Doubling the tesselation of triangles

Using the dualization method we get a the regular tiling of equilateral triangles from a grid of regular hexagons. Each point of the grid where three lines makes a triangle. Its sides are perpendicular to the grid lines.

Two hexagon grids. Note the 12-pointed star resulting from two hexagons in the lower left.

To get a tessellation of twelvefold rotational symmetry we use the dualization method and two hexagon grids rotated by 90 degrees with respect to each other. This is the same grid as for the projection method. But now each crossing of lines gives a new tile. We obtain the same Stampfli tiling as with the projection method.

Quasiperiodic tessellation resulting from the grid above. The star of two hexagons makes the star of 12 rhombs and triangles at the lower left.

This is due to the self-similarity of this tiling: the centers of the stars of twelve rhombs are the corner points of the tiling with larger lengths. Note that each pair of hexagons in the grid that forms a 12-pointed star makes a star of twelve rhombs in the tiling. This is exactly the condition used in the projection method to make one single point of the tiling.

The dualization method has been used by Stampfli to discover this tiling. However the projection method is much easier to use. Unfortunately, the iterative method cannot generate this tiling.

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