954.40 Functions of the Coupler: In their cosmic roles as the basic allspacefilling complementarity pair, our regular tetrahedron and regular octahedron are also always identified respectively by the disparate numbers 1 and 4 in the column of relative volumes on our comprehensive chart of the topological hierarchies. (See Chart 223.64.) The volume value 4^{__}being 2^{2} also identifies the prime number 2 as always being topologically unique to the symmetrical octahedron while, on the same topological hierarchy chart, the uniquely asymmetrical allspacefilling octahedron, the Coupler, has a volume of 1, which volume1identity is otherwise, topologically, uniquely identified only with the nonallspacefilling regular symmetrical tetrahedron. 
954.45 As learned in Sections 953 and 954, one plusbiased Mite and one minus biased Mite can be facebonded with one another in three different allspacefilling ways, yet always producing one energyproclivitybalanced, sixquantamoduled, double isosceles, allspacefilling, asymmetrical tetrahedron: the Syte. The asymmetric octahedron can also be composed of four such balancedbias Sytes (4 As^{__}2 + , 2 ^{__}and 2 Bs^{__}1 + , 1 ). Since there are eight always onewayortheotherbiased Mites in each uniquely asymmetrical octahedron, the latter could consist of eight positively biased or eight negatively biased Mites, or any omnigeometrically permitted mixed combination of those 16 (2^{4}) cases. 
954.46
There are always 24 modules (16 As and 8 Bs^{__}of which
eight As are always
positive and the eight other As are always negative,
while the eight Bs consist of any of the
eight possible combinations of positives and negatives)^{5}
in our uniquely asymmetrical
octahedron. It is important to note that this 24 is
the same 24module count as that of the
24Amoduled regular tetrahedron. We have named the
uniquely asymmetrical octahedron
the Coupler.
(Footnote 5:

