We can vary the dualization method in many ways. Here we play with the Ammann-Beenker tiling and use different lengths for the lines generated by the first square grid and the second square grid. This produces squares of different sizes and parallelograms instead of rhombs.
In the beginning of the animation we put the lengths due to the second grid equal to zero. Thus we get just the usual tessellation of squares. Note that the sides of the squares are vertical and horizontal. Then we increase the length of the lines obtained from the second grid and small squares and parallelograms appear. In the middle we have the usual Ammann-Beenker tiling. Finally, we again have a square tessellation, but now the sides of the squares are parallel to the diagonals.
Morphing the Ammann-Beenker tiling from and to the square tessellation.