955.00 Modular Nuclear Development of Allspace-Filling Spherical Domains

955.01 The 144 A and B Quanta Modules of the rhombic dodecahedron exactly embrace one whole sphere, and only one whole sphere of closest-packed spheres as well as all the unique closest-packed spatial domains of that one sphere. The universal versatility of the A and B Quanta Modules permits the omni-invertibility of those same 144 Modules within the exact same polyhedral shell space of the same size rhombic dodecahedron, with the omni-inversion resulting in six Veth spheres symmetrically and intertangentially deployed around one concave, octahedral space center.

955.02 On the other hand, the vector equilibrium is the one and only unique symmetric polyhedron inherently recurring as a uniformly angled, centrially triangulated, complex collection of tetrahedra and half-octahedra, while also constituting the simplest and first order of nuclear, isotropically defined, uniformly modulated, inward-outward-and-around, vector-tensor structuring, whereby the vector equilibrium of initial frequency, i.e., “plus and minus one” equilibrium, is sometimes identified only as “potential,” whose uniform-length 24 external chords and 12 internal radii, together with its 12 external vertexes and one central vertex, accommodates a galaxy of 12 equiradiused spheres closest packed around one nuclear sphere, with the 13 spheres’ respective centers omnicongruent with the vector equilibrium’s 12 external and one internal vertex.

955.03 Twelve rhombic dodecahedra close-pack symmetrically around one rhombic dodecahedron, with each embracing exactly one whole sphere and the respective total domains uniquely surrounding each of those 13 spheres. Such a 12-around-one, closest symmetrical packing of rhombic dodecahedra produces a 12-knobbed, 14-valleyed complex polyhedral aggregate and not a single simplex polyhedron.

955.04 Since each rhombic dodecahedron consists of 144 modules, 13 X 144=1,872 modules.

955.05 Each of the 12 knobs consists of 116 extra modules added to the initial frequency vector equilibrium’s 12 corners. Only 28 of each of the 12 spheres’ respective 144 modules are contained inside the initial frequency vector equilibrium, and 12 sets of 28 modules each are ⁷/36ths embracements of the full 12 spheres closest packed around the nuclear sphere.

955.06 In this arrangement, all of the 12 external surrounding spheres have a major portion, i.e., \sfrac{29}{36}ths, of their geometrical domain volumes protruding outside the surface of the vector equilibrium, while the one complete nuclear sphere is entirely contained inside the initial frequency vector equilibrium, and each of its 12 tangent spheres have Vseths of one spherical domain inside the initial frequency vector equilibrium. For example, 12x7 = [⁸⁴/3g] = 2^J/s + 1 = 3V3 spheric domains inside the vector equilibrium of 480 quanta modules, compared with 144x3.333 rhombic dodecahedron spherics = 479.5 + modules, which approaches 480 modules.

955.07 The vector equilibrium, unlike the rhombic dodecahedron or the cube or the tetrakaidecahedron, does not fill allspace. In order to use the vector equilibrium in filling allspace, it must be complemented by eight Eighth-Octahedra, with the latter’s single, equiangular, triangular faces situated congruently with the eight external triangular facets of the vector equilibrium.

955.08 Each eighth-octahedron consists of six A and six B Quanta Modules. Applying the eight 12-moduled, 90-degree-apexed, or “cornered,” eighth-octahedra to the vector equilibrium’s eight triangular facets produces an allspace-filling cube consisting of 516 modules: one octahedron = 8 x 12 modules = 96 modules. 96 + 480 modules = 576 modules. With the 576-module cube completed, the 12 (potential) vertexial spheres of the vector equilibrium are, as yet, only partially enclosed.

955.09 If, instead of applying the eight eighth-octahedra with 90-degree corners to the vector equilibrium’s eight triangular facets, we had added six half-octahedra “pyramids” to the vector equilibrium’s six square faces, it would have produced a two-frequency octahedron with a volume of 768 modules: 6 \times 48 = 288 + 480 = an octahedron of 768 modules.

955.10 Mexican Star: If we add both of the set of six half-octahedra made up out of 48 modules each to the vector equilibrium’s six square faces, and then add the set of eight Eighth-Octahedra consisting of 12 modules each to the vector equilibrium’s eight triangular facets, we have not yet completely enclosed the 12 spheres occurring at the vector equilibrium’s 12 vertexes. The form we have developed, known as the “Mexican 14-Pointed Star,” has six square-based points and eight triangular-based points. The volume of the Mexican 14-Pointed Star is 96 + 288 + 480 = 864 modules.

955.11 Not until we complete the two-frequency vector equilibrium have we finally enclosed all the original 12 spheres surrounding the singlesphere nucleus in one single polyhedral system. However, this second vector equilibrium shell also encloses the inward portions of 42 more embryo spheres tangentially surrounding and constituting a second closest-packed concentric sphere shell embracing the first 12, which in turn embrace the nuclear sphere; and because all but the corner 12 of this second closest-packed sphere shell nest mildly into the outer interstices of the inner sphere shell’s 12 spheres, we cannot intrude external planes parallel to the vector equilibrium’s 14 faces without cutting away the internesting portions of the sphere shells.

955.12 On the other hand, when we complete the second vector equilibrium shell, we add 3,360 modules to the vector equilibrium’s initial integral inventory of 480 modules, which makes a total of 3,840 modules present. This means that whereas only 1,872 modules are necessary to entirely enclose 12 spheres closest packed around one sphere, by using 12 rhombic dodecahedra closest packed around one rhombic dodecahedron, these 13 rhombic dodecahedra altogether produce a knobby, 14-valleyed, polyhedral star complex.

