987.310 Irrationality of Nucleated and Nonnucleated Systems

987.311 The six great circles of Symmetry #3 interact with the three great circles of Symmetry # 1 to produce the 48 similar-surface triangles ADH

1. Eighth-Octa composed of six asymmetric tetrahedra. Each asymmetric tetrahedron is composed of one A Quanta Module and one B Quanta Module. The drawing is labeled to show the relationship of the A Modules and the B Modules. Vertex A is at the center of volume of the octahedron and F is at the midface of any of the octahedron’s eight triangular faces.

2. Proximate assembly of Eighth-Octa and Quarter-Tetra to be face bonded together as Octet.

3. Octet: (Oc-Tet = octahedron + tetrahedron.) An Eighth-Octa is face bonded with a Quarter-Tetra to produce the Octet. (See Sec. 986.430.) The Octet is composed of 12 A Quanta Modules and 6 B Quanta Modules. (Compare color plate 22.)

4. E. Duo-Tet Cube: Alternate assemblies of eight Octets form Duo-Tet Cube. Each Duo-Tet Cube = 3-tetravolumes.

5. Rhombic Dodecahedron: Two Duo-Tet Cubes disassociate their Octet components to be reassembled into the Rhombic Dodecahedron. Rhombic Dodecahedron = 6-tetravolumes.

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987.312 These 48 asymmetric tetrahedra combine themselves into 12 sets of four asymmetric tetra each. These 12 sets of four similar (two positive, two negative) asymmetric tetrahedra combine to define the 12 diamond facets of the rhombic dodecahedron of tetravolume-6. This rhombic dodecahedron s hierarchical significance is elsewhere identified as the allspace-filling domain of each closest-packed, unit-radius sphere in all isotropic, closest-packed, unitradius sphere aggregates, as the rhombic dodecahedrons domain embraces both the unit-radius sphere and that sphere’s rationally and exactly equal share of the intervening intersphere space.

987.313 The four great circles of Symmetry #2 produce a minimum nucleated system of 12 unit-radius spheres closest packed tangentially around each nuclear unit-radius sphere; they also produce a polyhedral system of six square windows and eight triangular windows; they also produce four hexagonal planes of symmetry that all pass through the same nuclear vertex spheres exact center.

987.314 These four interhexagonalling planes may also be seen as the tetrahedron of zero-time-size-volume because all of the latters equi-edge lengths, its face areas, and system volumes are concurrently at zero.

987.315 This four-great-circle interaction in turn defines the 24 equi-lengthed vectorial radii and 24 equi-lengthed vector chords of the VE . The 24 radii are grouped, by construction, in two congruent sets, thereby to appear as only 12 radii. Because the 24 radial vectors exactly equal energetically the circumferentially closed system of 24 vectorial chords, we give this system the name vector equilibrium. Its most unstable, only transitional, equilibrious state serves nature’s cosmic, ceaseless, 100-percent-energy-efficient, self regenerative integrity by providing the most expansive state of intertransformation accommodation of the original hierarchy of primitive, pre-time-size, “click-stop” rational states of energy-involvement accountabilities. Here we have in the VE the eight possible phases of the initial positive-negative tetrah edron occurring as an inter-double-bonded (edge-bonded), vertex-paired, self-inter-coupling nuclear system.

1\sfrac{1}{2}-tetravolumes of “substance” within the eight vertexes and 1-tetravolumes of complementation domain within the eight vertexes. The overall cubic domain consists of three tetravolumes: one outside-out (116) and one inside-out (16). The same star polyhedron appears within a vector equilibrium net at Fig. 1006.32.

1. Octahedron: tetravolume-4

2. Icosahedron; tetravolume-18.51229586

3. Vector equilibrium: tetravolume-20

4. Eight-faceted asymmetric Hourglass Polyhedron: tetravolume-116. These complex asymmetric doughnut-cored hexahedra appear within the star polyhedron at A.

987.316 With the nucleated set of 12 equi-radius vertexial spheres all closest packed around one nuclear unit-radius sphere, we found we had eight tetrahedra and six Half-octahedra defined by this VE assembly, the total volume of which is 20. But all of the six Half-octahedra are completely unstable as the 12 spheres cornering their six square windows try to contract to produce six diamonds or 12 equiangular triangles to ensure their interpatterning stability. (See Fig. 987.240.)

987.317 If we eliminate the nuclear sphere, the mass interattraction of the 12 surrounding spheres immediately transforms their superficial interpatterning into 20 equiangular triangles, and this altogether produces the self-structuring pattern stability of the 12 symmetrically inter arrayed, but non-spherically-nucleated icosahedron.

987.318 When this denucleation happens, the long diagonals of the six squares contract to unit-vector-radius length. The squares that were enclosed on all four sides by unit vectors w e re squares whose edgesbeing exactly unity had a diagonal hypotenuse whose length was the second root of twoergo, when VE is transformed to the icosahedron by the removal of the nuclear sphere, six of its\sqrt{2}-lengthed, interattractive-relationship lines transform into a length of 1, while the other 24 lines of circumferential interattraction remain constant at unit-vector-radius length. The difference between the second root of two (which is 1.414214, i.e., the difference is .414214) occurs six times, which amounts to a total system contraction of 2.485284. This in turn means that the original block-katex 24 + 8.485284 = 32.485284 overall unit-vector-lengths of containing bonds of the VE are each reduced by a length of 2.485284 to an overall of exactly 30 unit-vector-radius lengths.

