Quasiperiodic designs from superposition of waves

Quasiperiodic crystals have sharp diffraction patterns with a quasiperiodic structure. Thus their atomic density is made of a corresponding set of sinusoidal waves. Inspired by these ideas I mixed waves to get quasiperiodic designs.

Now, if you want to have something with a five-fold rotational symmetry you would need at least five waves with an angle of 360/5=72 degrees between them. Similarly, you could use n waves at regularly spaced angles and see what happens. With cosine functions we put

This function f defines the two-dimensional quasiperiodic structure.The wavenumber k is a suitable scaling factor and determines the size of the figure. The coefficients a and b define the direction of the waves. They are

and

We can use the values of f  in many ways. Here we simply show black where f is below zero and white where f is larger than zero. The lines between the black and white regions correspond to zeros of f, they are its nodal lines. I am showing the resulting quasiperiodic structure for n=10 with 10-fold rotational symmetry:

Quasiperiodic structure with 10-fold rotational symmetry resulting from sinusoidal waves.

The center of perfect 10-fold symmetry (x=y=0) lies at the left above the middle. You see the same small motif with some variations throughout the image. Actually, we get also the same picture with n=5 because pairs of waves coincide for n=10.

With only 5 waves we can use sine-functions instead of the cosine, thus

and we get a quasiperiodic structure of 5-fold symmetry. It can also be seen as a 10-fold rotational symmetry which alternates colors:

Quasiperiodic structure with 10-fold rotational symmetry alternating colors.

Here again the center of perfect symmetry lies at the left in the upper half. We seem to see long nearly straight lines which are only optical illusions. Note the pentagons made up of smaller ones just as in the earlier post “A quasiperiodic tiling with pentagrams“. The same design can also be obtained using n=10 and cosine-waves with alternating sign, then

Differences should arise, if we look at the exact magnitudes of f. But right now I am  confused.

Note added later (22nd september 2012): For experimentation we can use the more versatile form

where

and s=1 or 2. Use Φo=0 to have cosine waves and Φo=π/2 to have sine waves. For rotational symmetries of an odd number N we have to use s=2, but for even numbers N we can use n=N/2 and s=1 thus needing only half the computer time. Values of m different from zero give unpredictable results of lower symmetry.

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