955.13 The 3,840 modules of the two-frequency vector equilibrium entirely enclosing 13 whole nuclear spheres, plus fractions of the 42 embryo spheres of the next concentric sphere shell, minus the rhombic dodecahedron’s 1,872 modules, equals 1,968 extra modules distributable to the 42 embryo spheres of the two-frequency vector equilibrium’s outer shell’s 42 fractional sphere aggregates omnioutwardly tangent to the first 12 spheres tangentially surrounding the nuclear sphere. Thus we learn that 1,9681,872 = 96 = 1 octahedron.

955.14 Each symmetrical increase of the vector-equilibrium system “frequency” produces a shell that contains further fractional spheres of the next enclosing shell. Fortunately, our A and B Quanta Modules make possible an exact domain accounting, in whole rational numbersas, for instance, with the addition of the first extra shell of the two-frequency vector equilibrium we have the 3,360 additional modules, of which only 1,872 are necessary to complete the first 12 spheres, symmetrically and embryonically arrayed around the originally exclusively enclosed nucleus. Of the vector equilibrium’s 480 modules, 144 modules went into the nuclear sphere set and 336 modules are left over.

955.20 Modular Development of Omnisymmetric, Spherical Growth Rate Around One Nuclear Sphere of Closest-Packed, Uniradius Spheres: The subtraction of the 144 modules of the nuclear sphere set from the 480-module inventory of the vector equilibrium at initial frequency, leaves 336 additional modules, which can only compound as sphere fractions. Since there are 12 equal fractional spheres around each corner, with 336 modules we have ³³⁶/12ths. ³³⁶/12ths = 28 modules at each corner out of the 144 modules needed at each corner to complete the first shell of nuclear self-embracement by additional closest-packed spheres and their space-sharing domains.

955.21 The above produces ²⁸/144ths = 7/36ths present, and ¹¹⁶/144ths = ²⁹/36ths per each needed.

955.30 Possible Relevance to Periodic Table of the Elements: These are interesting numbers because the ²⁸/144ths and the ¹¹⁶/144ths, reduced to their least common denominator, disclose two prime numbers, i.e., seven and twenty-nine, which, together with the prime numbers 1, 2, 3, 5, and 13, are already manifest in the rational structural evolvement with the modules’ discovered relationships of unique nuclear events. This rational emergence of the prime numbers 1, 3, 5, 7, 13, and 29 by whole structural increments of whole unit volume modules has interesting synergetic relevance to the rational interaccommodation of all the interrelationship permutation possibilities involved in the periodic table of the 92 regenerative chemical elements, as well as in all the number evolvements of all the spherical trigonometric function intercalculations necessary to define rationally all the unique nuclear vectorequilibrium intertransformabilities and their intersymmetric-phase maximum aberration and asymmetric pulsations. (See Sec. 1238.11 for the Scheherazade Number accommodating these permutations.)

955.40 Table: Hierarchy of A and B Quanta Module Development of Omni-Closest-Packed, Symmetric, Spherical, and Polyhedral, Common Concentric Growth Rates Around One Nuclear Sphere, and Those Spheres’ Respective Polyhedral, Allspace-Filling, Unique Geometrical Domains (Short Title: Concentric Domain Growth Rates)

+:---------------------+:--------------------------------------+:------------+
| | A and B Quanta Module Inventory | Spherical |
| | | |
| | | Domains |
±---------------------±--------------------------------------±------------+
| Rhombic Dodecahedron | = 144 modules | = 1 |
| | | |
| Initial-Frequency | | |
±---------------------±--------------------------------------±------------+
| Vector Equilibrium | = 480 modules | = 3 Vs |
±---------------------±--------------------------------------±------------+
| Octahedron | = 96 modules | ²/₃ |
±---------------------±--------------------------------------±------------+
| Cube | = 72 modules | V« |
±---------------------±--------------------------------------±------------+
| Tetrahedron | = 24 modules | Vs |
±---------------------±--------------------------------------±------------+

955.41 Table: Spherical Growth Rate Sequence

1. Modular Development of Omnisymmetric Spherical Growth Rate Around One Nuclear Sphere.

2. Nuclear Set of Rhombic Dodecahedron:

144 modules 1 sphere

1. Vector Equilibrium, Initial Frequency:

480 modulesItself and 2Vs additional spheres

1. CubeInitial Frequency:

576 modules 4 spheres

1. Octahedron, Two-Frequency:

768 modules 5Vs spheres

1. Mexican 14-Point Star:

864 modules 6 spheres

1. Rhombic Dodecahedron, 12-Knobbed Star:

1,872 modules 13 spheres

1. Vector Equilibrium, Two-Frequency:

3,840 modules 26Vo spheres

955.50 Rhombic Dodecahedron at Heart of Vector Equilibrium: Nature always starts every ever freshly with the equilibrious isotropic-vector-matrix field. Energy is not lost; it is just not yet realized. It can be realized only disequilibriously.

955.51 At the heart of the vector equilibrium is the ball in the center of the rhombic dodecahedron.

955.52 Look at the picture which shows one-half of the rhombic dodecahedron. (See Illus. 955.52.) Of all the polyhedra, nothing falls so readily into a closest-packed group of its own kind as does the rhombic dodecahedron, the most common polyhedron found in nature.