987.319 This 2.485284 A excess of gravitational tensional-embracement capability constitutes the excess of intertransformative stretchability between the VE’s two alter natively unstable, omnisystem’s stable states and its first two similarly stable, omni triangulated states.

987.320 Because the increment of instability tolerance of most comprehensive intertransformative events of the primitive hierarchy is an irrational increment, the nucleus-void icosahedron as a structural system is inherently incommensurable with the nucleated VE and its family of interrational values of the octahedral, tetrahedral, and rhombic dodecahedral states.

987.321 The irrational differences existing between nucleated and nonnucleated systems are probably the difference between proton-nucleated and proton- neutron systems a nd nonnucleated-nonneutroned electron systems, both having identical numbers of external closest-packed spheres, but having also different over all, system-domain, volumetric, and system-population involvements.

987.322 There is another important systemic difference between VE’s proton-neutron system and the nonnucleated icosahedron’s electron system: the icosahedron is arrived at by removing the nucleus, wherefore its contraction will not permit the multilayering of spheres as is permitted in the multilayerability of the VE ergo, it cannot have neutron populating as in the VE ; ergo, it permits only single-layer, circumferential closest packings; ergo, it permits only single spherical orbiting domains of equal number to the outer layers of VE -nucleated, closest-packed systems; ergo, it permits only the behavioral patterns of the electrons.

987.323 When all the foregoing is comprehended, it is realized that the whole concept of multiplication of information by division also embraces the concept of removing or separating out the nucleus sphere (vertex) from the VE’s structurally unstable state and, as the jitterbug model shows, arriving omnisymmetrically throughout th e transition at the structural stability of the icosahedron. The icosa hedron experimentally evidences its further self-fractionation by its three different polar great-circle hemispherical cleavages that consistently follow the process of progressive self-fractionations as spin-halved successively around respective #5 , #6, and #7 axes of symmetry. These successive halvings develop various fractions corresponding in arithmetical differentiation degrees, as is shown in this exploratory accounting of the hierarchy of unit-vector delineating multiplication of information only by progressive subdividing of parts.

987.324 When the tetrahedron is unity of tetravolume-1 (see Table 223.64), then (in contradistinction to the vector-radiused VE of tetravolume-20)

• the vector-diametered VE = +2\sfrac{1}{2} or = -2\sfrac{1}{2}

• a rational, relative primitive prime number 5 tetravolume is also only realizable with half of its behavioral potentials in the presently-tune-in-able macrocosm and the other half of its total 5 behavioral potential existent in the presently-tune-out-able microcosm; thus,

• an overall +5 tetravolume potential -2\sfrac{1}{2}ergo, $+
  5 - 2\sfrac= +2\sfrac$ or

• an overall -5 tetravolume potential +2\sfrac{1}{2}ergo, -5 + 2\sfrac{1}{2}= -2\sfrac{1}{2}

987.325 The positive and negative tetrahedra, when composited as symmetrically concentric and structurally stable, have eight symmetrically interarranged ver texes defining the corners of what in the past has been mistakenly identified as a primitive polyhedron, popularly and academically called the “cube” or hexahedron. Cubes do not exist primitively because they are structurally unstable, having no triangularly-self-stabilizing system pattern. They occur frequently in nature’s crystals but only as the superficial aspect of a conglomerate complex of omni triangulated polyhedra.

987.326 This positive-negative tetrahedron complex defines a hexahedron of overall volume-31\sfrac{1}{2}inside and1\sfrac{1}{2} outside its intertrussed system’s inside-and-outside-vertex-defined domain.

• The three-great-circle symmetrical cleavaging (#1) of the primitive tetrahedron produces the vector-edged octahedron of tetravolume-4.

• The vector-radiused rhombic triacontahedron, with its .9994833324 unit-vector-radius perpendicular to its midface center produces a symmetrical polyhedron of tetra volume-5.

• With its 12 diamond-face-centers occurring at unit-vector-radius, the rhombic dodecahedron has a tetravolume-6.

The rhombic dodecahedron exactly occupies the geometric domain of each unit-vector-radius sphere and that sphere’s external share of the symmetrically identical spaces intervening between closest-packed unit-radius spheres of any and all aggregates of unit-radius, closest-interpacked spheres. In this closest-packed condition each sphere within the aggregate always has 12 spheres symmetrically closest packed tangentially around it. The midpoints of the 12 diamond faces of the rhombic dodecahedron’s 12 faces are congruent with the points of tangency of the 12 surrounding spheres. All the foregoing explains why unit-radius rhombic dodecahedra fill allspace when joined together.

987.327 Repeating the foregoing more economically we may say that in this hierarchy of omnisymmetric primitive polyhedra ranging from1through$2, 2\sfrac
{2}, 3, 4, 5, and 6$ tetravolumes, the rhombic dodecahedron’s 12 diamondface-midpoints occur at the points of intertangency of the 12 surrounding spheres. It is thus disclosed that the rhombic dodecahedron is not only the symmetric domain of both the sphere itself and the sphere’s symmetric share of the space intervening between all closest-packed spheres and therefore also of the nuclear domains of all isotropic vector matrixes (Sec. 420 ), but the rhombic dodecahedron is also the maximum-limit-volumed primitive polyhedron of frequency-1.