905.00 Equilibrium and Disequilibrium Modelability
Fig. 905.02 
905.02
Unity as two is inherent in life and the resulting
model is tetrahedral, the
conceptuality of which derives as follows:

905.10 Doubleness of Unity 
905.11 The prime number twoness of the octahedron always occurs in structuring doubled together as four^{__}i.e., 2^{2} ^{__}a fourness which is also doubleness of unity. Unity is plural and, at minimum, is two. The unity volume 1 of the tetrahedron is, in structural verity, two, being both the outwardly displayed convex tetrahedron and the inwardly contained concave tetrahedron. (See Chart 223.64 , columns 2, 12, and 15) 
905.12 The threegreatcircle model of the spherical octahedron only "seems" to be three; it is in fact "double"; it is only foldably produceable in unbroken (whole) greatcircle sheets by edgecombining six hemicircularly folded whole great circles (see Sec. 840 ). Thus it is seen that the octahedron^{__}as in Iceland spar crystals^{__}occurs only doubly, i.e., omnicongruent with itself, which is "quadrivalent." 
905.13 Among the three possible omnisymmetrical prime structural systems^{__}the tetrahedron, octahedron, and icosahedron^{__}only the tetrahedron has each of its vertexes diametrically opposite a triangular opening. (See Illus. 610.2.) In the octahedron and icosahedron, each vertex is opposite another vertex; and each of their vertexes is diametrically blocked against articulating a selfinsideouting transformation. In both the octahedron and the icosahedron, each of the vertexes is tensevectorrestrained from escaping outwardly by the convergent vectorial strength of the system's other immediately surrounding^{__}at minimum three^{__}vertexial event neighbors. But contrariwise, each of the octahedron's and icosahedron's vertex events are constrainingly impulsed inwardly in an exact centralsystem direction and thence impelled toward diametric exit and insideouting transformation; and their vertex events would do so were it not for their diametrically opposed vertexes, which are surroundingly tensevectorrestrained from permitting such outward egress. 
905.14 As a consequence of its uniquely unopposed diametric vertexing^{__}ergo permitted^{__}diametric exit, only the tetrahedron among all the symmetric polyhedra can turn itself pulsatingly insideout, and can do so in eight different ways (see Sec. 624 ); and in each instance, as it does so, onehalf of its combined concaveconvex unity "twoness" is always inherently invisible. 
905.17 Any and all of the icosahedron's vertexes pulsate individually and independently from the convex to concave state only in the form of local dimpling, because each onlyfromoutwardmotionrestrained vertex^{__}being free to articulate inwardly toward its system center, and having done so^{__}becomes abruptly fivevector restrained by its immediate neighboring vertexial event convergences; and the abrupt halting of its inward travel occurs before it reaches the system center. This means that one vertex cannot pulse inwardly more deeply than a local dimple similar to the popping in of a derby hat. (See Sec. 618.30 .) 
905.18 Both the coexisting concave and convex aspects of the icosahedron^{__}like those of the octahedron, but unlike those of the unique case of the tetrahedron^{__}are always visually obvious on the inside and outside of the only locally dimpledin, or nested in, vertex. In both the octahedron and the icosahedron, the concaveconvex, only inwardly pulsative selftransforming always produces visually asymmetrical transforming; whereas the tetrahedron's permitted insideouting pulsatively results only in a visible symmetry, the quasiasymmetry being invisibly polarized with the remainder of Universe outside the tetrahedron, which, being omniradially outward, is inferentially^{__}but not visually^{__}symmetrical; the only asymmetrical consideration of the tetrahedron's inside outing being that of an initial direction of vertexial exiting. Once exited, the visible remaining symmetrical tetrahedron is in verity the insideoutness of its previously visible aspects. (See Sec. 232.01 .) 
905.20 The vertexes are the unique, individual, ergo intime events; and the nonvertex voids are the outdividual, ergo out, timeless, sizeless nonevents. The both outwardly and inwardly escaping nonevents complement the embryo, localintime, specialcase, convergentevent, systemic pattern fixation of individual intercomplementary event identities. (See Sec. 524 .) 
905.22 Socalled edges and vectors are inherently only convergent or divergent interrelationships between multiplyidentifiable, pointtoable, vertex fixes. 
