In this post I am presenting a third method for producing quasiperiodic tilings. This is rather technical but also very important for the future work in this blog. Note that the projection method, the iteration method and the dualization method have all their merits. Each method gives results that cannot be obtained with other methods. In addition, one method may suggest things to do with an other method.
First we look at the dual grid of tessellations. I am just discussing the most important facts. More details can be found in Wolfram MathWorld and in Oracle ThinkQuest. For any tessellation of regular polygons we search for the centers of the polygons. Note that the center of a regular polygon is also the center of the circumscribing circle, which goes through all corners of the polygon. The lines going from the center cross the middle of the sides at right angles. The centers of the polygons can be joined by straight lines. This results in a new grid or tessellation, the dual of the original tessellation.
A regular tessellation has only one kind of regular polygons. Its dual is again a regular tessellation. The square grid has a square grid as its dual, a triangle grid has as its dual a hexagon grid and the hexagon grid has as its dual the triangle grid. Thus the dual of the dual of a regular tessellation is the original tessellation itself.
Semiregular tessellations are made of several different regular polygons. At each corner of the polygons there are the same sort of polygons in the same sequence. An example is the tessellation of octagons and squares with two octagons and one square meeting at each corner point. Its dual contains isosceles triangles with a right angle. Similarly, all semiregular tessellations have a dual consisting of one irregular polygon as tile. The dual of the dual is again the original tessellation.
To get a quasiperiodic tiling of eight or twelve-fold rotational symmetry one can double the symmetry of a tessellation with four or six-fold rotational symmetry using two copies of the dual tessellation. The copies are rotated by 45 or 30 degrees with respect to each other and combined to one grid. The intersection of the edges of the duals give rise to new points of the grid. There are many small irregular polygons due to the intersection of the polygons of the two duals. Thus we have to use a different procedure to find the dual of this combined dual grid. We note that each point of the combined grid should result in a polygon. Each line of the grid going from this point gives a side of the polygon. In the most simple case, the side of the polygon is perpendicular to the line of the grid and all sides have the same length. The resulting polygons are either rhombs or regular polygons.
As an example we consider the square grid. Its dual is a square grid and two such square grids make up the dual of the quasiperiodic tessellation.
At the center of the picture you see two overlapping squares. Their sides cross at angles of 45 degrees. This results in eight rhombs meeting at a point. Next, there are the corners of the squares, giving rise to eight squares. Then further intersections make again rhombs, fitting between the squares. Finally we obtain the Ammann-Beenker tiling, see the earlier post “The basic Ammann-Beenker tiling”. I suppose that Beenker used this method to obtain the tiling.
It is interesting to compare this with the projection method I presented earlier. Note that there too one uses the same grid. With the dualization method two squares in the shape of an eight-pointed star in the dual grid result in a star of eight rhombs in the tessellation. From the projection method we get just one single point. Yet we have the same tessellation in both methods because of self-similarity. Note that the centers of the stars of rhombs just give the same tessellation at an inflated scale. The projection method essentially determines the places where the dualization method produces stars of rhombs.
Clearly, the dualization method can only reasonably be used with the help of a computer. But this is also the case for the projection and the iteration methods. Care has to be taken to make shure that the computing time increases only proportionally to the area of the tessellation.