Quasiperiodic pattern from eight waves and the Ammann-Beenker tiling

Periodic structure of four waves shown in black and white together with a fitting square lattice (blue lines).

Using four waves at right angles we get a periodic structure of fourfold symmetry. A square grid of the same periodicity is easily fitted to this structure.

Now, together with an extra set of four waves rotated by 45 degrees we get a set of eight waves. They make a quasiperiodic structure of eightfold rotational symmetry as discussed in the previous post. On the other hand we can use the corresponding two square lattices to make an Ammann-Beenker tiling as shown earlier, see the post “An easy way to quasiperiodic tilings“. We now compare these two results:

Quasiperiodic structure from eight waves (in grey) together with a fitting Ammann-Beenker tiling (blue lines and yellow dots).

We see a nice agreement. The white spots that are surrounded by black are corner points of the Ammann-Beenker tiling. The rhombs and squares of the Ammann-Beenker tiling have all essentially the same pattern. For the rhombs there is only some marked asymmetry in the intensities. But we can think of the pattern we get from the waves as being an elaborate decoration of the Ammann-Beenker tiling. This decoration is a good camouflage. Obviously, this wave pattern could be produced by laser interference.

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