987.00 Multiplication Only by Division
[987.00987.416 Multiplication by Division Scenario]
987.010 Operational Scenario in Proof of Multiplication Only by Progressive Divisioning of Simplest Vectorially Structured Polyhedra 
987.011 Six equizeromagnitude, masstimesvelocityproduced vectors representing the six equienergetic, differently angled (i.e., differently directioned) cosmic forces that always cointeract to freshly reinitiate minimum local structuring in Universe, constitute the minimummaximum cosmic set of coordinates necessary to formulate a definitive system. A system is the first finite unitarily conceptual subdivision of finite but nonunitarily conceptual Scenario Universe. (See Sec. 400.011 and especially Fig. 401.) 
987.020 Topological Uniqueness 
987.022 The topological variables of systems are identified exclusively as the unique number of vertexes (points), faces (areas), and edges (lines) of the system considered. 
987.030 Finite Synergetics 
987.033
Synergetics progressively divides and progressively
discovers the
omnirational tetrahedralrelated volumes (see Table
1033.192
for table of tetravolume
values) and the other topological and angular characteristics
of the greatcirclespun,
hemisystem cleavages and their respective fractionation
resultants. This progressive
synergetic division and discovery describes the entire
primitive hierarchy of timeless
sizeless, omnisymmetrical, omniconcentric, omniintertransformative,
intercommensurable,^{7} systemic polyhedral structures.
(See cosmic hierarchy at Table
982.62)
(Footnote 7: Intercommensurable means the uniform proportional interequatability of two or more separate, volumetrically interrational, geometrical sets. These sets have different divisors, which are noninterrational but interproportionally constant and successively intertransformative.) 
987.040 Macromediomicro Mensuration Limits 
987.042 Micro tetra and macro icosa always and only coact as cosmic unity equaling at least two. This incommensurable pair serves as the two only separately rationalbut proportionately constant and interequatable^{__}mensuration reference limits in all geometrical, topological, chemical, and quantumcoordinate scientific interconsiderability. 
987.043 The medio octahedron serves as the average, betweenlimits, most structurally expedient, and most frequently employed of the three prime structural systems of Universe. It is significant that the limit case pair micro tetra and macro icosa are both prime numbers^{__}1 and 5^{__}whereas the medio octa is a second power of 2, the only prime even number in Universe: 2^{2} = 4. 
987.044 The selfregeneration of the nonsimultaneously and onlypartially overlappinglyepisoded, beginningless and endless Scenario Universe inherently requires in pure principle an eternal incommensurability of^{__}at minimum two^{__}overall symmetrical and concentric system intertransformative behaviors and characteristic phases. 
987.050 Intercommensurable Functions of Jitterbug 
987.051 The vector equilibrium of tetravolume20 = prime 5 × prime 2^{2}, is rationally coordinate with the tetrahedron representing the prime number 1 and with the octahedron representing the prime number 2. But the 20tetravolume (5 × 2^{2}) VE is inherently incommensurable with the icosahedron, which represents the prime number 5 compounded with swrt(2), even though the VE and icosa are concentrically and omnisymmetrically intertransformable (see Secs. 461.0206). 
987.052 The mathematical span between the second power of 2, (2^{2} = 4) and the second root of 2, (sqrt(2))^{__}which is the same proportional relationship as that existing between sqrt(2) and 1^{__}is the constant proportional accommodating median between tetra 1 and octa 2 and between the first two prime numbers: between the most primitive odd and even, between the most primitive yes and no of the primitive binary system^{__}ergo, of all computer mathematics. 
