To get a better idea how the dualization method works we look at the same hexagonal tiling as before. Here is an image, where the triangles of the dual tiling are shaded:
We now add a line to the original tiling. This line passes throughout the tiling and creates new grid-points wherever it intersects grid-lines, cutting them in two. The line is cut up in small parts, each of them becomes a new grid-line. Then the regular hexagons of the grid are cut into irregular polygons. The dual of this tiling has several new polygons. If we place the points of the dual at the center of their corresponding polygons of the original tiling we get irregular tiles, but we can easily see how the original tiling and its dual correspond:
The dual tessellation of triangles is now cut up in two parts. These part are joined by quadrilaterals. We adjust the positions of the corner points, such that the lines of the dual are perpendicular to the lines of the tiling and that all have the same length. Then we get again equilateral triangles and the quadrilaterals become rhombs:
Actually, you can add any line to any tiling to get a new dual tiling, as long as the line does not coincide with a line of the tiling. The tiles of the dual tiling appear in the new dual tiling and rhombs are added. I won’t give you a formal proof, but in the following posts you will find the computer code that does this.