1033.63 Prefrequency and Initial Frequency Vector Equilibrium 
1033.631 The primitive tetrahedron has four planes of symmetry^{__}i.e., is inherently fourdimensional. The cosmic hierarchy of relative tetravolumes (Sec. 982.62) is primitive, fourdimensional, and unfrequenced. 
1033.633 Compare Section 1053.84 and Table 1053.849. 
1033.64 Eightness Dominance 
1033.641 The quanta involvement sum of the polar pairings of octahedra would be dominant because it consists of 12 QuarterOctahedra (i.e., 12  8 = 4) = involvement dominance of four, whereas eight is the equilibrious totality vector of the 4><4: since the eightness is the interbalancing of four, the 12  8's excess four is an unbalanced four, which alone must be either the outsideout or the insideout four; ergo, one that produces the maximum primitive imbalance whose asymmetric proclivity invites a transformation to rectify its asymmetry. (Compare Sec. 1006.40.) 
1033.642 Thus the offbalance four invites the one quantum of six vectors released by the precessed octahedron's onequantum "annihilation"^{__}whose entropy cannot escape the Universe. 
1033.643 The vectorequilibrious maximum nothingness becomes the spontaneous syntropic recipient of the energy quantum released from the annihilation phase of the transformation. 
1033.65 Convergentdivergent Limits 
1033.651 Vector equilibrium is never a shape. It is either a tetravolume 0 nothingness or a tetravolume 20 nothingness. The only difference between space nothingness and matter somethingness is vector equilibrium. 
1033.652 Primitive, unfrequenced vector equilibrium is both the rationally interstaged, expansivecontractive, minimum 0, 1, 2, 3, 4, 5, 6 > to 20 to maximum 0, as well as the cosmicresonance occupant of the minimum and maximum event void existing between the primitive, systematic somethingnesses. 
1033.653 The vector equilibrium has four insideout and four outsideout self intercancelation, eightcongruent, zerovolume tetrahedra, as well as eight centrally single bonded tetrahedra of maximum zerovolume expansion: both invoke the cosmically intolerable vacuum voids of macromicronothingness essential to the spontaneous capture of one quantum's six vectors, which^{__}in the VE's maxistate^{__}structurally contracts the VE's 20ness of spatial Universe nothingness into the 20ness of icosahedral somethingness, just as the octaannihilated quantum provides the alwayseightinone, outsideout tetrahedron to fill the insideout "black hole" tetravoid. 
1033.654

1033.655 In the octahedron as the maximum conservation and quantumannihilability model of substance (Sec. 935) the precessing vector edge of the entropic octahedron drops out 1 tetra; 1 tetra = 6 vectors = 1 quantum of energy which^{__}as the entropically random element of radiation's nonformedness^{__}may be effortlessly reformed by reentering the vector equilibrium to produce the icosahedron and thus to form new substance or matter. 
1033.656 The vector equilibrium has 24 external vector edges: inserting the quantum set of six more makes 30 external edges whose omniintertriangulation resolves as the 30 edged icosahedron. The six added edges are inserted as contractive diagonals of the six square faces of the vector equilibrium . The contracted 30 edges = 5 energy quanta. Icosahedron = tetravolume5 . Icosahedron is the least dense of all matter. 
1033.657
As we approach absolute zero, taking all the energy
out of the system,^{5} the
chemical elements of which the apparatus parts consist
each have unique atomicfrequency
temperatures that are inherently different. This is
evident to anyone who, within the same
room temperature, has in swift succession touched glass,
plastic, leather, or whatever it
might be. Therefore, as in cryogenics we approach absolute
zero (for the whole system's
average temperature), the temperature of some of the
elemental components of the
experiment go through to the other side of zero, while
others stay on this side^{__}with the
whole aggregate averaging just short of right on absolute
zero. As a consequence of some
components going through to the other side of zero,
some of the most extraordinary
things happen, such as liquids flowing in antigravity
directions. This is the insideout
Universe.
(Footnote 5: See Secs. 205.02, 251.02, 427.01, and 443.02.) 
