Fig. 987.230 
987.230 Symmetries #1 & 3; Cleavages #1 & 2 
987.231 Of the seven equatorial symmetries first employed in the progression of self fractionations or cleavages, we use the tetrahedron's six midedge poles to serve as the three axes of spinnability. These three greatcircle spinnings delineate the succession of cleavages of the 12 edges of the tetracontained octahedron whose six vertexes are congruent with the regular tetrahedron's six midedge polar spin points. The octahedron resulting from the first cleavage has 12 edges; they produce the additional external surface lines necessary to describe the twofrequency, nontimesize subdividing of the primitive onefrequency tetrahedron. (See Sec. 526.23, which describes how four happenings' loci are required to produce and confirm a system discovery.) 
987.232 The midpoints of the 12 edges of the octahedron formed by the first cleavage provide the 12 poles for the further greatcircle spinning and Cleavage #2 of both the tetra and its contained octa by the six great circles of Symmetry #3. Cleavage #2 also locates the centerofvolume nucleus of the tetra and separates out the centerofvolume surrounding 24 A Quanta Modules of the tetra and the 48 B Quanta Modules of the two frequency, tetracontained octa. (See Sec. 942 for orientations of the A and B Quanta Modules.) 
Fig. 987.240 
987.240 Symmetry #3 and Cleavage #3 
Fig. 987.241 
987.241 Symmetry #3 and Cleavage #3 mutually employ the sixpolarpaired, 12 midedge points of the tetracontained octa to produce the six sets of greatcircle spinnabilities that in turn combine to define the two (one positive, one negative) tetrahedra that are intersymmetrically arrayed with the commonnuclearvertexed location of their eight equiinterdistanced, outwardly and symmetrically interarrayed vertexes of the "cube"^{__}the otherwise nonexistent, symmetric, squarewindowed hexahedron whose overall most economical intervertexial relationship lines are by themselves unstructurally (nontriangularly) stabilized. The positive and negative tetrahedra are internally trussed to form a stable eightcornered structure superficially delineating a "cube" by the most economical and intersymmetrical interrelationships of the eight vertexes involved. (See Fig. 987.240.) 
Fig. 987.242 
987.242 In this positivenegative superficial cube of tetravolume3 there is combined an eightfaceted, asymmetric hourglass polyhedron of tetravolumel½, which occurs interiorly of the interacting tetrahedra's edge lines, and a complex asymmetric doughnut cored hexahedron of tetravolume 1½, which surrounds the interior tetra's edge lines but occurs entirely inside and completely fills the space between the superficially described "cube" defined by the most economical interconnecting of the eight vertexes and the interior 1½tetravolume hourglass core. (See Fig. 987.242E987.242E.) 
987.243 An illustration of Symmetry #3 appears at Fig. 455.11A. 
987.250 Other Symmetries 
987.251 An example of Symmetry #4 appears at Fig. 450.10. An example of Symmetry #5 appears at Fig. 458.12B. An example of Symmetry #6 appears at Fig. 458.12A. An example of Symmetry #7 appears at Fig. 455.20. 
987.300 Interactions of Symmetries: Spheric Domains 
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 similarsurface triangles ADH and AIH at Fig. 987.21ON. The 48 similar triangles (24 plus, 24 minus) are the surfacesystem set of the 48 similar asymmetric tetrahedra whose 48 central vertexes are congruent in the one^{__}VE's^{__}nuclear vertex's center of volume. 
Fig. 987.312 
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 tetravolume6. This rhombic dodecahedron's hierarchical significance is elsewhere identified as the allspacefilling domain of each closestpacked, unitradius sphere in all isotropic, closestpacked, unitradius sphere aggregates, as the rhombic dodecahedron's domain embraces both the unitradius 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 unitradius spheres closest packed tangentially around each nuclear unitradius 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 sphere's exact center. 
987.314 These four interhexagonalling planes may also be seen as the tetrahedron of zerotimesizevolume because all of the latter's equiedge lengths, its face areas, and system volumes are concurrently at zero. 
987.315 This fourgreatcircle interaction in turn defines the 24 equilengthed vectorial radii and 24 equilengthed 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, 100percentenergy efficient, selfregenerative integrity by providing the most expansive state of intertransformation accommodation of the original hierarchy of primitive, pretimesize, "clickstop" rational states of energyinvolvement accountabilities. Here we have in the VE the eight possible phases of the initial positivenegative tetrahedron occurring as an interdoublebonded (edgebonded), vertexpaired, selfintercoupling nuclear system. 
987.316 With the nucleated set of 12 equiradius vertexial spheres all closest packed around one nuclear unitradius 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 Halfoctahedra 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 selfstructuring pattern stability of the 12 symmetrically interarrayed, but nonsphericallynucleated icosahedron. 
987.318
When this denucleation happens, the long diagonals
of the six squares
contract to unitvectorradius length. The squares that
were enclosed on all four sides by
unit vectors were squares whose edges^{__}being exactly
unity^{__}had a diagonal hypotenuse
whose length was the second root of two^{__}ergo, when VE
is transformed to the
icosahedron by the removal of the nuclear sphere, six
of its sqrt(2)lengthed,
interattractiverelationship lines transform into a
length of 1, while the other 24 lines of
circumferential interattraction remain constant at unitvectorradius
length. The difference
between the second root of two (which is 1.414214 
1, i.e., the difference is 0.414214)
occurs six times, which amounts to a total system contraction
of 2.485284. This in turn
means that the original
24 + 8.485284 = 32.485284
overall unitvectorlengths of containing bonds of the
VE are each reduced by a length of
2.485284 to an overall of exactly 30 unitvectorradius
lengths.