905.30 Hierarchy of Pulsating Tetrahedral Arrays 
905.31 Among the exclusively, three and only, prime cosmic structural systems^{__}the tetra, octa, and icosa^{__}only the tetrahedron's pulsative transforming does not alter its overall, visually witnessable symmetry, as in the case of the "cheese tetrahedron" (see Sec. 623.00). It is important to comprehend that any one of the two sets of four each potential vertexial insideouting pulsatabilities of the tetrahedron^{__}considered only by themselves^{__}constitutes polarized, but only invisible, asymmetry. In one of the two sets of four each potential insideoutings we have threeofeachtothreeoftheother (i.e., trivalent, triangular, basetobase) vertexial bonding together of the visible and invisible, polarized pair of tetrahedra. In the other of the two sets of four each potential inside outings we have onevertextoonevertex (i.e., univalent, apextoapex) interbonding of the visible and invisible polarized pair of tetrahedra. 
905.32 Because each simplest, ergo prime, structural system tetrahedron has at minimum four vertexes (pointtoable, systemic, eventpatterned fixes), and their four complementary system exitouts, are symmetrically identified at midvoid equidistance between the three other convergent event identity vertexes; and because each of the two sets of these four halfvisible/halfinvisible, polarpaired tetrahedra have both threevertex tothreevertex as well as singlevertextosinglevertex insideout pulsatabilities; there are eight possible insideouting pulsatabilities. We have learned (see Sec. 440 ) that the vector equilibrium is the nuclearembracing phase of all eight "empty state" tetrahedra, all with common, central, singlevertextosinglevertex congruency, as well as with their mutual outwardradiusends' vertexial congruency; ergo the vector equilibrium is bivalent. 
905.33 The same vector equilibrium's eight, nuclearembracing, bivalent tetrahedra's eight nuclear congruent vertexes may be simultaneously outwardly pulsed through their radiallyopposite, outward, triangular exits to form eight externally pointing tetrahedra, which thus become only univalently, i.e., onlysinglevertex interlinked, and altogether symmetrically arrayed around the vector equilibrium's eight outward "faces." The thus formed, eightpointed star system consisting of the vector equilibrium's volume of 20 (tetrahedral unity), plus the eight starpointarrayed tetrahedra, total volumetrically to 28. This number, 28, introduces the prime number seven factored exclusively with the prime number two, as already discovered in the unitytwoness of the tetrahedron's always and only, cooccurring, concaveconvex inherently disparate, behavioral duality. This phenomenon may be compared with the 28ness in the Coupler accounting (see Sec. 954.72). 
905.35 The jitterbug shows that the bivalent vector equilibrium contracts to the octahedral trivalent phase, going from a twentyness of volume to a fourness of volume, 204, i.e., a 5:1 contraction, which introduces the prime number five into the exclusively tetrahedrally evolved prime structural system intertransformabilities. We also witness that the octahedron state of the jitterbug transforms contractively even further with the 60 degree rotation of one of its triangular faces in respect to its nonrotating opposite triangular face^{__}wherewith the octahedron collapses into one, flattenedout, twovector length, equiedged triangle, which in turn consists of four onevectoredged, equiangled triangles, each of which in turn consists of two congruent, onevectorlong, equiedged triangles. All eight triangles lie together as two congruent sets of four small, onevector long, equiedged triangles. This centrally congruent axial force in turn plunges the two centrally congruent triangles through the inertia of the three sets of two congruent, edge hinged triangles on the three sides of the congruent pair of central triangles which fold the big triangle's corners around the central triangle in the manner of the three petals folding into edge congruence with one another to produce a tetrahedrally shaped flower bud. Thus is produced one tetrahedron consisting of four quadrivalently congruent tetrahedra, with each of its six edges consisting of four congruent vectors. The tetrahedron thus formed, pulsatively reacts by turning itself insideout to produce, in turn, another quadrivalent, fourtetrahedra congruence; which visibletovisible, quadrivalent tetrahedral insideouting/outsideinning is pulsatively regenerative. (See Illus. 461.08.) 
905.40 As we jitterbuggingly transform contractively and symmetrically from the 20volume bivalent vector equilibrium phase to the 8volume quadrivalent octahedral phase, we pass through the icosahedral phase, which is nonselfstabilizing and may be stabilized only by the insertion of six additional external vector connectors between the 12 external vertexes of the vector equilibrium travelling toward convergence as the six vertexes of the trivalent 4volume octahedron. These six vectors represent the edge vectors of one tetrahedron. 
