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.