Modified Ammann-Beenker tiling. The dual lines of the diagonal square grid have an angle of 64 degrees to the grid lines.

I simply experimented around with changing angles of the dual lines. Then I had the idea to treat the two single grids differently. For the Ammann-Beenker tiling I got thus especially interesting results. For the square grid with horizontal and vertical lines I still used right angles between the grid and the dual lines. Then I varied this angle for the second grid with diagonal lines. This squashes one half of the rhombs, which become thinner, and expands the other half making them thicker. To see better what happens, I am showing rhombs in dark blue if they belong to a star of eight rhombs and else in dark grey. Clearly, the eight-fold rotational symmetry breaks down.

Finally, with an angle of 45 degrees, all lines are either vertical or horizontal. The rhombs have disappeared or they have become squares. We get a square lattice. Its coloring is quasiperiodic and has four-fold rotational symmetry.

Square lattice with quasiperiodic coloring derived from the Ammann-Beenker tiling.

But the double grid has not changed. The stars of rhombs have now become large squares made of four smaller squares arising out of four of the eight rhombs. They are shown here in dark blue. Now, because of the self-similarity of the Ammann-Beenker tiling, these squares are the corner points of an Ammann-Beenker tiling of larger scale. It is easy to see the rhombs and squares of this tiling in the figure above. Thus this tiling of squares includes a hidden Ammann-Beenker tiling of higher eight-fold rotational symmetry.

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I had my mother make a quilt of this, shown at https://www.pinterest.com/pin/507851295450640704/ . I have found more degrees of freedom for Ammann tiling lattice, described at https://www.pinterest.com/pin/507851295458682724/ and demonstrated at https://www.pinterest.com/pin/507851295458682717/ . I enjoy the patterns formed by rhombs “disappearing” and quilters might find them easier than the typical nonperiodic tilings.