1033.180 Vector Equilibrium: Potential and Primitive Tetravolumes 
1033.181 The potential activation of tetravolume quantation in the geometric hierarchy is still subfrequency but accounts for the doubling of volumetric space. The potential activation of tetravolume accounting is plural; it provides for nucleation. Primitive tetravolume accounting is singular and subnuclear. 
1033.182 When the isolated single sphere's vector equilibrium of tetravolume 2 l/2 is surrounded by 12 spheres to become a nuclear sphere, the vector equilibrium described by the innermosteconomicallyinterconnecting of the centers of volume of the 12 spheres comprehensively and tangentially surrounding the nuclear sphere^{__}as well as interconnecting their 12 centers with the center of the nuclear sphere^{__}has a tetravolume of 20, and the nuclear group's rhombic dodecahedron has a tetravolume of 48. 
1033.183 The tetravolume6 rhombic dodecahedron is the domain of each closest packed, unitradius sphere, for it tangentially embraces not only each sphere, but that sphere's proportional share of the intervening space produced by such unitradiussphere closest packing. 
1033.184
When the timesizing is initiated with frequency^{2},
the rhombic
dodecahedron's volume of 6 is eightfolded to become
48. In the plurality of closest
packedsphere domains, the sphereintospace, spaceintosphere
dual rhombic
dodecahedron domain has a tetravolume of 48. The total
space is 24^{__}with the vector
equilibrium's EighthOctahedra extroverted to form the
rhombic dodecahedron. For every
space there is always an alternate space: This is where
we get the 48ness of the rhombic
dodecahedron as the macrodomain of a sphere:

1033.185 The 12 spheric domains around one nuclear sphere domain equal 13 rhombic dodecahedra^{__}nuclear 6 + (12 × 6) = tetravolume 78. 
1033.192
Table: Prime Number Consequences of Spinhalving of
Tetrahedron's
Volumetric Domain Unity

1033.20
Table: Cosmic Hierarchy of Primitive Polyhedral Systems:
The constant
octave system interrelationship is tunable to an infinity
of different frequency keys:

1033.30 Symmetrical Contraction of Vector Equilibrium: Quantum Loss 
1033.31 The six square faces of the vector equilibrium are dynamically balanced; three are oppositely arrayed in the northern hemisphere and three in the southern hemisphere. They may be considered as three^{__}alternately polarizable^{__}pairs of half octahedra radiantly arrayed around the nucleus, which altogether constitute three whole "internal" octahedra, each of which when halved is structurally unstable^{__}ergo, collapsible^{__}and which, with the vector equilibrium jitterbug contraction, have each of their six sets of halfoctahedra's four internal, equiangular, triangular faces progressively paired into congruence, at which point each of the six halfoctahedra^{__}ergo, three quanta^{__}has been annihilated. 
1033.32 In the alwaysomnisymmetrical progressive jitterbug contraction the vector equilibrium^{__}disembarrassed of its disintegrative radial vectors^{__}does not escape its infinite instability until it is symmetrically contracted and thereby structurally transformed into the icosahedron, whereat the six square faces of the halfoctahedra become mildly folded diamonds ridgepoled along the diamond's shorter axis and thereby bent into six ridgepole diamond facets, thus producing 12 primitively equilateral triangles. Not until the six squares are diagonally vectored is the vector equilibrium stabilized into an omnitriangulated, 20triangled, 20tetrahedral structural system, the icosahedron: the structural system having the greatest system volume with the least energy quanta of structural investment^{__}ergo, the least dense of all matter. 
1033.33
See Sec.
611.02
for the tetravolumes per vector quanta
structurally invested
in the tetra, octa, and icosa, in which we accomplish^{__}

1033.35 The six new vector diagonals of the three pairs of opposing halfoctahedra become available to provide for the precession of any one of the equatorial quadrangular vectors of the halfoctahedra to demonstrate the intertransformability of the octahedron as a conservation and annihilation model. (See Sec. 935.) In this transformation the octahedron retains its apparent topological integrity of 6V + 8F = 12E + 2, while transforming from four tetravolumes to three tetravolumes. This tetrahelical evolution requires the precession of only one of the quadrangular equatorial vector edges, that edge nearest to the massinterattractively precessing neighboring mass passing the octahedron (as matter) so closely as to bring about the precession and its consequent entropic discard of one quantum of energy^{__}which unbalanced its symmetry and resulted in the three remaining quanta of matter being transformed into three quanta of energy as radiation. 
