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Cuboctahedron as Vector Equilibrium
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 cuboctahedronhereafter referred to by Fuller's term, vector equilibrium or VEcan the radius be of unit length.
Again, in order for all vectors to be exactly the same length, the angles between themboth radial and circumferentialare 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 mustin 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.
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.
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 vectorexactly 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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