In my earlier post “Doubling the tessellation of hexagons and triangles” I found a tiling with a rather high density of stars of twelve rhombs. I called it the L2-tiling.
To see if the L2-tiling is self-similar I let the computer draw a large piece of it. Then it searched for all stars of twelve rhombs and found their centers. They in turn gave the corner points of a new tiling:
This tiling seems to be the Stampfli tiling. Thus the L2-tiling is not self-similar and it is somehow related to the Stampfli tiling.
Is self-similarity important ? Well, there are many quasiperiodic tilings that are not self-similar. These do have not fractal character. They cannot be made with iterative methods because the iterative method always makes self-similar structures. Still these tilings are interesting. And studying the self-similarity may result in a new tiling if there is in fact no self-similarity.