Dave Computes Fibonacci Bands in Aperiodic Patterns

In this post, I'll show that many aperiodic patterns are based on Fibonacci words. Everyone knows the Fibonacci sequence goes 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, etc. Each value is the sum of the two previous values. We can generate the sequence starting from 0 and 1, as follows.

        0
        1
1 + 0 = 1
1 + 1 = 2
2 + 1 = 3
3 + 2 = 5
5 + 3 = 8

Now, instead of treating the values as numbers to be added, think of them as strings to be concatenated. Thus, 1 + 0 gives 10, and 10 + 1 gives 101, producing Fibonacci words like this.

                   0
                   1
1        + 0     = 10
10       + 1     = 101
101      + 10    = 10110
10110    + 101   = 10110101
10110101 + 10110 = 1011010110110

Notice that the lengths of these Fibonacci words correspond to Fibonacci numbers. And each word starts with the bits of the previous word. We can think of these as converging on an infinite Fibonacci word, which begins

10110101101101011010110110101101101 ...

These sequences appear in any Penrose-based aperiodic tiling, as in this picture I drew.

Because these tiles are small and aligned with a grid, it's easy to see that they form horizontal bands. The bands shown in blue are thinner, and the bands shown in yellow and green are wider. On the left, I've marked each of the thin blue rows with a 0, and the wider rows with a 1. Reading down from the top, we get the sequence 110110101101011011, which appears in the infinite Fibonacci word from earlier.

10110101101101011010110110101101101 ...

Penrose-based tilings like this exhibit fivefold rotational symmetry, so we should be able to find these thinner and wider bands in each of the other four directions, and indeed we can. The following shows the exact same picture, but colored to highlight bands running in a diagonal direction.

Reading the bits down from the top-left, we find this sequence also appears in the infinite Fibonacci word.

10110101101101011010110110101101101 ...

These same Fibonacci bands occur in every Penrose tiling, although they can be harder to see.

When I learned of the so-called "hat" tile, I immediately noticed the same Fibonacci bands in these tilings. I've highlighted bands in yellow on this image from the authors of An Aperiodic Monotile.

With Fibonacci bands appearing in the two most interesting families of aperiodic tilings, I decided to write a program to start with Fibonacci bands and generate aperiodic patterns from them. The images below show horizontal and vertical Fibonacci bands. Black bands represent 0s and white bands represent 1s in a Fibonacci word.

Here's an image with both horizontal and vertical bands. Red is used where two 0s meet, black where a 0 and 1 meet, and white where two 1s meet.

And another with diagonal bands.

Devoting fewer pixels per diagonal band generates some nice patterns, due to round-off effects.

Here's that last aperiodic pattern again, this time using only black and white. There is a lot going on in it.

By changing the relative thickness of the original diagonal Fibonacci bands, other interesting aperiodic patterns emerge. (In both images below, black pixels occur where a 0 and a 1 meet. In the left, bands of 1s are 2 pixels wide. On the right, bands of 0s are 2 pixels wide, and a larger portion of the pattern is shown.)

So far, we've only generated patterns using bands running in 2 directions. Using bands running in 5 directions, we can generate Penrose-like aperiodic patterns with fivefold symmetry. Just as every pixel in the images above is part of two different bands, every pixel in an aperiodic pattern with fivefold symmetry is part of five different bands, as in the following image.

In the image above, 40.1% of the pixels are colored green. These correspond to places where two 0-bands cross three 1-bands. Here's a breakdown of the different colored squares in the image.

0 Bands  1 Bands  Color   Percent

5        0        black    3.1%

4        1        white    6.4%

3        2        red     12.4%

2        3        green   40.1%

1        4        blue    32.7%

0        5        yellow   5.2%

Here's the same image in black-and-white, with black corresponding to three 0-bands and two 1-bands.

And the same image again, this time with black corresponding to zero or one 0-bands and four or five 1-bands.

Clearly these are fascinating patterns. Because of how we generated them, we know they're aperiodic and exhibit fivefold symmetry. But are they Penrose patterns? By adjusting the relative widths of 0- and 1-bands, and by moving one set of bands a few pixels this way, another set of bands a few pixels that way, I arrived at the following pattern. The left image uses the earlier coloring rules. In the right image, all the blue and yellow regions have been colored white, and the rest black.

Coloring it as follows, we can see yellow stars, green boats, and blue diamonds in a clear Penrose tiling.

What about threefold symmetry and hat tilings? We'll explore those topics in the next posts.