Dualization method for the Stampfli tiling

I am now showing step by step how to get the Stampfli tiling with the dualization method. In the next post you will find the code, which you could change to make other quasiperiodic tilings.

First, we combine two hexagon tilings to get a grid of twelve-fold rotational symmetry using the method discussed in “Combination of grids“. It looks like this:

only gridHere, the green line shows the boundary used to cut off the grid, see”Testlimits“. Then we find the topology of the dual tiling, as discussed in “Tilings and their duals” and in “irregular tilings and their duals“. If we put the points of the dual at the centers of their grid polygons we can see how the dual matches its grid:

tilingPlusDualThe red lines show the dual. These tiles are irregular and we have to adjust the angles and lengths of the dual lines. This gives the final tiling. I tried to match the dual to the grid as closely as possible:

correctDualMatchColoring the polygons of the dual we can see better the Stampfli tiling:

tilingpartI have used this method first to discover the Stampfli tiling. But it can create many other tilings too. As an example, a large family of tilings results from combining bundles of parallel lines.

 

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

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