Quasiperiodic design with five-fold rotational symmetry in color

Wavevectors (frequency and direction) used for red, green and blue. The black dot is the origin.

An easy way to get color is using three different sets of waves to control the three basic colors red, green and blue. The green color is the brightest one and thus used for the waves of lowest frequencies. The other frequencies are higher by the golden ratio (1.618). There are two different sets. One is parallel to the set of lowest frequency (shown in blue) and the other is rotated by 36 degrees (shown in red). To get five-fold symmetry I used the sine function. Actually this gives a ten-fold rotational symmetry with a two-color symmetry because of the antisymmetry of the sine function sin(-x)=sin(x).

The lines in the figure are parallel to the axis of the surrounding five-dimensional space. They show how the wavevectors are corners of rhombs. The rhombs form a rosette inside a regular polygon with ten sides.

Quasiperiodic design of five-fold rotational symmetry. The center of perfect rotational symmetry is at the lower right.

Note that pentagrams appear in different sizes. There is a great similarity with the results of my earlier post “A quasiperiodic design with pentagrams“.

With the same set of waves we can also get a quasiperiodic design of ten-fold rotational symmetry. We simply use cosine functions instead of sine functions. Note that the cosine function is symmetric cos(-x)=cos(x). Then we would get the same waves for blue and red giving a design in magenta and green only. To get more colors I used an opposite sign for the blue waves, replacing cos(x) by -cos(x). Thus red and blue are separated as above.

Quasiperiodic design of ten-fold rotational symmetry. The center of perfect symmetry is in the same place as above.

Here we see many dots arranged as corners of a pentagon or of a decagon (regular polygon with ten sides). The decagons are overlapping, sharing a common rhomb. This rhomb is part of the dissection shown in the figure for the wavevectors.

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