Isotropic Vector Matrix

The omnidirectional array of identical-length vectors joining the centers of closest-packed equiradius spheres — "isotropic" meaning everywhere the same — which Fuller treats as the natural coordinate framework of Universe.

The Isotropic Vector Matrix (IVM) is disclosed when the centers of equiradius spheres in closest packing are joined by the most economical lines — the geodesic vectorial lines between adjacent sphere centers (Synergetics §420.01). Because closest-packed spheres touch tangentially, every such connecting line equals the sphere diameter, so all the vectors are of identical length and all their intersecting angles are the same. Fuller reads each vector as an energy condition; equal vectors mean "everywhere the same energy conditions," which is what isotropic names, and this omnisameness prescribes an everywhere state of equilibrium (§420.01, §420.03). Remove the spheres and leave only the lines, and what remains is a pure vector lattice — an array of equilateral triangles extending in every direction (§420.01).

The matrix is generated by closest packing, the twelve-around-one nesting of unit spheres repeated to fill all space. Around any one nuclear vertex, twelve vectors converge, each at 60 degrees to the next adjacent one, and the pattern repeats identically at every internal convergence (§421.01, §423.01). Every vector runs from one nuclear center to another, composed of two radial halves belonging to the two tangent spheres it joins, so the whole aggregate is a "universal vector system in dynamic equilibrium" in which identical nuclear systems and layer growths recur about every sphere (§421.02, §421.10). The nucleus is thus incidental to the lattice geometry itself but central to its regeneration (§422.02).

The IVM partitions all space into just two "clear-space" polyhedra: the regular tetrahedron and the regular octahedron, operating as complementary space fillers in alternation (§422.01). This single omnitriangulated octahedron-tetrahedron space frame is what Fuller names the octet truss — the IVM realized as structure (§422.01, §422.02). All the polygons formed by the interacting vectors reduce to equilateral triangles and squares, the squares appearing only as equatorial cross sections of the octahedra; and all such polyhedra reduce ultimately to non-further-reducible triangulation, i.e., trusses (§420.07, §420.08). Because loads applied to any node distribute radially and hexagonally through the whole three-way grid, the octet truss diffuses concentrated force throughout the system like the tension skin of a pneumatic tire (§422.10).

Fuller offers the IVM as nature's coordinate system in place of the XYZ cube. It is 60-degree-coordinated rather than 90-degree-coordinated, and where the cube's radials and hypotenuses are incommensurable, the IVM's radial and circumferential vectors are congruent and integratable (§420.02, §423.03). This yields an omnirational accounting system that becomes inherently irrational only when arbitrarily forced onto a 90-degree basis (§420.02, §420.07). In this framework the tetrahedron is taken as the unit of volume — tetrahedron = 1 — whereupon the octahedron is exactly 4 and the cube exactly 3, whole-number volumes throughout the synergetics hierarchy (§420.02; cf. §973.30, §982.62). Integrating arithmetical identities such as n² and n³ with the 60-degree system produces coincidences with the topological inventories of systems, which is why Fuller claims the IVM "makes possible fourth- and fifth-power modeling" (§423.04). The lattice is a precessionally nonredundant vector-tensor relationship whose accountings are rational, radially and circumferentially, to all chemical, biological, electromagnetic, thermodynamic, gravitational, and radiational behaviors of nature (§424.01).

The IVM is intimately bound to two other synergetics constructs. The vector equilibrium (VE, or cuboctahedron) is the IVM's nuclear or local expression: the tetrahedron's exclusively edge-congruent agglomeration around any one nuclear point produces the VE, which Fuller calls the minimum, prime, nucleated structural system of Universe, just as the non-nucleated tetrahedron is the minimum prime structural system (§421.05, §421.21). Every vertex of the IVM is thus a potential VE center from which the matrix can be radiantly generated or regenerated at any selectable (tunable) wavelength (§426.02, §426.40). And the two clear-space cells — the tetrahedron and octahedron — are themselves compounded from just two rational fractional sub-units, the A and B Quanta Modules, which are the least-common-denominator volumetric quanta into which the IVM's polyhedra resolve (§422.04; cf. §920).

See Also

• Vector Equilibrium (Vector Equilibrium) — the IVM's nuclear expression
• Closest Packing of Spheres (Closest Packing of Spheres) — generates the IVM
• A and B Quanta Modules (A and B Quanta Modules) — the volumetric sub-units of the IVM cells
• Synergetics (Synergetics) — Fuller's system
• Octet Truss (Octet Truss) — the IVM realized as a structural space frame

Sources

• Synergetics — Fuller's primary text; Isotropic Vector Matrix at §420.00–426.47 (esp. §420.01 generation from closest packing, §420.02 four-dimensional 60-degree omnirational coordination, §420.07–420.08 triangles and squares, §421 function of nucleus, §422 octet truss and A/B Quanta Modules, §423 60-degree coordination, §424 complementary symmetry), with volumetric-unity relations at §973.30 ff.
• Fuller's Coined & Redefined Terms