905.41 The 28volume, univalent, nucleusembracing, tetrahedral array extends its outer vertexes beyond the bounds of the nucleusembracing, closestpacked, omnisymmetrical domain of the 24volume cube formed by superimposing eight Eighth Octahedra, asymmetrical, equianglebased, threeconvergent90degreeangleapexed tetrahedra upon the eight outward equiangular triangle facets of the vector equilibrium. We find that the 28ness of freespace expandability of the univalent, octahedral, nucleus embracement must lose a volume of 4 (i.e., four tetrahedra) when subjected to omniclosestpacking conditions. This means that the dynamic potential of omniinterconnected tetrahedral pulsation system's volumetric embracement capability of 28, upon being subjected to closestpacked domain conditions, will release an elsewhere structurallyinvestable volume of 4. Ergo, under closestpacked conditions, each nuclear array of tetrahedra (each of which is identifiable energetically with one energy quantum) may lend out four quanta of energy for whatever tasks may employ them. 
905.42 The dynamic vs. kinetic difference is the same difference as that of the generalized, sizeless, metaphysically abstract, eternal, constant sixnessofedge, foumess ofvertex, and fournessofvoid of the onlybymindconceptual tetrahedron vs. the only intimesized, specialcase, brainsensed tetrahedron. This generalized quality of being dynamic, as being one of a plurality of inherent systemic conditions and potentials, parts of a whole set of eternally cooccurring, complex interaccommodations in pure, weightless, mathematical principle spontaneously producing the minimum structural systems, is indeed the prime governing epistemology of wave quantum physics. 
905.43 In consideration of the tetrahedron's quantum intertransformabilities, we have thus far observed only the expandablecontractable, variablebondingpermitted consequences. We will now consider other dynamical potentials, such as, for instance, the axial rotatabilities of the respective tetras, octas, and icosas. 
905.44 By internally interconnecting its six vertexes with three polar axes: X, Y, and Z, and rotating the octahedron successively upon those three axes, three planes are internally generated that symmetrically subdivide the octahedron into eight uniformly equal, equiangletrianglebased, asymmetrical tetrahedra, with three convergent, 90 degreeanglesurrounded apexes, each of whose volume is oneeighth of the volume of one octahedron: this is called the EighthOctahedron. (See also Sec. 912.) The octahedron, having a volume of four tetrahedra, allows each EighthOctahedron to have a volume of onehalf of one tetrahedron. If we apply the equiangledtriangular base of one each of these eight EighthOctahedra to each of the vector equilibrium's eight equiangle triangle facets, with the EighthOctahedra's three90degreeanglesurrounded vertexes pointing outwardly, they will exactly and symmetrically produce the 24volume, nucleus embracing cube symmetrically surrounding the 20volume vector equilibrium; thus with 8 × 1/2 = 4 being added to the 20volume vector equilibrium producing a 24volume total. 
905.45 A nonnucleusembracing 3volume cube may be produced by applying four of the EighthOctahedra to the four equiangled triangular facets of the tetrahedron. (See Illus. 950.30.) Thus we find the tetrahedral evolvement of the prime number three as identified with the cube. Ergo all the prime numbers^{__}1, 2, 3, 5, 7^{__}of the octave wave enumeration system, with its zeronineness, are now clearly demonstrated as evolutionarily consequent upon tetrahedral intertransformabilities. 
905.46 Since the tetrahedron becomes systematically subdivided into 24 uniformly dimensioned A Quanta Modules (one half of which are positive and the other half of which are negatively insideout of the other); and since both the positive and negative A Quanta Modules may be folded from one whole triangle; and since, as will be shown in Sec. 905.62 the flattenedout triangle of the A Quanta Module corresponds with each of the 120 disequilibrious LCD triangles, it is evidenced that five tetrahedra of 24 A Quanta Modules each, may have their sumtotal of 120 A Modules all unfolded, and that they may be edgebonded to produce an icosahedral spherical array; and that 2 1/2 tetrahedra's 60 A Quanta Modules could be unfolded and univalently (singlebondedly) arrayed to produce the same spheric icosahedral polyhedron with 60 visible triangles and 60 invisible triangular voids of identical dimension. 
