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.