## Morphing the tiling of octagons and squares – a new twist

In an earlier post I have shown how the tiling of octagons and squares  transforms to the Ammann-Beenker tiling by using different lengths for the dual lines.  Here I am presenting another modification of the dualization method. The dual lines do not have to be perpendicular to their generating grid lines. Instead, they can have any angle. Changing the angle by the same amount for all lines simply rotates the resulting tiling. This is trivial.

Really interesting are grids with distinct sets of grid lines. An example is the grid of the tessellation with octagons and squares with the sets of grid lines shown in blue or black. We rotate the dual lines to the right or to the left depending on the color of their grid lines. First we look at the single grid. The sides of the octagons belong to both sets and twist in different directions. Thus the angles between the sides of the octagons change. Some angles become larger and others smaller. This distorts the octagon. It becomes a large square and then a star. The square remains unchanged, because all its sides belong to blue grid lines.

Morphing the semiregular tessellation by rotating the sides of its tiles.

Note that after this distortion, the tessellation is not mirror-symmetric anymore. This is important when we double the grid to get the quasiperiodic tiling.

Morphing the quasiperiodic tiling by rotating the sides of its tiles.

Look at the tiling around the rings of octagons. At the start there is a mirror symmetry. But changing the angles destroys the mirror symmetry. The rhombs inside the rings of octagons are mostly inclined in the same direction. We see something like wheels rotating all in the same direction. Thus there is no mirror image of these regions anywhere in the tiling. This quasiperiodic tiling is distinct to its mirror image. They are enantiomorphic. I hope that I got this right as I am not too good in crystallography.

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