1033.658 When the "black hole" phenomenon is coupled with the absolutezero phenomenon, they represent the specialcase manifests of synergetics' macromicro generalization extremes^{__}i.e., both minimaxi, zeronothingness phases, respectively. 
1033.659 Here are both the macro and microdivergenceconvergencelimits in which the fourdimensional transformative and conversion behaviors are quite different from the nonscientificallydemonstrable concept of arbitrary cutoffs of exclusively onedimensional infinity unlimits of linear phenomena. The speed of fourdimensional light in vacuo terminates at the divergent limit. The gravitational integrity of insideout Reverse Universe becomes convergently operative at the macrodivergence limits. 
1033.66 Terminal Reversings of Evolution and Involution 
1033.661 In selecting synergetics' communication tools we avoid such an unresolvable parallellinear word as equals. Because there are neither positive nor negative values that add or detract from Universe, synergetics' communication also avoids the words plus and minus. We refer to active and passive phases. Parallel equivalence has no role in an alternatively convergentdivergent Universe. Inflection is also a meaningless two dimensional linear word representing only a shadow profile of a tetrahelical wave. 
1033.662 In fourdimensional conversion from convergence to divergence^{__}and vice versa^{__}the terminal changing reverses evolution into involution^{__}and vice versa. Involution occurs at the system limits of expansive intertransformability. Evolution occurs at the convergent limits of system contraction. 
1033.663 The macromicronothingness conversion phases embrace both the maximumsystemcomplexity arrangements and the minimumsystemsimplicity arrangements of the constant set of primitive characteristics of any and all primitive systems. A single special case system embraces both the internal and external affairs of the single atom. A plurality of special case systems and a plurality of special case atoms may associate or disassociate following the generalized interrelationship laws of chemical bonding as well as of both electromagnetics and massinterattractiveness. 
1033.664
Primitive is what you conceptualize sizelessly without
words. Primitive has
nothing to do with Russian or English or any special
case language. My original 4D
convergentdivergent vector equilibrium conceptualizing
of 192728^{6} was primitive ><
Bow Tie: the symbol of intertransformative equivalence
as well as of complementarity:
convergence >< divergence >< Also the symbol of syntropyentropy, and of wave and octave, 4, 3, 2, 1, +1, +2, +3, +4 
1033.665
Minimum frequency = two cycles = 2 × 360°.
Two cycles = 720° = 1 tetra = 1 quantum of energy. Tetrahedron is the minimum unitytwo experience. 
1033.666
The center or nuclear sphere always has two polar
axes of spin independent
of surface forming or intertransforming. This is the
"plus two" of the spheric shell growth
around the nucleus. NF^{2} + 2, wherefore in four primitive
cosmic structural systems:

1033.70 Geometrical 20ness and 24ness of Vector Equilibrium 
1033.701 The maximum somethingness of the VE's 20ness does not fill allspace, but the 24tetravolume Duotet Cube (short name for the doubletetrahedron cube) does fill allspace; while the tetravolume4ness of the exterior octahedron (with its alwayspotential onequantum annihilability) accommodates and completes the finite energypacking inventory of discontinuous episodic Physical Scenario Universe. 
1033.702 The three interior octahedra are also annihilable, since they vanish as the VE's 20ness contracts symmetrically to the quadrivalent octahedron jitterbug stage of tetravolume 4: an additive 4tetravolume octahedron has vanished as four of the VE's eight tetrahedra (four insideout, four outsideout) also vanish, thereby demonstrating a quantaannihilation accomplished without impairment of either the independent motion of the system's axial twoness or its convergentdivergent, omniconcentric symmetry. 
1033.703 The four of the 24ness of the Duotet Cube (which is an f^{2} cube: the double tetrahedron) accounts for the systemic fourdimensional planes of fourdimensional symmetry as well as for the everregenerative particle fourness of the quark phenomena characterizing all highenergysystembombardment fractionability. 
1033.704 24 × 4 = 96. But the number of the selfregenerative chemical elements is 92. What is missing between the VE 92 and the f^{2} Duotet Cube's 96 is the fourness of the octahedron's function in the annihilation of energy: 92 + 4 = 24 × 4 = 96. The four is the disappearing octa set. The 24 is the secondpower 24 unique indig turnabout increment. (See Fig. 1223.12.) 
