In the last post “doubling the tessellation of squares and triangles” I have shown a quasiperiodic tiling with an unusual mirror symmetry. Earlier in “Morphing the tiling … a new twist” I got a tiling which is not at all mirror symmetric. Instead it has a distinct mirror image. For both I have used the dualization method. Here I use the iteration method to make a quasiperiodic tiling without mirror symmetry.

Decomposition of the rhomb destroying the mirror symmetry.

Decomposition of the square.

Some time ago I presented a tiling of octagons, squares and rhombs based on the iteration method. Then I showed how the octagons can be decomposed in squares and rhombs to do without the octagons. Thus I got a quasiperiodic tiling of squares and rhombs with rather different properties than the Ammann-Beenker tiling. Recently I noticed that one can rotate the decomposition of the octagon to get other tilings. In the case I am showing here, this destroys the mirror-symmetry of the rhomb.

This iterative scheme produces such a quasiperiodic tiling of eight-fold rotational symmetry:

Quasiperiodic tiling without mirror symmetry.

Note the rhombs that form shapes in the form of the letter S. You will not find a mirror image of this shape. Then you can also discover stars of eight rhombs surrounded by triplets of rhombs that always point in clockwise direction. Thus this tiling is not mirror symmetric. Its mirror image is an entirely different tiling.

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*Related*

You might be interested in a modification of the Ammann-Beenker tiling

that keeps one of the mirrors while turning the other into a quasiperiodic

glide line: see “Aperiodic Tilings with Nonsymmorphic Space Groups p2^jgm”,

Acta Crystallographica A44, 678-688 (1988). (E-mail me if you need a

copy; I’m the first author and currently at the University of South Florida.)