As you can see from my earlier post “kaleidoscopic images with local rotational symmetry” you cannot get seemless images of eight-fold rotational symmetry with a kaleidoscope of mirrors. This is also the case for 12-fold rotational symmetry and for other symmetries that do not allow periodic images. But we know that quasiperiodic tilings of these symmetries are possible. Then, why not try to make kaleidoscopic images on the basis of these tilings ?
This is quite easy. We simply cut the tiles into small pieces. The same section of the original image is then repeated in each piece, such that we get seemless local mirror symmetries. For regular polygons we use mirror lines going from the corner points and the midpoints of the edges to the center. This gives us triangles with a right angle. Its two sides meeting at the right angle define the mapping between the original image and the kaleidoscopic image. Rhombs are a bit more difficult to treat. We use the two diagonals as mirror lines. Additional mirror lines go out from the midpoints of the sides. They are perpendicular to the sides and end where they intersect the diagonal mirror lines. You can see this in the figure at the left. Depending on the original image one easily recognizes the tiles. But sometimes parts of the Voronoi diagram (see “The Voronio diagram of quasiperiodic tilings“) are more prominent.
Here is another kaleidoscopic decoration of the Ammann-Beenker tiling:
The Socolar tiling gives too a nice kaleidoscope:
Finally, I can’t resist using the Stampfli tiling: