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Amy C. Edmondson
A Fuller Explanation
Chapter 7, Vector Equilibrium
pages 90 through 93

Cuboctahedron as Vector Equilibrium

We can understand the symmetry of the plane by observing that although any polygon can be made to have equal edge lengths, only the regular hexagon can have edges equal in length to the distance between the polygon's center and its vertices. In the same way, although there are many regular and seniiregular polyhedra with equal edge lengths, there is only one spatial configuration in which the length of each polyhedral edge is equal to that of the radial distance from its center of gravity to any vertex: the cuboctahedron (Fig. 7-5). (3) This shape therefore is the only one that allows the requisite arrangement of vectors to demonstrate equilibrium.

Vector equilibrium

Fig. 7-5. Vector equilibrium.
Click on thumbnail for larger image.

Cuboctahedron, twist cuboctahedron
Fig. 7-6. (a) Cuboctahedron; (b) twist cuboctahedron.
Click on thumbnail for larger image.

      We first saw the cuboctahedron as the degenerate truncation of both the cube and the octahedron, but at that point in our investigation we were only looking at surface topology. Now diving into the interior shape, we discover this unique property of equivalence. Table IV compares the radial lengths of various familiar polyhedra given unit edge lengths. Only in the cuboctahedron—hereafter referred to by Fuller's term, vector equilibrium or VE—can the radius be of unit length.

Table IV
Unit-edge polyhedronRadiusCentral AngleAxial Angle

      Again, in order for all vectors to be exactly the same length, the angles between them—both radial and circumferential—are necessarily equal. In Figure 7-5, the VE is shown with both radial and edge vectors. Radial vectors connect the twelve vertices to the system's center, thereby forming twenty-four radiating equilateral triangles, corresponding to each polyhedral edge and pointing inwardly. We should not be surprised to find an array of equilateral triangles in the VE, for this is the only polygon with equal distances and angles between all points. And, as vectors incorporate both magnitude and direction, an equilibrium of vectors must—in Fuller's terminology-balance both angle and frequency. Sixty-degree angles are inevitable.

      What if we had anticipated the necessity of 60-degree angles? We could then have started this part of the investigation by specifying the angle between radial vectors and looking for the resulting implications of that choice. The discovery then would be that the necessary 60-degree gaps in a spatial array generate exactly twelve vectors, just as six is the outcome in a plane. Had we started thus-with the choice of angles-we would have had to check the resulting vector lengths, to find out that indeed they are all the same. In either case, the end result is extremely satisfying.

VE: Results

Our first encounter with the vector equilibrium in Chapter 4 —then we called it the "cuboctahedron" —illuminated its direct relationship to the octahedron and the cube. We now elaborate on our description of the VE, and before the end of this investigation we shall know almost everything about this extremely important shape. Let's begin here with its major characteristics.

      Above all, it is the "omnidirectional arrangement of forces." This equivalence is unique to the VE.

Eight radiating tetrahedra alternate with six half octahedra

Fig. 7-7. Eight radiating tetrahedron
alternate with six half octahedron.
Click on thumbnail for larger image.

      Secondly, this shape bears an interesting relationship to other familiar polyhedra. Its twelve radii form eight symmetrically arrayed regular tetrahedra—corresponding to the VE's eight triangular faces. Figure 7-7 emphasizes the tetrahedra, which radiate outward edge to edge, creating six cavities in the shape of square-based pyramids. Again, because of the uniform edge lengths everywhere, these cavities are actually perfect half octahedra, corresponding to the six square faces of the VE, which in turn correspond to the six faces of the cube, as was revealed by degenerate truncation in Chapter 4.

      Thirdly, "the pattern of this nuclear equilibrium discloses four hexagonal planes symmetrically interacting and symmetrically arrayed... around the nuclear center" (981.11). If you look closely at Figure 7-8 the four hexagons are clearly visible: one parallel to the horizon, one in the plane of the page, and two more, slanted to the right and to the left, at 60 degrees to the horizon. As we might have expected, the vector equilibrium consists—in a way exclusively—of hexagons. The symmetrical properties of hexagons with respect to the plane are evident (refer back to Fig. 7-4), and so the discovery of intersecting hexagons in a spatial equilibrium of vectors is not surprising.

Four hexagonal cross-sections of the Vector Equilibrium (VE)

Fig. 7-8. Four hexagonal
cross-section of VE.
Click on thumbnail for larger image.

      However, intuition cannot as easily predict the number of hexagons. An array of equivalent vectors (taking into consideration both magnitude and angular orientation) is achieved by exactly four evenly spaced intersecting hexagons. Thus the existence of four fundamental planar directions ("dimensions"?) describes one aspect of the inherent shape of space.

      These hexagons are exactly parallel to the four faces of the tetrahedron; having the same angular orientation, they are identical mathematical planes. The only difference is that in the VE they intersect at a common center, while in the minimum system they together enclose space.

      Also fascinating is the fact that each of the twelve radiating vectors is perfectly aligned with an opposite vector—exactly 180 degrees apart. Thus the twelve can be seen as six intersecting lines with a positive and negative direction (each line twice the length of the original unit vector)—just as the XYZ axes are three lines intersecting to define six directions: three positive and three negative, evenly spaced with intervening angles of 90 degrees. Once again, these six intersecting lines are parallel to the tetrahedron's edges. It was not at all obvious from our initial requirements for a vector equilibrium display that the resulting radial lines would be collinear pairs, nor that these six (double-length) vectors would each lie in the same plane as two others, producing four precisely defined hexagons.

      Our goal was to create a radial display of evenly spaced unit vectors. In so doing, we arrive at two fundamental observations about the order inherent in space: the existence of four distinct planes of symmetry and six linear elements. Both aspects are first exhibited in nature's choice of minimum system and secondly reinforced by her unique equilibrium configuration.

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