A Spiraling Rhombic Tessellation

In the late 1980s, using the BASIC programming language, I contrived to make a certain (infinite) set of rhombic tilings with a spiral structure. I used nested loops. Attempts to re-create similar tilings using the GDM have not succeeded, but some of the resulting tilings are of some interest, anyway.

Mirrored Worlds

One can use the free ray-tracing software, POV-Ray, to create virtual kaleidoscopes. If merely two vertical mirrors are used, hinged along a common edge, depending upon the angle between them, any object between the mirrors is repeated by reflection, perhaps many times.



If three vertical mirrors are used, erected upon the sides of a 30-60-90 triangle, one can obtain tessellations of the plane.



I have posted some animations involving such kaleidoscopes on YouTube, see My YouTube

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.

The GDM in Dimension Three

The Generalized Dual Method, or GDM, itself generalizes to higher Euclidean spaces. In 3-space, as mentioned in my last, an arrangement of planes, expressed in the form {x,y,z,d}, leads to a tessellation of zonohedra. And although we may give the GDM planes in arbitrary, "general" position, more typically we use a set of symmetry vectors in the 3-space.

For instance, we could pick the edge-centers of the Platonic Icosahedron; there are thirty edges, but they occur in oppposite pairs. This gives problems to the GDM, so we pick one out of each of the fifteen pairs, it doesn't matter which one, to obtain fifteen independent vectors.

Were we to make merely fifteen planes, and let them all pass through the origin, that is, each plane of the form {x,y,z,0}, we'd obtain a "degenerate" tiling consisting of one undivided zonohedron, the Archimedean Truncated Icosidodecahedron.

But, suppose we make thirty planes, fifteen at distance 0 as above, the other fifteen at distance 1. We obtain a Truncated Iocosidodecahedron of edge 2, composed of 176 separate zonohedra of edge 1; and, perforce, since fifteen of the planes do intersect at one point (those with distance 0), there is one Truncated Icosidodecahedron of edge 1 among the 176 zonohedra.

But it is not symmetrically situated within the tiling. It is entirely to one side, although, of course, entirely within the Truncated Icosidodecahedron of edge 2.

The animation above, made using the software, Mathematica, builds up the tiling zonohedron by zonohedron, beginning at the innermost and ending with the outermost.

Recent Animations Part II

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.

Experiments in Animation

Lately I have been using the fabulous software, Mathematica, to create zonogonal tilings, and then to export the tilings to the free ray-tracing software, POV-Ray. This is nice enough in itself, but since I have been using spline curves in POV-Ray to animate objects and/or camera location, it must follow that I zoom around over the tilings, that I make a spotlight follow along the spline curve, and so on and so forth.

The tilings arise from my implementation of the "generalized dual method," in which an arrangement of lines leads to a zonogonal tiling. Although the lines may be randomly oriented, typically one chooses a set of symmetry vectors, and creates a "grid" of some number of lines perpendicular to each of the vectors.

Here is another example of such a tiling.



Well, I wished to add more images here, but Blogger seems to degenerate into an infinite loop of inactivity. After waiting twenty minutes for an image to upload, enough is enough.