954.57 We now understand why the K points are the kinetic switchoffendon points of Universe. 
954.72 There being three axes^{__}the X, Y, and M sets of obversereverse, polar viewed systems of eight^{__}each eight has 28 relationships, which makes a total of three times 28 = 84 integral axially regenerated, and 8 facetoface regenerated KtoK couplings, for a total of 92 relationships per Coupler. However, as the inspection and enumeration shows, each of the three sets of 28, and one set of 8 unique, holdortransmit potentials subgroup themselves into geometrical conditions in which some provide energy intertransmitting facilities at four different capacity (quantum) magnitudes: 0, 1, 2, 4 (note: 4 = 2^{2}), and in three axial directions. The XX' axis transmits between^{__}or interconnects^{__}every spheric center with one of its 12 tangentially adjacent closestpacked spheres. 
954.73 The YY' axis transmits between^{__}or interconnects^{__}any two adjacent of the six octahedrally and symmetrically interarrayed, concave vector equilibria conformed, `tweenspace, volumetric centers symmetrically surrounding every unitradius, closest packed sphere. 
954.74 The MM' axis interlinks, but does not transmit between, any two of the cubically and symmetrically interarrayed eight concave octahedra conformed sets of `tweenspace, concave, empty, volumetric centers symmetrically surrounding every unit radius, closestpacked sphere in every isotropic vector matrix of Universe. 
954.75 The eight KtoK, facetoface, couplings are energizingly interconnected by one Mite each, for a total of eight additional interconnections of the Coupler. 
954.76 These interconnections are significant because of the fact that the six concave vector equilibria, YY' axisconnected `tweenspaces, together with the eight concave octahedral `tweenspaces interconnected by the MM' axis, are precisely the set of spaces that transform into spheres (or convex vector equilibria) as every sphere in closestpacked, unitradius, sphere aggregates transforms concurrently into either concave vector equilibria `tweenspaces or concave octahedra `tweensphere spaces. 
954.77 This omniintertransformation of spheres into spaces and spaces into spheres occurs when any single force impinges upon any closestpacked liquid, gaseous, or plasmically closestpacked sphere aggregations. 
954.78 The further subdivision of the A Modules into two subtetrahedra and the subdividing of the B Modules into three subtetrahedra provide every positive Mite and every negative Mite with seven plusorminus subtetrahedra of five different varieties. Ergo 92 × 7 = 644 possible combinations, suggesting their identification with the chemical element isotopes. 
955.00 Modular Nuclear Development of AllspaceFilling 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 closestpacked spheres as well as all the unique closestpacked spatial domains of that one sphere. The universal versatility of the A and B Quanta Modules permits the omniinvertibility of those same 144 Modules within the exact same polyhedral shell space of the same size rhombic dodecahedron, with the omniinversion resulting in six l/6th 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 halfoctahedra, while also constituting the simplest and first order of nuclear, isotropically defined, uniformly modulated, inwardoutward andaround, vectortensor structuring, whereby the vector equilibrium of initial frequency, i.e., "plus and minus one" equilibrium, is sometimes identified only as "potential," whose uniformlength 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 closepack 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 12aroundone, closest symmetrical packing of rhombic dodecahedra produces a 12knobbed, 14valleyed complex polyhedral aggregate and not a single simplex polyhedron. 
955.04 Since each rhombic dodecahedron consists of 144 modules, 13 × 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., 29/36ths, 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 7/36ths of one spherical domain inside the initial frequency vector equilibrium. For example, 12 × 7 = 84/36 = 2 1/3 + 1 = 3 1/3 spheric domains inside the vector equilibrium of 480 quanta modules, compared with 144 ' 3.333 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 EighthOctahedra, with the latter's single, equiangular, triangular faces situated congruently with the eight external triangular facets of the vector equilibrium. 
955.08 Each eighthoctahedron consists of six A and six B Quanta Modules. Applying the eight 12moduled, 90degreeapexed, or "cornered," eighthoctahedra to the vector equilibrium's eight triangular facets produces an allspacefilling cube consisting of 576 modules: one octahedron = 8 × 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 eighthoctahedra with 90degree corners to the vector equilibrium's eight triangular facets, we had added six halfoctahedra "pyramids" to the vector equilibrium's six square faces, it would have produced a two frequency octahedron with a volume of 768 modules: 6 × 48 = 288 + 480 = an octahedron of 768 modules. 
955.10 Mexican Star: If we add both of the set of six halfoctahedra made up out of 48 modules each to the vector equilibrium's six square faces, and then add the set of eight EighthOctahedra 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 14Pointed Star," has six squarebased points and eight triangularbased points. The volume of the Mexican 14Pointed Star is 96 + 288 + 480 = 864 modules. 
955.11 Not until we complete the twofrequency vector equilibrium have we finally enclosed all the original 12 spheres surrounding the singlesphere nucleus in one single polyhedral system. However, this second vectorequilibrium shell also encloses the inward portions of 42 more embryo spheres tangentially surrounding and constituting a second closestpacked concentric sphere shell embracing the first 12, which in turn embrace the nuclear sphere; and because all but the corner 12 of this second closestpacked 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, 14valleyed, polyhedral star complex. 
955.13 The 3,840 modules of the twofrequency 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 twofrequency 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 1,968  1,872 = 96 = 1 octahedron. 
955.14 Each symmetrical increase of the vectorequilibrium 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 numbers^{__}as, for instance, with the addition of the first extra shell of the twofrequency 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 ClosestPacked, Uniradius Spheres: The subtraction of the 144 modules of the nuclear sphere set from the 480module 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 selfembracement by additional closestpacked spheres and their spacesharing domains. 
955.21 The above produces 28/144ths = 7/36ths present, and 1l6/144ths = 29/36ths per each needed. 
955.30 Possible Relevance to Periodic Table of the Elements: These are interesting numbers because the 28/l44ths and the 116/144ths, reduced to their least common denominator, disclose two prime numbers, i.e., seven and twentynine, 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 intersymmetricphase maximum aberration and asymmetric pulsations. (See Sec. 1238 for the Scheherazade Number accommodating these permutations.) 
955.40
Table: Hierarchy of A and B Quanta Module Development
of Omni
ClosestPacked, Symmetric, Spherical, and Polyhedral,
Common Concentric
Growth Rates Around One Nuclear Sphere, and Those Spheres'
Respective
Polyhedral, AllspaceFilling, Unique Geometrical Domains
(Short Title: Concentric
Domain Growth Rates)

955.41
Table: Spherical Growth Rate Sequence

Next Section: 955.50 