970.00 First- and Third-Power Progressions of Vector Equilibria

970.01 Operational Note: In making models or drawing the concentric growth of closest-packed sphere-shells, we are illustrating with great-circle cross sections through the center of the vector equilibrium; i.e., on one of its symmetrically oriented four planes of tetrahedral symmetry; i.e., with the hexagonally cross-section, concentric shells of half-KEs.

970.02 Your eye tends quickly to wander as you try to draw the closest- packed spheres’ equatorial circles. You have to keep your eye fixed on the mid-points of the intertriangulated vectorial lines in the matrix, the mid-points where the half-radiuses meet tangentially.

970.03 In the model of 10F^{2}+ 2, the green area, the space occupied by the sphere per se, is really two adjacent shells that contain the insideness of the outer shell and the outsideness of the inner shell. These combine to produce tangentially paired shellsergo, two layers.

970.10 Rationality of Planar Domains and Interstices: There is a 12F^{2}+ 2 omniplanar-bound, volumetric-domain marriage with the 10F^{2}+ 2 strictly spherical shell accounting. (See tables at Sec. 955.40 and at Sec. 971.00.)

970.11 Both the total inventories of spheres and their planar-bound domains of closest-packed sphere VE shells, along with their interstitial, “concave” faceted, exclusively vector equilibrium or octahedral spaces, are rationally accountable in nonfractional numbers.

970.12 Synergetics’ isotropic-vector-matrix, omnisymmetric, radiantly expansive or contractive growth rate of interstices that are congruent with closest-packed uniradius spheres or points, is also rational. There is elegant, omniuniversal, metaphysical, rational, whole number equating of both the planar-bound polyhedral volumes and the spheres, which relationships can all be discretely expressed without use of the irrational number pi (\pi), 3.14159, always required for such mathematical expression in strictly XYZ coordinate mathematics.

970.13 A sphere is a convexly expanded vector equilibrium, and all inter-closest-packed sphere spaces are concavely contracted vector equilibria or octahedra at their most disequilibrious pulsative moments.

970.20 Spheres and Spaces: The successive (20F^{3}) -20 (F-1)^{3} layer-shell, planar-bound, tetrahedral volumes embrace only the tangential inner and outer portions of the concentrically closest-packed spheres, each of whose respective complete concentric shell layers always number 10F^{2}+ 2. The volume of each concentric vector-equilibrium layer is defined and structured by the isotropic vector matrix, or octet truss, occurring between the spherical centers of any two concentric-sphere layers of the vector equilibrium, the inner part of one sphere layer and the outer part of the other, with only the center or nuclear ball being both its inner and outer parts (see drawings section).

970.21 There is realized herewith a philosophical synergetic sublimity of omnirational, universal, holistic, geometrical accounting of spheres and spaces without recourse to the transcendentally irrational pi(\pi). (See drawings section.)