In the last post “Doubling the tessellation of octagons and squares” I have used a grid of squares (in black) with diagonals (in blue). The blue lines are distinct and cannot be mapped onto the black lines by the symmetries of the grid, namely the rotation and mirror symmetries. Thus the dualization can give different lengths for the sides of polygons resulting from blue or black grid lines. The octagon then becomes irregular as its sides have two different lengths.
You see this in the following animation for the periodic tiling with four-fold rotational symmetry. It begins with length equal to zero for the lines due to the blue lines in the grid. Thus the octagons are actually squares and the squares due to the intersection of blue lines vanish. This results in a square lattice. This length due to black lines gradually increases and the length of the sides due to the black lines decreases. In the middle both lengths are equal and we get a tiling of octagons and squares. Finally the length due to the black lines vanishes and we get again squares instead of octagons. The checkerboard pattern results from the different color used for octagons and squares.
Morphing the periodic tiling of squares and octagons.
The different lengths can too be used for the doubled grid, see the next animation. Here we have first a Ammann-Beenker tiling from the degenerate octagons that seem to be squares. In the middle we see the quasiperiodic tiling with octagons, squares and rhombs presented in the last post. Finally, there is an Ammann-Beenker tiling with squares of two different colors. I do not understand yet the symmetries related to these two colors.
Morphing the quasiperiodic tiling of squares, octagons and rhombs.
To make these animations I have used Avidemux. Much thanks to the creators of this extraordinary software.