These zonogonal tilings arise from an "arrangement of lines," and the lines, usually, are taken perpendicular to a set of symmetrically-disposed vectors. In the algorithm itself, the lines are represented in the form {x,y,d}, where {x,y} is a unit vector, and "d" is the distance of a line perpendicular to that vector, from the origin.
For instance, were {x,y,d} to be {1,0,0}, it would denote a line coincident with the y axis, passing through the origin. Were {x,y,d} to be {1,0,1} it would represent a line parallel to the y axis, and one unit away from the origin, in the direction {1,0}.
Note that we could represent one and the same line by, for instance, {1,0,1} and {-1,0,-1}.
This confuses my algorithm--my implementation, in Mathematica, of the Generalized Dual Method, or GDM.
Hence I strive to avoid such cases.
Suppose the "symmetrically disposed vectors" are the vectors from the center to the vertices of a regular polygon of n sides; an n-gon. If "n" is even, then any one vertex is diametrically opposed to another, and we are confronted with the confusion noted above. In such a case it suffices to use only the first (n/2) of the n vectors.
That is, if n=8, we can use the four unit vectors which make angles with the x axis of 0, 45, 90, and 135 degrees, and obtain zonogonal tilings with loci of 8-fold symmetry.
I should say that a zonogon is a convex polygon with central symmetry. Hence a zonogon has an even number of sides, and every vertex, every side, has a diametrically opposed counter-vertex, and counter-side. All regular convex n-gons, n even, are zonogons; but a zonogon need not be regular. A rhomb of whatever shape is a zonogon, and so is a parallelogram.
If the GDM implies duality, what does that mean?
Consider the duality between the Platonic Cube and Octahedron. We could say that every vertex of the cube corresponds to a triangle of the octahedron, and that every face of the cube corresponds to a vertex of the octahedron; and vice versa.
Perhaps it is better to look within the plane. Take the regular tessellation of equilateral triangles: six meet at every vertex. Supposing a dual tiling exists, it must have only six-sided polygons. In fact, the dual tiling is the tiling of regular hexagons, in which three hexagons surround every vertex. We could obtain the tiling of hexagons directly by placing a dot at the exact center of each triangle in the triangular tessellation, and then connecting each such dot to its nearest neighbors.
Similarly, we could obtain the triangular tessellation directly from the hexagonal, by placing a dot in the exact center of each hexagon, and connecting all such dots to their nearest neighbors.
In the GDM, it is a less-precise and more generalized type of duality which obtains.
We form an arrangement of lines. If they are not all parallel, there are points of intersection.
Every point of intersection in the arrangement of lines leads to a zonogon in the tiling.
Suppose from now on that we have a typical arrangement of lines. They are not all parallel. Let it be quite a simple arrangement: let us merely produce the sides of a regular pentagon. There are five lines, and we see they intersect in ten points. Immediately we know that the corresponding zonogonal tiling has exactly ten zonogons.
Furthermore, by looking at the arrangement of lines, we know how many sides each zonogon in the related tiling has; for, if "k" lines intersect at some one point, to that point corresponds a zonogon with 2k sides, in the tiling.
For instance, if we take the same five lines as in the example above, but change their distance from the origin, we could force all five lines to intersect at the origin. Now there is only one point of intersection, hence only one zonogon in the tiling. Since five lines meet at that point, we know that the zonogon has ten sides. Since the five lines are symmetrically disposed, it is in fact a regular decagon.
We can take the same five directions in which lines lie, and add more lines in each direction, and contrive to force many subsets of the lines to intersect. An example, below.
The duality extends further: every open region, bounded or unbounded, cut out from the plane by the arrangement of lines, leads to a vertex in the tiling. And every line segment or ray, bounded or unbounded, cut out from the lines by one another, leads to an edge in the tiling.
Not only that, the GDM itself generalizes, to n-dimensional space. In three dimensions we take an arrangement of planes; each plane is expressed in the form {x,y,z,d}, where, typically, {x,y,z} is some unit vector, and "d" is the distance of the plane perpendicular to that unit vector, from the origin.
Similarly, in 4-space we make an arrangement of 3-spaces, each 3-space expressed as {x,y,z,w,d}; and so on.
I will end this post with an image from an animation I made yesterday, in which regular star dodecagons (which are not zonogons!) are flying over a zonogonal tiling--a tiling in which a sphere lies upon every vertex, and a cylinder upon every edge, and, moreover, all these spheres and cylinders are stretched out to five times their normal height.
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