987.319 This 2.485284 a excess of gravitational tensionalembracement capability constitutes the excess of intertransformative stretchability between the VE's two alternatively unstable, omnisystem's stable states and its first two similarly stable, omnitriangulated 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 protonnucleated and protonneutron systems and nonnucleatednonneutroned electron systems, both having identical numbers of external closestpacked spheres, but having also different overall, systemdomain, volumetric, and systempopulation involvements. 
987.322 There is another important systemic difference between VE's protonneutron 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 singlelayer, circumferential closest packings; ergo, it permits only single spherical orbiting domains of equal number to the outer layers of VEnucleated, closestpacked 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 the transition at the structural stability of the icosahedron. The icosahedron experimentally evidences its further selffractionation by its three different polar greatcircle hemispherical cleavages that consistently follow the process of progressive selffractionations 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 unitvector delineating multiplication of information only by progressive subdividing of parts. 
987.324
When the tetrahedron is unity of tetravolume1 (see
Table
223.64), then (in
contradistinction to the vectorradiused VE of tetravolume20)

987.325 The positive and negative tetrahedra, when composited as symmetrically concentric and structurally stable, have eight symmetrically interarranged vertexes 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 triangularlyselfstabilizing system pattern. They occur frequently in nature's crystals but only as the superficial aspect of a conglomerate complex of omnitriangulated polyhedra. 
Fig. 987.326 
987.326
This positivenegative tetrahedron complex defines
a hexahedron of overall
volume3^{__}1½ inside and 1½ outside its intertrussed system's
insideandoutsidevertexdefined domain.

987.327 Repeating the foregoing more economically we may say that in this hierarchy of omnisymmetric primitive polyhedra ranging from I through 2, 2 , 3, 4, 5, and 6 tetravolumes, the rhombic dodecahedron's 12 diamondfacemidpoints 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 closestpacked spheres and therefore also of the nuclear domains of all isotropic vector matrixes (Sec. 420), but the rhombic dodecahedron is also the maximumlimitvolumed primitive polyhedron of frequencyl. 
Next Section: 987.400 