Trying to get something new I used a grid of greater squares of length L=sqrt(2.) and a distance of 1 between their centers. This makes overlapping squares. Using two such grids as before (see “An easy way to quasiperiodic tilings“) resulted in these points of a tiling with 8-fold rotational symmetry:
Now, we have to join these dots by straight lines to make tiles. We use always the same length to get regular polygons and rhombs as tiles. We could get many small octagons if we use the smallest distance between the points. But this leaves out many points and gives us incomplete octagons. To use every point in the tiling we need a larger length, that joins every point with at least another point. This gets us overlapping octagons and squares. Thus I use semitransparent color to get:
The light blue areas are just covered by a single octagon. Two overlapping octagons give a dark blue shade. Note that two overlapping squares are at the place of smaller octagons. Small white gaps arise in places without octagons or squares.
We compare this picture with the original Amman-Beenker tiling and see that the overlapping squares are actually corner points of the Amman-Beenker tiling. Thus we can compare its squares and rhombs with parts of this picture. The white gaps appear in the squares of the tiling, which correspond always to identical parts of this picture. The rhombs of the Amman-Beenker tiling however have not always the same kind of decoration here. They can have two, three or four dark areas. Thus this new tiling is almost equivalent with the Amman-Beenker tiling, but it has same subtle variations.