Duality

The Generalized Dual Method, or GDM, can be used to make the famous Penrose tilings of 36- and 72-degree rhombs, and to explore quasicrystalline structures of infinite variety. In previous posts, I have mentioned how typical it is to choose some set of "symmetrical" vectors for one's "basis vectors." Weakly, I have described the imprecise duality which obtains between "arrangements of hyperplanes" and zonotopal tilings.

However, if the symmetry vectors are carefully chosen, one can obtain the same kind of exact duality as one is perhaps used to, in the plane, or in three dimensions. We can obtain truly periodic tilings, of several types.

For instance, let there be three basis vectors, {x,y}, in the form {cos[0],sin[0]}, {cos[120], sin[120]}, and {cos[240], sin[240]}. That is, each vector is perpendicular to one of the sides of an equilateral triangle.

Now let the distances "d" run through, say, the integers from -3 to 3. We obtain an arrangement of 3*7 lines, each line expressed as {x,y,d}.

In this case, our lines cut the plane into a region composed of equilateral triangles (surrounded by some other shapes). The dual tiling is the tiling of regular hexagons. Actually, we obtain a large hexagon tiled by a number of smaller hexagons and some rhombs.



Now let the distances "d" run through -3 to 3 as before, but add the fixed number .5 to each. Hence the distances now run from -2.5 to 3.5 in steps of 1.

When we examine this line arrangement, we see that we have obtained a region of the "Archimedean" tiling composed of regular hexagons and equilateral triangles. The dual of this tiling is a tiling by 60-degree rhombs.