1033.36 This transformation from four tetravolumes to three tetravolumes^{__}i.e., from four to three energy quanta cannot be topologically detected, as the Eulerean inventory remains 6V + 8F = 12E + 2. The entropic loss of one quantum can only be experimentally disclosed to human cognition by the conceptuality of synergetics' omnioperational conceptuality of intertransformabilities. (Compare color plates 6 and 7.) 
1033.40 Asymmetrical Contraction of Vector Equilibrium: Quantum Loss 
1033.41 The vector equilibrium contraction from tetravolume 20 to the tetravolume 4 of the octahedron may be accomplished symmetrically (as just described in Sec. 1033.30) by altogether collapsing the unstable six halfoctahedra and by symmetrical contraction of the 12 radii. The angular collapsing of the 12 radii is required by virtue of the collapsings of the six halfoctahedra, which altogether results in the eight regular tetrahedra being concurrently reduced in their internal radial dimension, while retaining their eight external equiangular triangles unaltered in their primevectoredge lengths; wherefore, the eight internal edges of the original tetrahedra are contractively reduced to eight asymmetric tetrahedra, each with one equiangular, triangular, external face and with three rightangle apexed and primevectorbaseedged internal isoscelestriangle faces, each of whose interior apexes occurs congruently at the center of volume of the symmetrical octahedron^{__}ergo, each of which eight regulartoasymmetrictransformed tetrahedra are now seen to be our familiar EighthOctahedra, each of which has a volume of l/2 tetravolume; and since there are eight of them (8 × 1/2 = 4), the resulting octahedron equals tetravolume4. 
1033.42 This transformation may also have been accomplished in an alternate manner. We recall how the jitterbug vector equilibrium demonstrated the fourdimensional freedom by means of which its axis never rotates while its equator is revolving (see Sec. 460.02). Despite this axis and equator differentiation the whole jitterbug is simultaneously and omnisymmetrically contracting in volume as its 12 vertexes all approach their common center at the same radial contraction rate, moving within the symmetrically contracting surface to pair into the six vertices of the octahedron^{__}after having passed symmetrically through that asyet12vectored icosahedral stage of symmetry. With that complex concept in mind we realize that the nonrotating axis was of necessity contracting in its overall length; ergo, the twovertextotwovertexbonded "pair" of regular tetrahedra whose mostremotelyopposite, equiangular triangular faces' respective centers of area represented the two poles of the nonrotated axis around which the six vertices at the equator angularly rotated^{__}three rotating slantwise "northeastward" and three rotating "southeastward," as the northeastward three spiraled finally northward to congruence with the three corner vertices of the nonrotating north pole triangle, while concurrently the three southeastwardslantwise rotating vertices originally situated at the VE jitterbug equator spiral into congruence with the three corner vertices of the nonrotating south pole triangle. 
Fig. 1033.43 
1033.43 As part of the comprehensively symmetrical contraction of the whole primitive VE system, we may consider the concurrent northtosouth polaraxis contraction (accomplished as the axis remained motionless with respect to the equatorial motions) to have caused the two original vertextovertex regular polar tetrahedra to penetrate one another vertexially as their original two congruent centerofVEvolume vertices each slid in opposite directions along their common polaraxis line, with those vertices moving toward the centers of area, respectively, of the other polar tetrahedron's polar triangle, traveling thus until those two penetrating vertices came to rest at the center of area of the opposite tetrahedron's polar triangle^{__}the planar altitude of the octahedron being the same as the altitude of the regular tetrahedron. (See Figs. 1033.43 and 1033.47.) 
1033.44 In this condition they represent the opposite pair of polar triangles of the regular octahedron around whose equator are arrayed the six other equiangular triangles of the regular octahedron's eight equiangular triangles. (See Fig. 1033.43.) In this state the polarly combined and^{__}mutually and equally^{__}interpenetrated pair of tetrahedra occupy exactly onehalf of the volume of the regular octahedron of tetravolume4. Therefore the remaining space, with the octahedron equatorially surrounding their axial core, is also of tetravolume2^{__}i.e., onehalf insideout (space) and onehalf insidein (tetracore). 