1033.71 We have three expendable interior octa and one expendable exterior octa. This fact accommodates and accounts both the internal and external somethingnessto nothingness annihilations terminally occurring between the 1 20 1 20 at the macroinvolution and microevolution initiating nothingness phases, between which the total outsideout 120 quanta and the total insideout 201 quanta intertransformabilities occur. 
1033.72 The final jitterbug convergence to quadrivalent tetravolume1 outsideout and tetravolume1 insideout is separated by the minimumnothingness phases. This final conversion is accomplished only by torquing the system axis to contract it to the nothingness phase between the threepetal, triangular, insideout and outsideout phases. (See Secs. 462.02, 464.01 and 464.02.) 
1033.73 The Quantum Leap: Between the maximum nothingness and the minimum nothingness we witness altogether five stages of the 4tetravolume octa vanishment in the convergent phase and five such 4tetravolume octa growth leaps in the divergent phase. These five^{__}together with the interior and exterior octa constitute seven octa leaps of four quanta each. The f^{2} of the inherent multiplicative two of all systems provides the eighth fourness: the quantum leap. (Compare Sec. 1013.60.) 
1033.74 It requires 24ness for the consideration of the total atomic behavior because the vector equilibrium is not allspacefillingly complete in itself. It requires the exterior, insideout, invisiblephase, eightwayfractionated, transformable octahedron superimposed on the VE's eight equiangular, triangular faces to complete the allspacefilling, two frequency Duotet Cube's eight symmetrically arrayed and mosteconomically interconnected corners' domain involvement of 24 tetravolumes. 
1033.741 The VE's involvement domain of 24 symmetrical, allspacefilling tetravolumes represents only one of the two alternate intertransformation domains of closestpacked, unitradius spheres transforming into spaces and spaces intertransforming into spheres: ergo, it requires 48tetravolumes to accommodate this phenomenon. To allow for each of these 48tetravolume domains to accommodate their respective active and passive phases, it requires 96tetravolumes. F^{2} tetravoluming, which is as yet primitive, introduces an allspacefilling, symmetrical cube of 192tetravolumes as an essential theater of omniatomic primitive interarrayings. 
1033.75 The total primitively nucleated Duotet Cube's doubletetra unique increment of allspace filling is that which uniquely embraces the whole family of local Universe's. nuclearly primitive intertransformabilities ranging through the 241 and the 124 cosmic hierarchy of rational and symmetrical "clickstop" holding patterns or minimumeffort selfstabilization states. 
1033.76 The Duotet Cube (the maxicube) occurring between micronothingness and macronothingness shows how Universe intertransformably accommodates its entropic syntropic energyquanta exportings and importings within the twofrequency, allspace filling minireality of specialcase Universe. Thus the entropicsyntropic, specialcase Physical Universe proves to be demonstrable within even the most allspacecrowding condition of the VE's maximumsomething 20ness and its exterior octahedron's even morethanmaximumsomething 4tetravolume nothingness. 
1033.77 This 24ness is also a requisite of three number behavior requirements as disclosed in the minmax variabilities of octave harmonics in tetrahedral and VE cumulative closestpacking agglomerations at holistic shell levels as well as in all second powering "surface" shell growths, as shown in three different columns in Fig. 1223.12. 
1033.80 Possible Atomic Functions in Vector Equilibrium Jitterbug 
1033.81 There can be nothing more primitively minivolumetric and omnisymmetrically nucleatable than 12 unitradius spheres closest packed around one such sphere, altogether conformed as the vector equilibrium as produced in multiplication only by division. We can multiply our consideration by endlessly dividing larger into smaller and smaller, ever more highly frequenced, closestpacked spheres. Conversely, the icosahedron is the configuration of nonnucleated, omnisymmetric, unitradius spheres closest packed circumferentially around a central space inadequate to accommodate one such unitradius sphere. The icosahedron may be identified as the miniconfiguration of the electron function as well as the second most volumetric, initial, convergentdivergent transformation, with only the vector equilibrium being greater. 
