The isotropic vector matrix has already been introduced; we just didn't know its name.

      If you can visualize the space-filling array of spheres in "cubic packing" described in the previous chapter, that's half the picture. Now, imagine interconnecting the centers of all spheres—and then eliminating the spheres. Two collinear radii meeting at the tangency point between adjacent spheres form one unit vector—the length of which is equal to the sphere's diameter (Fig. 9-1). The resulting array of vectors is the "isotropic vector matrix," a space-filling network of continuously alternating octahedra and tetrahedra. Reviewing the characteristics of cubic packing, we shall not be surprised to find that all the newly formed vertices (the spheres' centers) are identically situated. Two types of cells, one type of vertex.

Fig. 9-1. Unit vectors interconnect centers of adjacent unit-diameter closepacked spheres. Click on thumbnail for larger image.

      It's not hard to see how Fuller's search for a geometry of vectors led him to the isotropic vector matrix. "Since vectors... produce conceptual structural models of energy events, and since my hypothetical generalization of Avogadro's law requires that 'all the conditions of energy be everywhere the same,'" ponders Fuller, "what does this condition look like as structured in vectorial geometry?" His answer is ready: "Obviously all the vectors must be the same length and all of them must interact [sic] at the same angles" (986.131b).

      The isotropic vector matrix, or IVM, takes the VE a step further, consisting of identical lengths and angles, not for vectors surrounding just one point, but surrounding every point in an indefinite expanse. In Fuller's words, the IVM is "a multidimensional matrix in which the vertexes are everywhere the same and equidistant from one another" (222.25).

Fig. 9-2. (a) Single row of alternating tetrahedron and octahedron. (b) Planar expanse of alternating tetrahedra and octahedra. Click on thumbnail for larger image.

      It is not correct to conclude that the IVM consists of many vector equilibria packed together, for the VE by itself cannot fill space. To understand why not, we look at isolated sections of the IVM. As difficult as it is to visualize the overall matrix, a single row of alternating tetrahedra and octahedra, or even a planar expanse, can be easily envisioned (Fig. 9-2a, b). Separate planar layers are then stacked together in such a way that every octahedron is adjacent to a tetrahedron and vice versa. Figure 9-3 shows three layers of the resulting matrix.

Fig. 9-3. Isotopic vector matrix and Octet Truss. Click on thumbnail for larger image.

      Every node in the IVM—as the origin of twelve unit vectors radiating outwardly—is the center of a local vector equilibrium. The ends of these unit vectors define the twelve vertices of the VE. However, this does not mean that adjacent cuboctahedra pack together to produce a space-filling expanse. A symmetrical array can be created by bringing the square faces of adjacent vector equilibria together, but they are necessarily separated by octahedral cavities—framed by the triangular faces of eight converging VEs. The unavoidable octahedra between adjacent VEs provide yet another manifestation of the specificity of the shape of space. This array can be readily understood by observing in Figure 9-4 that a packing of vector equilibria is equivalent to a framework of cubes in which the corners have been chopped off, thus automatically carving out an octahedral cavity at every junction of eight boxes.

Fig. 9-4. Octahedral cavities between adjacent VEs. Click on thumbnail for larger image.

      The above observations provide information about the shapes and angles of the IVM—the most symmetrical arrangement of points in space—and therefore about the shape of space itself. These characteristics reveal the basis for the term "isotropic vector matrix": in Fuller's words,

      "isotropic" meaning "everywhere the same, isotropic vector" meaning every-where the same energy conditions."... This state of omnisameness of vectors....... prescribes an everywhere state of equilibrium." (420.01-3)

He calls the IVM "multidimensional" because it "accommodates" (or occupies) all spatial dimensions, and—consistent with his unorthodox interpretation of dimension—space is "multi-" rather than "three-dimensional." Vectors are directed in every possible direction, while deliberately maintaining equivalent lengths and angles. This equivalence is necessarily determined by the symmetry of space:

      This matrix constitutes an array of equilateral triangles that corresponds with the comprehensive coordination of nature's most economical, most comfortable, structural interrelationships employing 60-degree association and disassociation. (420.01)

      As seen in the earlier development of vector equilibrium, spatial "omnisymmetry" incorporates four planes of symmetry: four unique directions of equilateral triangles. Recalling the way cookies fit most economically on a baking sheet, we can feel quite comfortable with the triangular symmetry of the plane. The implication is that the shape of space can be described through four such continuous planes.