Tessellations Upon a Theme by Archimedes

The nine vectors which determine the Archimedean Truncated Cuboctahedron may be used to form tilings, using the Generalized Dual Method. Here two different tilings, each arising from 3*9=27 planes, are gradually built up, from an axis of 3-fold symmetry, outward. The first has a central region tiled by Truncated Cuboctahedra, Cubes, and Truncated Octahedra. The second has a central region tiled by Truncated Cuboctahedra and Octagonal Prisms.

Interpolation

Using Mathematica, one can read in a photo and then manipulate the color data.

Here I used interpolation functions to assign RGB colors to the tiles, according to tile centers.

Penrose Tilings

The fascinating tiling discovered by Sir Rogen Penrose in the middle 1970s may be constructed using the Generalized Dual Method, using an arrangement of lines. Let a set of equally-spaced parallel lines, one unit apart, be called a grid. Let a set of n non-zero, independent vectors be called a star. Let the star be the five unit vectors from center to vertices of a regular pentagon (of unit circumradious). Now create a grid perpendicular to each vector of the star, each grid centered upon the origin and containing one line passing through the origin. Now offset all five grids by 2/5, measured in the direction of each grid's star vector.

Above is an example of the resulting tiling. This strategy leads to a Penrose tiling with 5-fold symmetry governing the entire tiling. If the grids have inifnitely many lines, the entire plane is tiled. If the grids have an equal and finite number of lines, the Penrose tiling arises in a central region of a regular decagon tiled by 36- and 72-degree rhombs.

If the edges of the rhombs are suppressed, and the 36- and 72-degree rhombs colored differently, one obtains the effect seen above.

Nested Polar Zonohedra

An n-merous polar zonohedron can be dissected into exactly n-things-take-three-at-a-time rhombic hexahedra. These same hexahedra can be added to the exterior of the polar zonohedron to build up larger and larger (yet similar) polar zonohedra. The above animation shows several examples of such tilings.

Symmetrical Structures

With my collaborator, Mark Newbold, I have been refining techniques to sort zonogonal/zonohedral tessellations by area/volume. Mark has written a marvelous Mathematica "package" which implements the Generalized Dual Method (GDM) in space of "n" dimensions. The data structures are unlike those in my own, much clunkier methods, so it has taken some time to effect this sorting.

The GDM goes from an "arrangement of hyperplanes" (here "hyperplane" means "(n-1)-space") to a zonotopal tiling.

In 2D an arrangement of lines goes to a zonogonal tessellation; in 3D an arrangement of planes goes to a zonohedral tessellation. A line, plane, or hyperplane is usually defined by a unit normal vector and a distance from the origin. Given a unit vector, a family of parallel lines, planes, or hyperplanes, will arise, at the various distances. The set of normal vectors is called the "star." A star must span its n-space. If an n-space-spanning star has an infinite number of distinct hyperplanes for each vector of the star, the whole n-space is tessellated. But computers cannot construct infinite numbers of hyperplanes. Given a finite number of hyperplanes, a "bounding n-zonotope" arises, tiled by smaller n-zonotopes of various types.

This "bounding zonotope" is exactly similar to that zonotope "determined by" the vectors of the star.

Hence in 2D a "bounding zonogon" arises, tiled by smaller zonogons.

In 3D a bounding zonohedron arises, tiled by smaller zonohedra. For every point of intersection between hyperplanes, a single zonotope arises in the tessellation. If there is but one point of intersection, only one zonotope arises: a degenerate tessellation, as it were. For instance, if the star of vectors in 2D is formed by the two unit vectors along the positive x and y axes, and any single distance whatsoever is assigned to each vector (i.e., {1,0,Sqrt[2]}, {0,1,1000}), we obtain a single square as the "tessellation." Or if the 2D star is formed by the n vectors from the center to the vertices of a regular n-gon, and to each vector of the star we assign a single distance of zero, then all n lines intersect at the origin, and we obtain an undivided (regular) 2n-gon.

But more typically, more than one point of intersection arises, and for each point, a zonotope arises in the tessellation.

It is quite difficult to visualize arrangements of planes, in 3D, and also quite difficult to visualize the resulting zonohedral tessellations; one zonohedron obscures another. If we make them transparent, it is almost worse yet.

For many years I have struggled to develop methods to reveal the internal structure of zonohedral tessellations. The movie above shows how a very symmetrical tessellation of a certain rhombic enneacontahedron can be built up in various ways. Later, several zonogonal tessellations are built from the center, outward. This is not really necessary to see their "internal" structure; thankfully, we are not in the plane of the zonogonal tessellations: we look down upon that plane like gods.

If we were 4-dimensional beings, we could, godlike, look down upon the (perpendicular) 3-space of a finite zonohedral tessellation.

We would not only see every zonohedron in the tessellation, but all its faces at once, and its interior, in its true, unforeshortened aspect. That is, the volumes of things like zonohedra would be very easily discerned, looking from four dimensions, into three dimensions.

Lately I have been not only making animations and still images using the Generalized Dual Method, but writing about it. I am crafting a long article with many graphics, for an electronic publication. Writing about the GDM, and programming the various graphics, has led to a renewed confrontation between me and the Method.

It has always seemed mysterious, a kind of black box into which I drop arrangements of lines, or arrangements of planes, and out pops a zonogonal tiling, or a zonohedral space-filling. The actual mechanism by which this is accomplished is much less interesting to me than the plane and solid tessellations engendered.

Above is a generalization upon the famous Penrose tiling, with 8-fold symmetry around a central point. It was created using De Bruijn's system of adding real-valued "offsets" to each set of parallel lines, such that the offsets sum to an integer.

The GDM: Transformations

The Generalized Dual Method, or GDM, accepts as input a set of lines, and outputs a zonogonal tiling.

Usually, the lines are in "n" subsets, each subset containing some number of lines parallel to the sides of a regular n-gon.

Now suppose n-1 of the n subsets are held fixed, but the nth subset is moved, from beyond the farthest point of intersection to one side of the fixed n-1 subsets, to beyond the farthest point of intersection on the other side, of the n-1 subsets.

As the lines move, the tessellation is transformed.

Animation Frame

A single frame from a recent animation. Lately I have been making POV-Ray's virtual camera follow spline curves over zonogonal tilings. Unfortunately, these animations are too large to upload to YouTube, over my dial-up connection. Also, YouTube tends to drastically degrade the quality of any animation.

In this image, a wooden tube follows the spline curve itself, while the camera is somewhat above and to the side. The star polygon {12/5}, set within the regular compound {12/4}, also follows the spline curve, a little ahead of the camera.

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.