Tilings and their duals

I briefly discussed the dualization method in “The dualization method” but you will need more details to be able to understand my computer code. You can find some interesting ideas in Wolfram MathWorld and in Oracle ThinkQuest. In part I am presenting here the same concepts.

I have difficulties to find good names and their consistent use, because both the original tiling and its dual are tilings. In general I put “grid” before something that belongs to the original tiling and “dual” before something of the dual tiling. But note that the “dual” of something “dual” is again part of the original and thus called with “grid”.

We have to take into account that we can use only finite parts of a tiling. This requires special care. Else we will get disorder near the border. I do it like this:

dualIlluBesser

Hexagon tessellation and its dual: The grey lines show the limits of the region used. The blue lines and dots show the part of the hexagon tessellation that remains. The complete hexagon cells are shaded in pale blue. They give the part of the dual tessellation shown in red.

I only use those lines (in blue) of the tessellation that have points inside a limiting rectangle (grey lines). Then the computer searches for all complete polygons (shaded pale blue) formed by these lines. Thus we have grid-points, grid-lines and grid-polygons, making up the original tiling.

The dual tiling (red lines) similarly has dual-points, dual-lines and dual-polygons. Note that each grid-line (blue) is crossed perpendicularly by a dual-line (red). Each grid-line has as its dual a line of the dual tiling. Then, at the center of each hexagon there is a point of the dual tiling. Each grid-polygon has as its dual a point of the dual tiling. Similarly, we find that each triangle of the dual tiling has at ist center a point of the original tiling. So each grid-point has as its dual a polygon of the dual tiling. At the border, there are exceptions.

This goes also the other way around. The dual line of a dual-line is a grid-line. Each dual-point has a grid-polygon as its dual and every dual-polygon has a grid-point as its dual.

If this is too abstract, you might consider a mobile phone installation. A country is divided into different regions. This corresponds to the original tiling. The regions are the grid-polygons. In each region there is an antenna, which is essentially the dual-point to the grid-polygon. There are transmission lines between neighboring antennas. These are the dual-lines to grid-lines that limit the regions.

It took some time for me to realize that this already determines the topology of the dual tiling. We thus get for any tiling the number of points of the dual tiling and how these points are connected by lines. But the positions of the dual-points are still unknown.

For many tilings the dual-points are simply at the center of their grid-polygons. This is valid for regular and semiregular tessellations and probably for all tilings, that have only regular polygons as tiles. But if we superimpose two tilings we get irregular tiles and we have to find the position of the dual-points differently. We then use that the dual-lines are perpendicular to their grid-lines and that they have all the same length. From this we get the positions of the dual-points.

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