1033.82 The 20 triangular faces of the icosahedron may be considered as 10 pairs of regular tetrahedra interpenetrating as internal vertexes. The energetic functions of these 10 pairs (as described in Secs. 464 and 465) are a fourdimensional evolution like the triangles rotating in the cube, generating the double tetrahedra in the process. But according to synergetics' topological accounting it is necessary to extract one pair of double tetrahedra for the axis of spin: this leaves eight pairs of double tetra. 10^{__}2=8 is the same fundamental octave eightness as the eight EighthOctahedra that convert the eight triangular corners of the VE to the involvement domain of the nucleated cube. 
1033.83 At the outset of the VE jitterbug evolution there are two polar verticalaxis triangles^{__}if the top one points away from you, the bottom one on the table points toward you. Without itself rotating, this activepassive, triangularly poled, vertical axis permits the jitterbug evolution to rotate its equatorial components either clockwise or counterclockwise, providing for the production of two different icosahedra^{__}an active pair and a passive pair. But since there are four VE axes that can be jitterbugged in the same manner, then there are potentially eight different icosahedra to be generated from any one vector equilibrium. 
1033.84 It could be that the eight paired tetrahedra are the positrons while the eight icosahedra are the electrons. Comprehension involves all four axes available. 
1033.90 Spheres and Spaces 
1033.91 How can an object move through water, which is a noncompressible substance? It does so by the intertransformability of spheres becoming spaces and spaces becoming spheres. (See Sec. 1032.) This is one of the ways in which the octahedron annihilation works in allspacefilling accommodation of local transformative events. The vector equilibrium and the eight EighthOctahedra on the triangular facets combine to produce the primitively nucleated cube. 
1033.92 The octahedron annihilation model is uniformly fractionated and redeployed eight ways to function structurally as eight asymmetric tetrahedra at the eight corners of the vector equilibrium in an intertransformable manner analogous to the onequantum annihilating octahedron which^{__}in EighthOctahedra increments^{__}complements the 024tetravolume vector equilibrium furnished with eight corners. 
1040.00 Seven Axes of Symmetry
1041.00 Superficial Poles of Internal Axes 
1041.01
There are only three topological axes of crystallography.
They are:

1041.10 Seven Axes of Truncated Tetrahedron 
1041.11
The prime generation of the seven axes of symmetry
are the seven unique
perpendiculars to the faces of the seven possible truncations
of the tetrahedron:

1041.12
The seven unique axes of the three unique sets (4 +
4 + 6) producing the 14
planes of the truncated tetrahedron are also identifiable
with:

1041.13 Various high frequencies of modular subdividings of the tetrahedron produce a variety of asymmetrical truncatabilities of the tetrahedron. The dynamics of symmetry may employ any seven sets of the 56 foldablegreatcircle variations of planar orientation. Thus it follows that both the biological cell arrays and the bubble arrays display vast varieties of asymmetries in their 14 enclosing planes, so much so that this set of interidentifiability with the 14 topological characteristics of the tetrahedron, the prime structural system of Universe, has gone unnoticed until now. (See Sec. 1025.14) 
1042.00 Seven Axes of Symmetry 
1042.01 Whatever subdivisions we may make of the tetrahedra, octahedra, and icosahedra, as long as there is cutting on the axes of symmetry, the components always come apart in whole rational numbers, for this is the way in which nature chops herself up. 
1042.02 The four sets of unique axes of symmetry of the vector equilibrium, that is, the 12 vertexes with six axes; the 24 midedges with 12 axes; and the two different centers of area (a) the eight centers of the eight triangular areas with four axes, and (b) the six centers of the six square areas with three axes^{__}25 axes in all^{__}generate the 25 great circles of the vector equilibrium. These are the first four of the only seven cosmically unique axes of symmetry. All the great circles of rotation of all four of these seven different cosmic axes of symmetry which occur in the vector equilibrium go through all the same 12 vertexes of the vector equilibrium (see Sec. 450). 
