Tessellation of four-fold rotational symmetry made of triangles and squares.

Equilateral triangles do not fit well with four-fold rotational symmetry. Yet there exists a semiregular tessellation having both. It has unusual symmetries. The centers of the rotational symmetries are at the center of the squares. It is mirror symmetric but the mirror lines pass only through the triangles. Thus mirror symmetry and rotational symmetry are separated.

With the dualization method I doubled the this tessellation to get a quasiperiodic tiling with eight-fold rotational symmetry.

Quasiperiodic tiling with eight-fold rotational symmetry I got from doubling the tiling of squares and triangles.

Here we have some complicated symmetries. A bit to the right of the center you see a motif with eight-fold rotational symmetry if you neglect the subdivision of the octagon. It has a swirling shape and is not mirror symmetric. But looking sufficiently far away (outside the screen) you could find mirror images of this motiv.

This tiling also contains its own mirror image. In fact, there are two centers of perfect eight-fold rotational symmetry in this tiling. They are mirror-symmetric to each other but they are infinitely far away from each other. But then this tiling is not enantiomorphic as the twisted tiling of octagons squares and rhombs I discussed earlier.

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