## Another way to see designs from waves with 12-fold rotational symmetry

Three checkerboard patterns in red, green and blue.

In the last post I have shown the superposition of two wave patterns with hexagonal symmetry. This yields designs with 12-fold rotational symmetry related to the Stampfli tiling. Now, similarly as in “Quasiperiodic designs from waves and higher dimensional space” we get a two-dimensional hexagonal wave pattern from a cross-section of a simple sine-wave pattern f(x)=sin(u)+sin(v)+sin(w) in three-dimensional space. Thus the patterns presented in the last post could also be cross-sections of a simple sine-pattern in 6-dimensional space. Similarly, we can generate the Stampfli tiling from a projection of a six-dimensional lattice too.

This suggests that three wave patterns of four-fold rotational symmetry, rotated by 60 degrees with respect to each other give the same wave pattern as in the last post. But now we can explore different combinations and colorings. The small image shows three checkerboard patterns in the basic colors red, green and blue, rotated by 60 degrees. As in the last post, the large white dots correspond to corner points of the Stampfli tiling, but here this is much more difficult to see. Similarly, a projection from six-dimensional space is more complicated to do.

Combinations of these three patterns to a single one give us new images. Just some examples:

The product of the three basic waves determines the brightness.

The sum of the three basic waves gives the hue. Brightness results from the sum of pairwise products.

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