The grids for quasiperiodic tilings do not look good because a lot of irregular shapes arise from superimposing two simple grids, see the article “Doubling the tessellation of triangles“. But we can distort these double grids and get new interesting images.
First, we look at tilings of regular polygons and their generating grids. The lines of the grid are perpendicular to the sides of the tiles and connect the centers of the tiles. Second, we note that each tile of the quasiperiodic tiling corresponds to exactly one intersection point of the double grid. Thus we get new coordinates for the points of the double grid by using the centers of the tiles. This distorts the double grid and the sides of the rhombs are not perpendicular to the lines of the grid anymore:
These distorted grids look like some intricate trellis and bring to mind islamic geometric art as seen in the beautiful book “Arabic Geometrical Pattern and Design” by J. Bourgoin. From the Stampfli tiling we get:
From the Ammann-Beenker tiling we also get a nice trellis. Here we can see how the straight lines are transformed into quasiperiodically wiggling lines:
Note added later (15th February):
The lines are more easily distinguished if the the two grids are drawn in different colors, like that :