tired of rhombs ?

Just only rhombs may become tiring. You want to have a quasiperiodic tiling of ten-fold rotational symmetry with other tiles ?

Well, we can easily find a different decoration of a tiling such as the one shown in “Dualization method for ten-fold rotational symmetry“. We use the idea of “beautifying the double grid” and put the positions of the grid points to the center of their dual polygons. The grid is then made only of a small set of different polygons. Most of them are irregular, but we get interesting designs if we draw them in different colors.

The grid points are dual points of the dual polygons. Thus we simply have to calculate the center of each dual polygon and use it for the position vector of its dual point, which is a grid point. In “Class Polygon – the code” there is a method to calculate the center of a polygon, but not its dual point. Thus we need some new code. Note that all dual lines of the sides of a polygon have the dual point of the polygon as one endpoint. Thus the dual point of a polygon is simply the common endpoint of the dual lines of two sides. It goes like this:

// find dualPoint of polygon and set its coordinates to center of polygon
 
 void setDualPointToCenter(Polygon polygon){
   polygon.dualPoint=polygon.sides[0].dualLine.commonEndPoint(polygon.sides[1].dualLine);
   polygon.dualPoint.vec=polygon.center();
 }
 
 void setGridPointsToCenter(){
   for(int i=0;i<grid.dualPolygons.polygons.length;i++){
     setDualPointToCenter(grid.dualPolygons.polygons[i]);
   }
 }

Still, this does not change the position of the grid points at the boundary of the image, because they do not have a dual polygon. To get a nice image we could simply cut off the border. Or you could first set the position vector of all grid points equal to null. Then you run the above code. Finally, you have to delete all grid polygons with undefined endpoints before you begin drawing.

The tiling with 10-fold rotational symmetry is thus changed into something like this:tiredOfRhombs

The center of ten-fold rotational symmetry lies at the upper right. Here we can use more colors because we have a larger variety of polygons.

This entry was posted in Quasiperiodic design, Tilings and tagged , , , , , . Bookmark the permalink.

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