1042.03 The set of 15 great circles of rotation of the 30 midedgepolared axes of the icosahedron, and the set of 10 great circles of rotation of the icosahedron's midfaces, total 25, which 25 altogether constitute two of the three other cosmic axes of symmetry of the seveninall axes of symmetry that go through the 12 vertexes of the icosahedron, which 12 represent the askewedly unique icosahedral rearrangement of the 12 spheres of the vector equilibrium. Only the set of the seventh axis of symmetry, i.e., the 12vertex polared set of the icosahedron, go through neither the 12 vertexes of the icosahedron's 12 corner sphere arrangement nor the 12 of the vector equilibrium phase 12ball arrangement. The set of three axes (that is 12 vertexes, 30 midedges, and 20 centers of area) of the icosahedron produce three sets of the total of seven axes of symmetry. They generate the 25 twelveicosavertextransiting great circles and the six nontransiting great circles for a total of the 31 great circles of the icosahedron. These are the last three of the seven axes of symmetry. 
1042.04 We note that the set of four unique axes of symmetry of the vector equilibrium and the fifth and sixth sets of axes of the icosahedron all go through the 12 vertexes representing the 12 spheres either (a) closestpacked around a nuclear sphere in the vector equilibrium, or (b) in their rearrangement without a nuclear sphere in the icosahedron. The six sets of unique cosmic symmetry transit these 12 spherical center corner vertexes of the vector equilibrium and icosahedron; four when the tangential switches of the energy railway tracks of Universe are closed to accommodate that Universe traveling; and two sets of symmetry when the switches are open and the traveling must be confined to cycling the same local icosahedron sphere. This leaves only the seventh symmetry as the one never going through any of those 12 possible sphereto sphere tangency railway bridges and can only accommodate local recycling or orbiting of the icosahedron sphere. 
1042.05
The seven unique cosmic axes of symmetry describe all
of crystallography.
They describe the all and only great circles foldable
into bow ties, which may be
reassembled to produce the seven, greatcircle, spherical
sets (see Secs.
455
and
457).

1043.00 Transformative Spherical Triangle Grid System 
1043.01 All the great circles of all the seven axes of symmetry together with all great circletrajectory interactions can be reflectively confined and trigonometrically equated with only one of the icosahedral system's 120 similar rightspherical triangles (of 90, 60, and 36 degrees, in contradistinction to the rightplanar triangle of 90, 60, and 30degree corners). (See Sec. 905.60.) The rational spherical excess of six degrees (of the icosahedron's 120^{__}60 plus and 60 minus^{__}similar tetrahedral components) is symmetrically distributed to each of the three central and three surface angles of each of the 120 tetrahedral components of the spherical icosahedron. 
1043.02 This sixness phenomenon tantalizingly suggests its being the same transformative sixness as that which is manifest in the cosmically constant sixfoldedness of vectors of all the topological accountings (see Secs. 621.10 and 721); and in the sixness of equieconomical alternative degrees of freedom inherent in every event (see Sec. 537.10); as well as in the minimum of six unique interrelationships always extant between the minimum of four "star events" requisite to the definitive differentiation of a conceptual and thinkable system from out of the nonunitarily conceptual but inherently finite Universe, because of the latter's being the aggregate of locally finite, conceptually differentiable, minimumsystem events (see Secs. 510 and 1051.20). 
1044.00 Minimum Topological Aspects
[1044.001044.13 Minimum Topology Scenario]
1044.01
Euler + Synergetics: The first three topological aspects
of all minimum
systems^{__}vertexes, faces, and edges^{__}were employed by
Euler in his formula V + F = E +
2. (See Table
223.64
and Sec.
505.10.) Since synergetics'
geometry embraces nuclear and
angular topology, it adds four more minimum aspects
to Euler's inventory of three:

1044.02 Euler discovered and developed the principle of modern engineering's structural analysis. He recognized that whereas all statically considered objects have a center of gravity, all dynamically considered structural components of buildings and machinery^{__}no matter how symmetrically or asymmetrically conformed^{__} always have a uniquely identifiable neutral axis of gyration. Euler did not think of his topology as either static or dynamic but as a mathematically permitted abstraction that allowed him to consider only the constant relative abundance of vertexes, faces, and edges isolated within a local area of a nonsystem. (The local consideration of the constant relative abundance of vertexes, faces, and edges applies to polyhedra as well as to cored through polyhedra.) 
