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:
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: