Why the Rhombic Dodecahedron is a Shadow of the 4-Dimensional Hypercube
(Contributed February 13, 1997 by Russell Towle)
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When confronted with the abstraction of higher space, we often imagine something mystical or magical, or perhaps recall H.G. Wells' assignment of Time as the fourth dimension. Then again, there are cosmologies recently advanced which invoke a 10-dimensional space at the time our universe began, subsequently thought to have collapsed or foreshortened into but five or six dimensions. I have my doubts about such theories, and when I expressed these to my childhood friend, Dr. Michio Kaku, he buried me beneath a barrage of highly technical mathematical terms to which I simply have no clue.
However, it is within the simpler confines of Euclidean n-dimensional space that we may understand how it is that a rhombic dodecahedron is a solid shadow of a 4-cube. Foremost among 20th-century geometers attacking the problems presented by these higher Euclidean spaces is Henry Scott McDonald Coxeter, Professor Emeritus of Mathematics at the University of Toronto. His classic work, "Regular Polytopes," is still in print, under the Dover aegis. There is certainly much of a very difficult nature, impervious to casual reading, in this great book. Yet is a wondrous jewel and a tremendous achievement. Buckminster Fuller idolized Professor Coxeter, declaring, "He has completed the work which Euclid began." It is no commonplace to have made a substantial contribution to human culture; Coxeter has done just this.
Now, "polytope" is a generalization upon the terms, "polygon" and "polyhedron." A polygon is a 2-polytope, and a polyhedron, a 3-polytope. Many of the properties of polytopes may be deduced by analogy, in fact, one need not be a trained mathematician to explore higher space and, in fact, rather important contributions to the subject have been made by amateurs. Now, a single point is a zero-dimensional polytope; a line segment is "bounded by" two points, at either end; a polygon is bounded by line segments; a polyhedron is bounded by polygons; and a 4-polytope is bounded by polyhedra. By analogy, the general n-dimensional polytope is bounded by (n-1)-dimensional polytopes.
Some things, however, do not generalize to higher spaces. For instance, we may obtain an infinitude of regular convex polygons in two dimensions, but only five regular convex polyhedra (the Platonic solids) exist in three dimensions. Curiously, six regular convex polytopes exist in four dimensions, but, in five-space and all higher spaces whatsoever, only three regular polytopes can exist. These exist in all n-spaces, and are known as the simplex, cross polytope, and measure polytope. This last is usually referred to as a hypercube, and the special term "measure polytope" is given to hypercubes of unit edge length, since they provide the measure of n-dimensional content. That is, with the unit square we measure surface area, with the unit cube we measure volume, and so on.
Space does not permit a detailed description of these three universal regular polytopes. The simplex is the "simplest" polytope which can exist in a space, i.e., of all polygons the simplest is a triangle, of all polyhedra the simplest is a tetrahedron. We may "coordinatize" space of n dimensions, by employing a frame of Cartesian axes, just as we use x, y, and z coordinates to specify every point in three dimensions. If we "stop off" the Cartesian frame at unit distances from the origin, we obtain what is called a "cross." The endpoints of a cross are the vertices of a cross polytope; in two dimensions, this is a square, in three, the Platonic octahedron.
The third universal polytope, the hypercube, is what relates most directly to Mark's rhombic dodecahedron. This dodecahedron is probably best referred to as Kepler's rhombic dodecahedron, being first formally described by Johannes Kepler, although it exists in nature as a crystalline form, and the cells of the honeycomb are also rhombic dodecahedra of this type. An infinitude of other rhombic dodecahedra exist. In fact, many of these can also be solid shadows of 4-cubes, as is Kepler's. But first things first.
An orthogonal projection, sometimes also called a parallel projection, is commonly thought of as a matter of dropping perpendiculars from a polyhedron, say, onto a plane. The "projectors" are parallel lines falling onto a plane at right angles. In terms of Cartesian coordinates, such a projection is easily obtained: consider the vertices of a polyhedron, each defined by a triplet of real numbers, {x,y,z}. We obtain the orthogonal projections onto the coordinate planes by simply dropping any one of the three coordinates of each vertex: by removing all the z coordinates, we project onto the xy plane, by removing all the y coordinates, we project onto the xz plane, by removing all the x coordinates, we project onto the yz plane.
Unlike a perspective projection, which introduces vanishing points, an orthogonal projection sends parallel lines into parallel lines. It is important to recognize that whether perspective is introduced or not, foreshortening occurs; unless, for instance, a polygonal face is parallel to the plane of projection, it will be variously squished in the projection. If we know which vertices connect to which to form the polygons bounding a polyhedron, then we merely connect up their projected images to form what is called a "wire-frame" projection. When a finite convex polyhedron is projected in this way, some sort of convex polygon forms the perimeter of the projection; this, with its interior, is what is called the orthogonal "shadow" of the polyhedron.
Now, these means of forming orthogonal projections and shadows generalize to all higher spaces; and therefore we may obtain "solid shadows" of 4-polytopes in particular, and n-polytopes in general. Kepler's rhombic dodecahedron is just this: a solid shadow of a 4-cube. Now, to make this more intelligible, imagine a cube floating above a plane, with light coming from an infinite distance above the plane: the cube casts a polygonal shadow. As we rotate the cube in its 3-space, the orthogonal shadow in the 2-space (the plane) changes continuously. If we measure its surface-area while it changes through all possible shapes, we shall find that its square shadow has the least area, its regular-hexagon shadow, the most area.
By analogy, as we rotate a 4-cube in its 4-space, its solid shadow, a polyhedron, changes continuously in our 3-space. Now, an n-cube is bounded by (n-1)-cubes, in the way that a regular cube is bounded by square faces. A 4-cube is bounded by just eight 3-cubes. Retreating again to projecting a cube onto a plane, we readily see that all six faces of the cube fall into coincidence, when the cube projects into a square. Similarly, all eight cubes of the 4-cube fall into coincidence, when it projects into a cube. This is one of the four most symmetrical shadows of the 4-cube, obtained when the 3-space of projection is at right angles to a line connecting the centers of opposite cubes bounding the 4-cube. It is called the "cell-first" projection, since the (n-1)-polytopes bounding an n-polytope are called "cells." Of all polyhedral shadows of the 4-cube, this has the least volume.
Now, to complete the analogy between the plane shadows of a cube, and the polyhedron shadows of a 4-cube, it is possible to choose a 3-space of projection such that it is at right angles to a line connecting opposite vertices of the 4-cube. We then obtain the shadow of greatest volume, Kepler's rhombic dodecahedron. The distinguishing mark of this projection is that it is isometric; the edges of the hypercube are equally foreshortened under orthogonal projection. Hence also the eight cubes bounding the hypercube are equally foreshortened, into eight rhombic hexahedra which interpenetrate to compose the rhombic dodecahedron. By comparison, note how the isometric projection of a cube onto a plane sends the six square faces of the cube into six equal 120-degree rhombs, which overlap.
Finally, it is worth noting that hypercubes close-pack to fill space, just as we may fill the plane with equal squares (each sharing line segments with adjacent squares), or fill 3-space with equal cubes (each sharing square faces with adjacent cubes). In four dimensions, the space-filling of hypercubes, by analogy, means that each hypercube shares its cubes with adjacent hypercubes. Suppose we project the entire space-filling of 4-cubes into 3-space, and choose the 3-space such that it is at right angles with a line connecting opposite vertices of any one of the 4-cubes. We obtain a space-filling of interpenetrating rhombic dodecahedra, each one sharing its rhombic hexahedra with "adjacent" dodecahedra.
Much more might be said about the solid shadows of hypercubes, which form without exception a special type of polyhedron called a "zonohedron." However, this is not higher space, and I must restrain my remarks within a reasonable compass.