A long time ago in “Crazy graph paper” I have shown a morphing between the square lattice and the quasiperiodic Ammann-Beenker tiling of eight-fold rotational symmetry. We can do similar morphs with mapping functions using waves.

The wave vectors (1,0) and (0,1) define designs of square symmetry that are aligned with the coordinate axis. We get the same designs rotated by an angle of 45 degrees from the wave vectors (1/√2,1/√2) and (1/√2,-1/√2). All wave vectors together give quasi-periodic designs with 8-fold rotational symmetry. A morphing arises if we combine these wave packages with varying weights.

To create an example I used a constant coefficient for the first set of waves. The weight for the second set vanishes at the border of the image and is equal to the weight of the first set at the center. Thus at the border we get a square lattice and the center has locally eight-fold rotational symmetry:

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