I am using the same basic method as for the Amman-Beenker tiling, see my post “How to find these corner points of the tiles“, but now with smaller squares. The distance between the centers of the squares remains equal to 1, but the length of their sides is just L=sqrt(0.5)=0.714. Rotating the squares around their centers by 45 degrees we get again a square grid.
The gaps between the squares have the same shape. But only the squares with a blue dot really count. To get a quasiperiodic tiling we add a second copy of the same grid, but rotated by 45 degrees. The corner points of the new tiling arise from two squares that make a star with eight points as before. Note that both squares should have a dot as center. Thus the new tiling has just half of the points of Amman-Beenker tiling.
As before (see “Twofold color symmetry in translation“), we can distinguish two different colors of the points. A larger part of the tiling has then the points
Here all rhombs have disappeared and we now have many regular octagons. They can have two distinct colors, depending on the color of the lower left corner point. Eliminating singular points I get the tesselation
Here we see some intricate combination of twofold color symmetry in rotation and in translation. Maybe some “hard-boiled” mathematician can explain us more. I am simply a “hard-boiled” hacker who hacks only his own programs and gets strange results.