We continue with the dualization method. Here again I am not presenting something really new. I am just trying to put things together.
It was probably Socolar who first used the dualization method to get a quasiperiodic tiling of 12-fold rotational symmetry. A grid of equilateral triangles results in the regular tessellation of hexagons with 6-fold rotational symmetry. Each point of the grid gives a hexagon. As before, two such grids, rotated and superimposed, make a tiling of 12-fold rotational symmetry. It is known as the Socolar tiling.
Note that hexagons come always in pairs. This is typical for the Socolar tiling. It has twelvefold rotational symmetry because you can take any finite part of the tiling, rotate it by 30 degrees and you will find the same arrangement of tiles somewhere near by. The distance you have to go is roughly proportional to its size.
Now we want to see whether it is self-similar. Similar as for the Stampfli tiling with the stars of rhombs we look out for a large symmetric arrangement of tiles. This is easily found as the rings of twelve hexagons, squares and rhombs put together.
We now check the double grid and look at the points where the lines of a single grid meet. They are the centers of hexagons of the grids. The length of the sides are equal to the distance between the points. A ring arises if two points are so close together that their hexagons form a complete twelve-pointed star.
This is similar as for the Stampfli tiling, see “Doubling the tessellation of triangles“, but the lengths are different. For the Stampfli tiling the sides of the hexagons are smaller by 1/sqrt(3). Thus we can use the same projection method, see “How to calculate the corner points …“, to find the centers of the rings. We have simply to use L=sqrt(3) instead of L=1. The resulting points define a tiling, which is quite different from the Socolar tiling
This tiling has irregular hexagons as tiles. They resemble shields and thus it is called the shield tiling. It is somehow equivalent to the Socolar tiling, as discussed by Gaehler. But the shield tiling itself is self-similar if one looks at rings of twelve hexagons, twelve squares and 36 triangles. Their distance is 7+4*sqrt(3)=(2+sqrt(3))*(2+sqrt(3))=13.9 times the side of a tile. This is the same self-similarity ratio as for the Stampfli tiling. In the image above you can thus discover such rings at the corners of a square (upper right), triangle (upper left and in the middle) and an irregular hexagon (at the left).