Quasiperiodic designs and moiré patterns

Superimposing two square grids which are rotated by a small angle we get a strong moiré pattern, see a very interesting Wikipedia article. This pattern is roughly periodic. It weakens if the angle becomes very large. At an angle of 45 degrees there is no noticeable moiré effect and thus the grid for the Ammann-Beenker tiling is rather uniform.

Now, the grid for the tessellation of Octagons and Squares, see my earlier post “Doubling the tesselation of Octagons and Squares“, has added diagonal lines. Thus there are parallel lines in both single grids of the quasiperiodic tiling. Their periodicities are different by a factor of sqrt(2)=1.41 resulting in moiré patterns:

Double grid for the quasiperiodic tiling of eight-fold symmetry with octagons, squares and rhombs. Note the moiré pattern.

The moiré is quite atypical. We see darker lines with different spacings distributed quasiperiodically. Nearly white spots appear. Often, they are corner points of octagons. This vaguely suggests some quasiperiodic designs and tilings.

Note added 2 days later:

Obviously, bright spots arise if the two single grids are “in phase” similarly as in the projection method. Thus bright spots here correspond often to corner points of the tiling. But the concept of a “bright spot” is not well-defined and doesn’t give us a tiling. Extending the idea we can get quasiperiodic designs from mixing sinusoidal waves, see some of the next posts.

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