Imagine one vertex within the IVM framework, which will be called 0-for origin. A unit vector (L = 1) pointing in any of the twelve directions away from 0 ends at a vertex which we shall call A. A second unit vector emanating from 0 is given a different orientation, in a direction 60 degrees away from vector OA, and arrives at vertex B. The distance between A and B is also unit length (Fig. 9-5a). As simple and repetitive as this observation might seem, it is the essence of "synergetics accounting," as opposed to "algebraic accounting."

Fig. 9-5. (a) 60-degree axes. (b) 90-degree axes. Click on thumbnail for larger image.

      The same procedure applied to a 90-degree framework, also using unit vectors, places vertices A and B an irrational-number distance apart. Unit increments along the x and y axes create points labeled simply ((0,1) and (1, 0)}, which are separated by a (not so simple) irrational Ö2 or 1.41421... units. Furthermore, if pathways between vertices are to follow along the network of vectors, the square grid disallows the shortest route between A and B. Observe in Figure 9-5b that to get from A to B along the prescribed grid requires traveling two units, despite the fact that they are separated by only 1.414... units. As energy always takes the shortest route, argues Fuller, the XYZ system clearly does not serve to illuminate the events of physical reality. In contrast, the most expedient route from A to B in the triangular grid happens to be directly along the unit-length vector connecting the two points.

      Irresolvable numbers do exist within the IVM, as there are square cross-sections (corresponding to the square faces of the VE); however, the irrationals are not part of the fundamental orientation of the system. To Fuller, a simple procedure, like the one described above, ought to yield simple ("omnirational") results. If a frame of reference is itself convoluted, its ability to describe and measure other phenomena will be all the more so.


Cells: "Inherent Complementarity"

Fig. 9-6. (a) Tetrahedra do not fit together. (b) Tetrahedra rearranged vertex to vertex. (c) A second row of tetrahedra is placed above the first, such that all tetrahedra meet vertex to vertex; the arrangement automatically creates octahedral cavities. (d) Alternating tetrahedra and octahedra can fill space indefinitely. Click on thumbnail for larger image.

Let's back up and start again. The goal is to establish a symmetrical and complete spatial array of vectors, and one logical approach might be to start with space's minimum system. We gather a number of unit-vector tetrahedra and place them on the ground side by side, and it is immediately apparent that they will not pack together to produce a continuous expanse (Fig. 9-6a). Awkward gaps between adjacent tetrahedra cannot be filled by regular, or symmetrical, shapes, precluding an isotropic array. This constraint is not new: we saw in the previous chapter that tetrahedra cannot fill space; however, vector models display the shapes more clearly than sphere packings.

      Still in pursuit of a space-filling array, we now rearrange the tetrahedra so that they meet vertex to vertex, each with one edge along a continuous line, in multiple adjacent rows (Fig. 9-6b). Notice what happens when we interconnect the tetrahedral peaks (as would be the result if a second layer of tetrahedra were placed on top of the first): precise octahedral cavities emerge in between all tetrahedra, automatically completing the isotropic vector matrix with its alternating two shapes (Fig. 9-6c, d).

      The reverse is also true: octahedra cannot themselves fill space, but when arranged edge to edge (not face to face or vertex to vertex) the emergence of by-product tetrahedra reconfirms the persistent pairing. The developed matrix with its unit vectors and equivalent points of convergence thus depicts the inherent complementarity of space—meaning inevitable co-occurrence of octahedra and tetrahedra. Fuller draws a parallel between this and other inseparable pairs such as electron-proton, concave-convex, male-female, and tension-compression. "Inherent complementarity of Universe" applies to the entire phenomenon of interdependent partners, whether atomic constituents or polyhedral space-fillers. (Chapter 12 will further elaborate on Fuller's interpretation of the significance of "inherent complementarity.")

      In conclusion, the development of both the VE and the IVM—whether through closest packing of spheres or by symmetrical arrangement of vectors—supports a sense of the balance of octahedral and tetrahedral symmetries inherent in space. Both configurations build themselves—in response to spatial constraints.

We now step inside the IVM to complete our investigation of this omnisymmetrical network of vectors. The centers of closepacked spheres constitute the vertices of most regular and semiregular polyhedra. We looked at some of them in the previous chapter, and with the use of toothpicks instead of Ping Pong balls, the outlines of these shapes can be more easily discerned.

      We have already observed that vertices in the IVM fall into triangular patterns in four distinct planar directions. Through our experience with cubic packing, we know to look for an additional three planes of symmetry, characterized by a square distribution of vertices. Neighboring octahedra share the edge between them, and thus the cross-sections of individual octahedra join together, forming the square pattern of graph paper in three orthogonal directions. Figure 9-7 highlights an IVM squared plane, by omitting certain lines; half octahedra shown without tetrahedral edges clarify the square aspect of the omni-triangulated matrix.

Fig. 9-7. Half octahedra shown without tetrahedral edges, to emphasize a square cross section of the IVM. Click thumbnail for larger image.

The combination of these two simple shapes in the IVM yields surprisingly many different angles and potential shapes. Our attention tends to be focused on the surface characteristics of the tetrahedron and octahedron, and so we observe only triangles and 60-degree angles. However, the interior structure introduces distinct new elements, such as the square octahedral cross-sections discussed above. The next step is to list other interior angles, for a sense of the range of possible shapes contained within the matrix.

Fig. 9-8 Click on thumbnail for larger image.

The dihedral angle (angle between two faces) in a regular tetrahedron is approximately 70° 32'. The tetrahedron is unique in that any two edges at a given vertex are part of a common face. Every other polyhedron has interior angles in addition to surface angles between edges, thus adding to the range of shapes incorporated into each system. For example, any two nonadjacent edges at an octahedron vertex meet at 90-degree angles, thus forming square cross-sections. The octahedron dihedral angle is 109° 28', which—as the supplement (1) to the tetrahedron's 70° 32'—results in perfectly flush surfaces shared by adjacent octahedra and tetrahedra, allowing the continuous planes of the IVM (Fig. 9-8).

      Both dihedral angles at first appear to be such irregular numbers that this exact geometric fit is surprising—especially when we recall our first encounter with the two shapes. Remember that we simply surrounded vertices by three triangles, then by four, allowing the systems to close off with as many triangles as necessary. (Refer to Chapter 4.) The process provided no basis for predicting the exact complementarity of the two polyhedral systems. The coincidence continues with the addition of central nodes, as we shall see next.