905.47
Conversely, 60 positive and 60 negative A Quanta Modules
could be folded
from the 120 A Module triangles and, with their "sharpest"
point pointed inward, could be
admitted radially into the 60positive60negative tetrahedral
voids of the icosahedron.
Thus we discover that the icosahedron, consisting of
120 A Quanta Modules (each of
which is 1/24th of a tetrahedron) has a volume of 120/24
= 5 The icosahedron volume is 5
when the tetrahedron is 1; the octahedron 2^{2} ; the cube
3; and the starpointed, univalent,
eighttetrahedra nuclear embracement is 28, which is
4 × 7; 28 also being the maximum
number of interrelationships of eight entities:

905.48 The three surrounding angles of the interior sharpest point of the A Quanta Module tetrahedron are each a fraction less than the three corresponding central angles of the icosahedron: being approximately onehalf of a degree in the first case; one whole degree in the second case; and one and threequarters of a degree in the third case. This loosefit, volumetricdebit differential of the A Quanta Module volume is offset by its being slightly longer in radius than that of the icosahedron, the A Module's radial depth being that of the vector equilibrium's, which is greater than that of the icosahedron, as caused by the reduction in the radius of the 12 balls closestpacked around one nuclear ball of the vector equilibrium (which is eliminated from within the same closestradially packed 12 balls to reduce them to closest surfacepacking, as well as by eliminating the nuclear ball and thereby mildly reducing the system radius). The plus volume of the fractionally protruded portion of the A Quanta Module beyond the icosahedron's surface may exactly equal the interior minus volume difference. The balancing out of the small plus and minus volumes is suggested as a possibility in view of the exact congruence of 15 of the 120 spherical icosahedra triangles with each of the spherical octahedron's eight spherical equiangle faces, as well as by the exact congruence of the octahedron and the vector equilibrium themselves. As the icosahedron's radius shortens, the central angles become enlarged. 
905.49 This completes the polyhedral progression of the omniphasebond integrated hierarchies of^{__}1234, 8^{__}symmetrically expanded and symmetrically subdivided tetrahedra; from the 1/24thtetrahedron (12 positive and 12 negative A Quanta Modules); through its octavalent 8in1 superficial volume1; expanded progressively through the quadrivalent tetrahedron; to the quadrivalent octahedron; to the bivalent vector equilibrium; to the univalent, 28volume, radiant, symmetrical, nucleusembracing stage; and thence exploded through the volumeless, flatoutoutfolded, doublebonded (edgebonded), 120AQuantaModuletriangular array remotely and symmetrically surrounding the nuclear volumetric group; to final dichotomizing into two such flatout half (positive triangular) film and half (negative triangular) void arrays, singlebonded (corner bonded), icosahedrally shaped, symmetrically nuclearsurrounding systems. 
905.50 Rotatability and Split Personality of Tetrahedron 
905.51 Having completed the expansivecontractive, couldbe, quantum jumps, we will now consider the rotatability of the tetrahedron's sixedge axes generation of both the two spherical tetrahedra and the spherical cube whose "split personality's" fourtriangle defining edges also perpendicularly bisect all of both of the spherical tetrahedron's four equiangled, equiedged triangles in a threeway grid, which converts each of the four equiangled triangles into six rightangle spherical triangles^{__}for a total of 24, which are split again by the spherical octahedron's three great circles to produce 48 spherical triangles, which constitute the 48 equilibrious LCD Basic Triangles of omniequilibrious eventless eternity (see Sec. 453). 
905.52 The spherical octahedron's eight faces become skewsubdivided by the icosahedron's 15 great circles' selfsplitting of its 20 equiangular faces into sixeach, right spherical triangles, for an LCD spherical triangle total of 120, of which 15 such right triangles occupy each of the spherical octahedron's eight equiangular faces^{__}for a total of 120^{__}which are the same 120 as the icosahedron's 15 great circles. 
905.53 The disequilibrious 120 LCD triangle = the equilibrious 48 LCD triangle × 2½. This 2½ + 2½ = 5; which represents the icosahedron's basic fiveness as split generated into 2½ by their perpendicular, midedgebisecting 15 great circles. Recalling the six edge vectors of the tetrahedron as one quantum, we note that 6 + 6 + 6/2 is 1 + 1 + 1/2 = 2½ ; and that 2½ × 6 = 15 great circles. (This halfpositive and half negative dichotomization of systems is discussed further at Sec. 1053.30ff.) 
