We reconsider the projection method for the Ammann-Beenker tiling, see the post “An easy way to quasiperiodic tilings” and have a look at the square lattices. Each can be made with two sets of periodically spaced parallel lines at right angles. We used two square lattices at angles of 45 degrees to get the grid for the Ammann-Beenker tiling. Thus we can think that this grid is the sum of four sets of parallel lines at angles of multiples of 45 degrees. We can generalize this procedure.
Using n sets of lines at angles of (180/n) degrees we get a grid with a 2n-fold rotational symmetry. Then we look for stars with 2n points resulting from these lines. The centers of these stars are then points of a quasiperiodic structure.
Now, to give an example I am using n=5. With five sets of lines at angles of 36 degrees I get a ten-fold rotational symmetry. All this lines put together looks like a big mess. But programming the computer to find stars with ten points I got this:
Here, the red lines highlight the ten-pointed stars and the small blue circles show their centers. We recognise immediately, that same of these centers lie at the corners of pentagons. At a larger scale we see that we have indeed a quasiperiodic structure of ten-fold rotational symmetry:
Similarly as for the Ammann-Beenker tiling we have difficulties to appreciate this result seeing dots only. We need lines and colored tiles. This I will do later. In the following I will first show how the computer finds the points.