## Patterns of waves with eight- and twelve-fold rotational symmetry

As discussed in the previous post “Quasiperiodic designs from superposition of waves” we get a quasiperiodic structure with eight-fold rotational symmetry using eight waves (n=8). Surprisingly, cosine waves of the same sign or alternating signs give us essentially the same result as the pattern only changes place. We have eight-fold rotational symmetry of the usual kind together with its two-color version. As before I use white for values of f above zero and black for values below zero:

Quasiperiodic pattern with eight-fold rotational symmetry resulting from eight cosine waves.

Here, we see often the same distance between the white and black spots. They correspond to the corners of squares, rhombs and octagons. How does this fit to the Ammann-Beenker tiling ?

Using twelve waves (n=12) we get different results for cosine waves of the same sign or for alternating signs:

Pattern of twelve-fold rotational symmetry obtained from twelve cosine waves of the same sign.

Twelve cosine waves with alternating sign give a twelve-fold rotational symmetry with alternating colors.

Can we find here some features of the Stampfli tiling ? Clearly, some spots indicate triangular, square and rhombic shapes.

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