars are tuned just right. The animation below shows that when the horizontal bars are shifted upward, different sized shapes appear and disappear, often leaving gaps.
Here is that same image again, this time colored black wherever two or three 0-bands meet (the blues and greens in the original image).
Again, all the triangles are the same (larger) size, and they appear in two different orientations. Again, this only occurs when the Fibonacci bars are tuned just right. The animation below shows that when the horizontal bars are shifted upward, different shapes appear and disappear.
In the animation below, the black pixels occur where three 0-bands or three 1-bands meet (reds and blues in the original image).
There are some really interesting moments in this animation, like the ones shown below (colored to highlight shapes). Each one seems like a set of tiles with its own rules.
And what if we tune the position of the horizontal bands just right? We get the following complex fractal image (colored to show shapes). The shapes in this image range in size from a lowly dot (blue), to a triangle (magenta), to arbitrarily large intestinal shapes (like the green one we see only part of).
I discovered a wonderful and illuminating method for drawing these patterns by hand. We start with an equilateral triangle whose side-lengths correspond to a Fibonacci number. Here, I've used 8, so I'll call it an 8-triangle.
On each corner of the triangle, append a smaller triangle whose side-lengths correspond to the next smaller Fibonacci number (5-triangles, in this case).
On each corner of these smaller triangles, append a triangle whose side-lengths correspond to the next smaller Fibonacci number (3-triangles, in this case). It gets a little trickier here. Not only are the corners of each 5-triangle surrounded by 3-triangles, but the original 8-triangle is also surrounded by 3-triangles--one on each side, 5 units away.
The process continues, adding 2-triangles, and then 1-triangles.
Finally, we add a second round of 1-triangles. (Some of these will appear inside 2-triangles, but I've chosen to omit those from my drawing.)
Coloring in the 2- and 1-triangles gives us the following familiar image.
Let's return to the stage when we had drawn the first round of 1-triangles but not the second round.
This time we'll look at each of the 1-triangles. We'll add a rhombus to all three sides of the 1-triangle, producing the following propeller shape.
When we do this for every 1-triangle, we get an image like this:
And here's the same image with all the propellers colored in.
In a pattern like this, but made infinitely large (with arbitrarily large Fibonacci triangles), the black portion would form one contiguous region, and the remaining white portion would form a second contiguous region. Here is the same pattern, with different markings.
These patterns correspond perfectly to the meta-tilings introduced in the paper An Aperiodic Monotile, like the one I've drawn below.
Before we come back to the metatiles, here's an alternative to the earlier propellers. This time, I added three arms to each of the 1-triangles in the first round, like so:
Here's the pattern that results. Again, in an infinite pattern, the black lines would all form a single connected curve, and the white regions would join to form a single contiguous region.
In the next post, we'll use Fibonacci triangles to create a pattern that's intimately related to the "hat" tilings.
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