1033.45 At this octahedronforming state two of the eight vertices of the two polar axis tetrahedra are situated inside one another, leaving only six of their vertices outside, and these six^{__}always being symmetrically equidistant from one another as well as equidistant from the system center^{__}are now the six vertices of the regular octahedron. 
1033.46 In the octahedronforming state the three polarbase, cornertoapex connectingedges of each of the contracting polaraxis tetrahedra now penetrate the other tetrahedron's three nonpolar triangle faces at their exact centers of area. 
1033.47 With this same omnisymmetrical contraction continuing^{__}with all the external vertices remaining at equal radius from the system's volumetric center^{__}and the external vertices also equidistant chordally from one another, they find their two polar tetrahedra's mutually interpenetrating apex points breaking through the other polar triangle (at their octahedralforming positions) at the respective centers of area of their opposite equiangular polar triangles. Their two regulartetrahedrashaped apex points penetrate their former polaropposite triangles until the six midedges of both tetrahedra become congruent, at which symmetrical state all eight vertices of the two tetrahedra are equidistant from one another as well as from their common system center. (See Fig. 987.242A.) 
1033.48 The 12 geodesic chords omniinterconnecting these eight symmetrically omniarrayed vertices now define the regular cube, onehalf of whose total volume of exactly 3tetravolumes is symmetrically cored by the eightpointed star core form produced by the two mutually interpenetrated tetrahedra. This symmetrical core star constitutes an insidein tetravolume of l 1/2, with the surrounding equatorial remainder of the cubedefined, insideout space being also exactly tetravolume 1 1/2. (See Fig. 987.242A.) 
1033.490 In this state each of the symmetrically interpenetrated tetrahedra's eight external vertices begins to approach one another as each opposite pair of each of the tetrahedra's six edges^{__}which in the cube stage had been arrayed at their mutual mid edges at 90 degrees to one another^{__}now rotates in respect to those midedges^{__}which six mutual tetrahedra's midedge points all occur at the six centers of the six square faces of the cube. 
1033.491 The rotation around these six points continues until the six edgelines of each of the two tetrahedra become congruent and the two tetrahedra's four vertices each become congruent^{__}and the VE's original tetravolume 20 has been contracted to exactly tetravolume 1. 
1033.492 Only during the symmetrical contraction of the tetravolume3 cube to the tetravolume 1 tetrahedron did the original axial contraction cease, as the two opposing axis tetrahedra (one insideout and one outsideout) rotate simultaneously and symmetrically on three axes (as permitted only by fourdimensionality freedoms) to become unitarily congruent as tetravolume1^{__}altogether constituting a cosmic allspace filling contraction from 24 to 1, which is three octave quanta sets and 6 × 4 quanta leaps; i.e., six leaps of the six degrees of freedom (six insideout and six outsideout), while providing the prime numbers 1,2,3,5 and multiples thereof, to become available for the entropicsyntropic, exportimport transactions of seemingly annihilated^{__}yet elsewhere reappearing^{__}energy quanta conservation of the eternally regenerative Universe, whose comprehensively closed circuitry of gravitational embracement was never violated throughout the 241 compaction. 
1033.50 Quanta Loss by Congruence 
1033.51
Euler's Uncored Polyhedral Formula:

1033.52 Although superficially the tetrahedron seems to have only six vector edges, it has in fact 24. The sizeless, primitive tetrahedron^{__}conceptual independent of size^{__}is quadrivalent, inherently having eight potential alternate ways of turning itself inside out^{__} four passive and four active^{__}meaning that four positive and four negative tetrahedra are congruent. (See Secs. 460 and 461.) 
1033.53 The vector equilibrium jitterbug provides the articulative model for demonstrating the always omnisymmetrical, divergently expanding or convergently contracting intertransformability of the entire primitive polyhedral hierarchy, structuring as you go in an omnitriangularly oriented evolution. 