905.54 We find that the split personality of the icosahedron's 15greatcircle splittings of its own 20 triangles into 120, discloses a basic asymmetry caused by the incompleteness of the 2½, where it is to be seen in the superimposition of the spherical icosahedron upon the spherical vector equilibrium. In this arrangement the fundamental prime number fiveness of the icosahedron is always split two ways: 2½ positive phase and 2½ negative phase. This halffiving induces an alternate combining of the half quantum on one side or the other: going to first three on one side and two on the other, and vice versa. 
905.55 This halfoneside/halfontheother induces an oscillatory alternating 120 degreearc, partial rotation of eight of the spherical tetrahedron's 20 equiangled triangles within the spherical octahedron's eight triangles: 8 × 2½ = 20. We also recall that the vector equilibrium has 24 internal radii (doubled together as 12 radii by the congruence of the fourgreatcircle's hexagonal radii) and 24 separate internal vector chords. These 24 external vector chords represent four quanta of six vectors each. When the vector equilibrium jitterbuggingly contracts toward the octahedral edgevector doubling stage, it passes through the unstable icosahedral stage, which is unstable because it requires six more edgevectors to hold fixed the short diagonal of the six diamondshaped openings between the eight triangles. These six equilength vectors necessary to stabilize the icosahedron constitute one additional quantum which, when provided, adds 1 to the 4 of the vector equilibrium to equal 5, the basic quantum number of the icosahedron. 
905.60 The Disequilibrium 120 LCD Triangle 
905.61
The icosahedral spherical greatcircle system displays: 12 vertexes surrounded by 10 converging angles; 20 vertexes surrounded by 6 converging angles; 30 vertexes surrounded by 4 converging angles

905.62 According to the Principle of Angular Topology (see Sec. 224 ), the 360 converging angle sinuses must share a 720degree reduction from an absolute sphere to a chorded sphere: 720°/360° = 2°. An average of 2 degrees angular reduction for each comer means a 6 degrees angular reduction for each triangle. Therefore, as we see in each of the icosahedron's disequilibrious 120 LCD triangles, the wellknown architects and engineers' 30°60°90° triangle has been spherically opened to 36°60°90° ^{__}a "spherical excess," as the Geodetic Survey calls it, of 6 degrees. All this spherical excess of 6 degrees has been massaged by the irreducibility of the 90degree and 60degree corners into the littlest corner. Therefore, 3036 in each of the spherical icosahedron's 120 surface triangles. 
905.63 In subsiding the 120 spherical triangles generated by the 15 great circles of the icosahedron from an omnispherical condition to a neospheric 120planarfaceted polyhedron, we produce a condition where all the vertexes are equidistant from the same center and all of the edges are chords of the same spherical triangle, each edge having been shrunk from its previous arc length to the chord lengths without changing the central angles. In this condition the spherical excess of 6 degrees could be shared proportionately by the 90°, 60°, 30° flat triangle relationship which factors exactly to 3:2:1. Since 6° = 1/30 of 180° , the 30 quanta of six each in flatout triangles or in the 120 LCD spherical triangles' 186 degrees, means one additional quantum crowded in, producing 31 quanta. 
905.64
Alternatively, the spherical excess of 6 degrees (one
quantum) may be
apportioned totally to the biggest and littlest corners
of the triangle, leaving the 60degree,
vector equilibrium, neutral corner undisturbed. As we
have discovered in the isotropic
vector matrix nature coordinates crystallographically
in 60 degrees and not in 90 degrees.
Sixty degrees is the vector equilibrium neutral angle
relative to which lifeintime
aberrates.

Table 905.65 
905.65 By freezing the 60degree center of the icosahedral triangle, and by sharing the 6degree, sphericalplanar, excess reduction between the 36degree and 90degree corners, we will find that the A Quanta Modules are exactly congruent with the 120 internal angles of the icosahedron. The minus 5° 16' closely approximates the one quantum 6 + of spherical excess apparent at the surface, with a comparable nuclear deficiency of 5° 16'. (See Table 905.65.) 
905.66 The Earth crustfault angles, steel plate fractionation angles, and ship's bow waves all are roughly the same, reading approximately 70degree and 110degree complementation. 



Next Section: 905.70 