1044.03 Euler's analysis failed to achieve the generalization of angles (whose convergence identified his corners), the complementary insideness and outsideness, and the convexityconcavity of all conceptual experience. Being content to play his mathematical game on an unidentified surface, he failed to conceive of systems as the initial, allUniverse separators into the tunably relevant, topologically considered set. Euler's lessthansystem abstraction also occasioned his failure to identify the spin axis of any and all systems with his axis of gyration of physical objects; thus he also failed to realize that the subtraction of two vertexes from all systems for assignment as polar vertexes of the spin axis was a failure that would necessitate the "plus two" of his formula V + F = E + 2. 
1044.04 Any and all conceptuality and any and all thinkaboutability is inherently systemic (see Secs. 905.0102). Systemic conceptuality and thinkaboutability are always consequent only to consideration. Consideration means bringing stars together so that each star may be then considered integrally as unity or as an infrasystem complex of smaller systems. 
1044.05
A system consists at minimum of four star events (vertexes)
with four
nothingness window facets and six lines of unique fourstar
interrelationships. As in
synergetics' 14 truncation faces, Euler's three aspects
result in 14 cases:
4 vertexes + 4 faces + 6 edges = 14 cases.

1044.06
Synergetics further augments Euler's inventory of three
topological aspects
(14 cases) with six additional and primitively constant
topological aspects:

1044.07 The total of nine minimum topological aspects consists of three from Euler (14 cases) plus synergetics' inventory of six additional aspects, with 12 angular cases and six nuclear cases for a total of 18 synergetics cases. The 14 Euler cases and the 18 synergetics cases provide a total of 32 minimum topological cases. 
1044.08 Topological analysis permits the generalization of all structuring in Universe as systemic. 
1044.09 What we speak of as substance^{__}a planet, water, steam, a cloud, a speck, or a pile of dust^{__}always has both insideness and outsideness. A substance is a single system or a complex of neighboring interbonded or criticalproximity systems. Substances have inherent insideness "volumes." 
1044.10 An Earthian observer can point in a describable compass direction and a describable angle of elevation toward the location in the sky where the contrails of two differently directioned jet air transports traveling at different altitudes appear to him to cross one another. Because they are flown at different altitudes, the "tohim" crossing does not mean that they touch one another; it is simply a moment when their two separate trajectories are nearest to one another. What the observer points to is a "nearesttoone another" moment. The observer points to an interrelationship event, which is not part of either contrail considered only by itself. This directionally identifiable interrelationship event is known as a "fix." (See Sec. 532.02.) 
1044.11 The four corner fixes of an environmental tetrahedron may be pointed toward with adequate communicability to visually inform others of a specific tetrahedral presence. This is accomplished as follows: Two sky fixes must have a most economical linear interrelatedness but no insideness. Three sky fixes define a triangle between whose three edgedefining, interrelationship lines is described a plane that has no altitude^{__}ergo, no insideness. Then the triangle described by the three sky fixes plus the position of the observer on the ground altogether describe the four corners of a tetrahedron that has six lines of observably inductable interrelatedness defining four triangular planes that observably divide all Universe into the included insideness and the excluded outsideness. 
1044.12 One fix does not have insideness. Two fixes define a noinsideness linear relationship. Three fixes define a noinsideness plane. Four fixes define an insideness including and outsidenessexcluding tetrahedron, which is the minimum cosmic system and which cannot have less than 32 unique and differentially describably generalized cases of the nine irreducibleinnumber unique topological aspects of the minimum system, but which in special frequenced cases may have more. 
1044.13 Although not enumerated topologically (because unconsidered and because nonsimultaneously considerable) there are^{__}in addition to the nine aspects and 32 cases^{__} two additional ultimate conceptual aspects of the complementary macro and microremainder of the physical Universe: all the asyetundiscovered^{__}ergo, unconsidered^{__}special cases as an epistemographic complementary to all the asyet undiscovered^{__}ergo, unconsidered^{__}generalized principles. 
Next Section: 1050.00 