Using the wrong harmonics …

If we combine sinusoidal waves making a square pattern, f(x,y)=cos(kx)+cos(ky) with other waves of higher frequency g(x,y)=cos(a kx)+cos(a ky) we should use an integer ratio a between the frequencies to get again the same periodicity. If the ratio a is rational we get again aperiodic wave, but now with a beat pattern. An irrational ratio a gives a quasiperiodic design as a result of the beat between the two frequencies k and a times k. An example:

Mixing waves making periodic square patterns give together an quasiperiodic design. Here the ratio between the frequncies is 1+sqrt(2).

What does the idea “quasiperiodic design” mean ? Intuitively, this means that a finite part of the design is repeated approximately somewhere else. The distance between the original part and its copy increases if we want increased accuracy.

This changes if we look at waves that fit to quasiperiodic tilings and have a rotational symmetry incompatible with periodicity. If the ratio a  between the frequencies is integer we get a beat pattern in their sum. Obviously, we have again a quasiperiodic design. But it does not fit the tiling. An example:

Result of two sets of waves, each fitting the Ammann-Beenker tiling. The ratio between their frequencies is a=2. This causes beats. Their sum does not fit the tiling.

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