Stampfli-tiling and related designs from waves

Hexagonal pattern from the sum of three waves.

Three sinusoidal waves make a hexagonal pattern for f(x,y)=cos(x)+cos(x/2+sqrt(3) y/2)+ cos(x/2+sqrt(3) y/2), see the figure at left. Using this and the same pattern rotated by 90 degrees we get patterns of 12-fold rotational symmetry. I found it interesting to draw one hexagonal pattern in green and the other rotated pattern in magenta (red plus blue), as shown below. The white dots with small color borders arise from nearly matching green and magenta dots. Looking now at “A tiling of 12-fold rotational symmetry from two hexagon grids” we see that these dots correspond to the corner points of the Stampfli tiling. Thus these quasiperiodic designs from a superposition of six waves correspond to the Stampfli tiling. This is similar to the agreement between designs of eight-fold rotational symmetry obtained from waves and the Ammann-Beenker tiling, see “Quasiperiodic pattern from eight waves and the Ammann-Beenker tiling“.

Two hexagonal patterns rotated by 90 degrees with respect to each other.

We put the two hexagon waves together to get nicer images. We use the hue and brightness controls. Instead of using just sums, we can also use the product of the two waves or the absolute value of the difference or something else. Let us be creative. Just two examples:

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