From two-color to single color ten-fold rotational symmetry

In “Quasiperiodic designs from superposition of waves” I showed two different designs with ten-fold rotational symmetry, one with a two-color symmetry and another with a single color symmetry. In spite of this great difference they are actually cross-sections of the same periodic pattern in 5-dimensional space.

I am getting mathematical, trying to explain this for the general case of an n-dimensional space. The periodic function f is simply

.

The drawing plane is given by the points

where the parameters x and y determine the points inside the plane. The vectors a and b fix the orientation of the plane and c its position. To get a quasiperiodic design with n-fold rotational symmetry we place the drawing plane perpendicularly to the diagonal in space, thus

and

In the drawing plane we then see the function values

For n=5 and c=0 we get a quasiperiodic design with a two-color ten-fold rotational symmetry. If all components of the vector c are equal to π/2 we get a simple ten-fold rotational symmetry as the sine-functions are effectively becoming cosine-function. These are the two cases shown in the earlier post. In the animation below I am using intermediate values for the components of c, going continuously from 0 to π/2. Thus I get a continuous transition between the two symmetries.

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