1033.54 As we explore the interbonding (valencing) of the evolving structural components, we soon discover that the universal interjointing of systems^{__}and their foldability^{__}permit their angularly hinged convergence into congruence of vertexes (single bonding), vectors (double bonding), faces (triple bonding), and volumetric congruence (quadribonding). Each of these multicongruences appears only as one vertex or one edge or one face aspect. The Eulerean topological accounting as presently practiced^{__}innocent of the inherent synergetical hierarchy of intertransformability^{__}accounts each of these multicongruent topological aspects as consisting of only one of such aspects. This misaccounting has prevented the physicists and chemists from conceptual identification of their data with synergetics' disclosure of nature's comprehensively rational, intercoordinate mathematical system. 
1033.55 Only the topological analysis of synergetics can account for all the multicongruent^{__}doubled, tripled, fourfolded^{__}topological aspects by accounting for the initial tetravolume inventories of the comprehensive rhombic dodecahedron and vector equilibrium. The comprehensive rhombic dodecahedron has an initial tetravolume of 48; the vector equilibrium has an inherent tetravolume of 20; their respective initial or primitive inventories of vertexes, vectors, and faces are always present^{__}though often imperceptibly so^{__}at all stages in nature's comprehensive 481 convergence transformation. 
1033.56 Only by recognizing the deceptiveness of Eulerean topology can synergetics account for the primitive total inventories of all aspects and thus conceptually demonstrate and prove the validity of Boltzmann's concepts as well as those of all quantum phenomena. Synergetics' mathematical accounting conceptually interlinks the operational data of physics and chemistry and their complex associabilities manifest in geology, biology, and other disciplines. 
1033.60 Primitive Dimensionality 
1033.601 Defining frequency in terms of interval requires a minimum of three intervals between four similar system events. (See Sec. 526.23.) Defining frequency in terms of cycles requires a minimum of two cycles. Size requires time. Time requires cycles. An angle is a fraction of a cycle; angle is subcyclic. Angle is independent of time. But angle is conceptual; angle is angle independent of the length of its edges. You can be conceptually aware of angle independently of experiential time. Angular conceptioning is metaphysical; all physical phenomena occur only in time. Time and size and specialcase physical reality begin with frequency. Pretimesize conceptuality is primitive conceptuality. Unfrequenced angular topology is primitive. (See Sec. 527.70.) 
1033.61 Fifth Dimension Accommodates Physical Size 
1033.611 Dimension begins at four. Fourdimensionality is primitive and exclusively within the primitive systems' relative topological abundances and relative interangular proportionment. Fourdimensionality is eternal, generalized, sizeless, unfrequenced. 
1033.612 If the system is frequenced, it is at minimum linearly fivedimensional, surfacewise sixdimensional, and volumetrically sevendimensional. Size is special case, temporal, terminal, and more than fourdimensional. 
1033.613 Increase of relative size dimension is accomplished by multiplication of modular and cyclic frequencies, which is in turn accomplished only through subdividing a given system. Multiplication of size is accomplished only by agglomeration of whole systems in which the whole systems become the modules. In frequency modulation of both single systems or wholesystem agglomerations asymmetries of internal subdivision or asymmetrical agglomeration are permitted by the indestructible symmetry of the four dimensionality of the primitive system of cosmic reference: the tetrahedron^{__}the minimum structural system of Universe. 
1033.62 Zerovolume Tetrahedron 
1033.621
The primitive tetrahedron is the fourdimensional,
eightinone, quadrivalent,
alwaysandonlycoexisting, insideout and outsideout
zerovolume whose four great
circle planes pass through the same nothingness center,
the fourdimensionally
articulatable inflection center of primitive conceptual
reference.

1033.622 Thus the tetrahedron^{__}and its primitive, insideout, outsideout intertransformability into the prime, whole, rational, tetravolumenumbered hierarchy of primitivestructuralsystem states^{__}expands from zerovolume to its 24tetravolume limit via the maximumnothingness vectorequilibrium state, whose domain describes and embraces the primitive, nucleated, 12aroundone, closestpacked, unitradius spheres. (See cosmic hierarchy at Sec. 982.62.) 
Next Section: 1033.63 