The Voronio diagram of quasiperiodic tilings

In the post “Beautifying the double grid” I have shown how to get an interesting trellis by distorting the grid of a quasiperiodic tiling. Here I am showing Voronoi diagrams of the corner points of tilings, which make nice trellis too.

You can find a good discussion of Voronoi diagrams in Wikipedia. I am trying to give a brief explanation. First, you have some given points in the plane. Then you want to know for each of them all points in the plane which are closest to it. This defines regions around each point. These boundary between these regions make up the Voronoi diagram.

socolarMitTilingKlein

Socolar tiling with its Voronoi diagram drawn in yellow (with JPEG coloring artifacts)

For a quasiperiodic tiling of regular polygons and rhombs we can easily get to Voronoi diagram. We simply have to find the middle of the sides of the tiles. There we draw perpendicular lines. For regular polygons these lines meet at the center and that’s it. In rhombs pairs of lines meet at two intersection points and we then connect these points. You can see this in the figure at the left. It shows the Socolar tiling, which is the double of the tessellation of hexagons, and its Voronoi diagram. The dual of the Voronoi diagram is again the tiling except that the rhombs are cut into triangles. TheVoronoi diagram gives nicer trellis as the angles are less acute.

Doubling the regular tessellations give interesting trellis:

socolar

Voronoi diagram of the Socolar tiling.

stampfli

Voronoi diagram of the Stampfli tiling.

amannBeenker

Voronoi diagram of the Ammann-Beenker tiling.

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