Finding polygons

In the earlier post “projection method for 10-fold rotational symmetry” I have shown how to get a quasiperiodic design of dots from sets of parallel lines. You can easily recognize dots that are corners of regular pentagons. But how do we get that with the computer ?

We do similarly as for the Ammann-Beenker tiling, see “Putting the dots and lines together“, and search for lines connecting the points. Here, the smallest distance between the points is the side of the pentagons and thus seems to be the right length of the lines.

In other cases, we rather see rhombs. Their small diagonal is this distance. Basic trigonometry gives that the length of the sides of the rhomb is larger than the small diagonal by 0.5/sin(π/2n) for 2n-fold rotational symmetry with rhombs of an acute angle of 360/(2n) degrees. This increases the length of the lines.

For the chosen length we let the computers search for all pairs of points that have this distance. These pairs are then connected by lines using the code of the class TPoint and the class Tiling. This gives us a quasiperiodic network. The meshes of the network are polygons of the quasiperiodic structure.

To find the meshes or polygons we use that the connections are ordered according to their angles. We use a connection to go from one point (or node) to another one. Arriving at the new node we choose its next connection to the left to go to a further node. Then we continue until we come back to the starting point. If all the angles between the connections have the same value we got a regular polygon. If we got alternating twice the same acute and obtuse angles then we have a rhombus. This is easy to program, see the future post “Finding polygons – the code“.

Here is the result of this procedure. Pentagons are shown in orange and decagons in green. The small white circles are single points. I am using a larger scale than in “projection method for 10-fold rotational symmetry“:

pentagonsymmetryNote that the center of perfect 10-fold rotational symmetry lies near the upper right corner. You can discover an inflating sequence of pentagonal structures going to the lower left. Obviously, this is not a tiling because of gaps.

At first sight this structure seems to be self-similar, but this is not so evident as for the iterative tiling with pentagons, see “Quasiperiodic tiling with pentagons“, or pentagrams, see “A quasiperiodic tiling with pentagrams“.

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