Often, you can put together two periodic patterns of different length and you get a new pattern, which is periodic too. The length of the period of the joint pattern is the least common multiple of the period lengths of the two patterns. As an example, if one length is 6 cm and the other is 10 cm, then the period of the joint pattern is 30 cm. Then, the ratio of the lengths has to be a rational number which is the ratio of two integers. In our example it is 10cm/6cm=5/3=1.6666…
Let us now look how the quasiperiodic Amman-Beenker tiling is built. We use the post “An easy way to quasiperiodic tilings“. Where we have joined two square grids with distance 1 between the centers of the squares. One has the sides of the squares parallel to the x-axis and thus a period length of 1 in this direction. The other has the diagonals parallel to the x-axis and thus a period length of sqrt(2.), which is also the ratio of the period lengths. But the square root of 2 is an irrational number and cannot be approximated by any rational number. Thus this pattern of the square grids and the resulting tiling are not periodic.
But we might want to have a periodic tiling for printing purposes and we approximate the square root of two by a close rational number, such as 142/100. Then we change the distance between the centers of the squares of the second square by a corresponding factor. This factor is close to one, in this case 1.42/sqrt(2)=1.0041. The error is less than one percent. Thus we first expect only a small change in the tiling and hope that much of the eight-fold rotational symmetry would stay.
But my results show that the tiling is strongly perturbed. In addition, gaps arise and tiles can overlap:
Surprisingly, only the small stars of eight rhombs remain from the eight-fold rotational symmetry. The resulting tiling depends strongly on the relative position of the two square grids. In this case, squares and gaps with the shape of irregular hexagons form large straight lines. They correspond to the periodicity of the tiling.
Here we see that a small change can have a dramatic effect.