987.053
The 2 as constant proportional equity median is clearly
evidenced as each of
the VE's six square unitlengthvectoredged faces jitterbuggingly
transform into the two
unitvectoredged equilateral triangles. We recall that
the diagonal of each square was the
hypotenuse of a rightangleapexed, unitvectoredged
isosceles triangle whose
hypotenuse jitterbuggingly contracted in length to the
length of each of the unitvector
edges. We have the wellknown formula for the second
power of the hypotenuse equaling
the sum of the second powers of the right triangle's
rightangle sides, and since the right
angleapexed isosceles triangle's sides were of unitvector
length = 1, the second powers
of both equal 1. The sum of their second powers was
2, and the length of the square's
hypotenuse diagonal = sqrt(2). Ergo, the total linear alteration
of the VE icosa
was the contraction of sqrt(2) 1. This
introduces one of nature's
most
profound incommensurability equations, wherein
2:sqrt(2) = sqrt(2):1

987.054
Proportionately expressed this equation reads:
VE:icosa = 2:sqrt(2) = sqrt(2):1
Fractionally expressed the equation reads:

987.055 In the jitterbug, as the 20tetravolume VE contracts symmetrically through the icosahedral phase with a tetravolume of 18.51229587, and then ever symmetrically contracts to the bivalent octahedral phase of tetravolume4, the sixmembered axis of the concentric system does not rotate while the other 18 nonaxis "equatorial" members rotate around the axis. (Fig. 460.08.) 
987.056 As the system contraction continues beyond the octahedron stage of tetravolume4, the axis also torques and contracts as the octahedron either (1) contracts symmetrically and rotationally into the regular tetrahedron of tetravolume1 (or counterrotates into the alternate regular tetrahedron of tetravolume1), or (2) flattens by contraction of its axis to form zerovolume, edgecongruent pair of triangular patterns; thereafter the triangle's three corners are foldable alternately into the quadrivalent positive or the quadrivalent negative regular tetrahedron of tetravolume1. 
987.057 Since all 24 internal radiation vectors had been removed before the jitterbugging, leaving only the 24 external gravitation vectors, the transformation is systematically comprehensive and embraces all the complex unities of the VE and icosa and their onlyproportionallyequatable, separately rational, geometrical membership sets. Though the tetra and icosa are incommensurable with each other, the octahedron is transformatively commensurable with either. 
987.058 The inherent volumetric incommensurability of VE and icosa (and their respective four and threeuniquesymmetricalgreatcirclesystem sets), compounded with the ability of the octahedron to intertransformably interconnect these two otherwise incommensurables, produces the energetic oscillations, resonances, and intertransformings of the eternally regenerative Universe. This eternal disquietude regeneration of Universe is also accommodated by the fact that the tetra and VE are a priori incommensurable with the icosa. Despite this the rhombic triacontahedron of tetravolume5 (as a product of the icosa's 15greatcircle cleavaging), while under the oscillatory pressuring, is volumetrically and rationally coordinate with the tetrahedron and the state we speak of as matter^{__}and when it is under the negative tensive pressure of the oscillatory Universe, it transforms from matter to radiation. (See scenario of T and E Modules at Sec. 986.) 
987.060 Isotropic Limits 
987.061 Cosmic regeneration, metaphysical and physical, involves phases of maximum asymmetry or of random pattern uniqueness. The selfregeneration propagated by the eternal war of incommensurability occurs at the medio phase of Universe; the propagation commences at the middle and proceeds syntropically outward or recedes syntropically inward from the maxientropic center in both macro and micro directions with the ultrasyntropic isotropic macrophase being manifest in the interspacing of the galaxies and with the infra syntropic, isotropic microphase being manifest kinetically in timesize as "cosmic background radiation" and statically (timelessly, sizelessly) in the closest packing of unit radius spheres, like the aggregates of atoms of any one element. 
987.062 The median turbulence and kaleidoscopically nonrepetitive, random, individually unique, local patterning events occur between the four successive, symmetrically orderly, "clickstop" phases of the hierarchy of primitive polyhedra: VE, icosa, octa, and tetra. Between the four maximum symmetrical phases the (overall symmetrical, internally asymmetrical) evolutionary events of Universe are empirically and operationally manifest by the VE jitterbugging: they are infinitely different as multiplication only by division is infinitely employed. Timesize infinity is embraced by primitive finity. 
987.063 In the VE jitterbug the local patterning events of Universe rotate outwardly to the macro isotropicity of VE, which can rotate beyond macro to converge symmetrically again through the central phase of the icosa octa transformation. The maximum incommensurability occurs between the latter two, whereafter octa transforms to tetra. The tetra occurs at the microphase of radiation isotropicity and itself transforms and rotates via the negative tetra, expanding again through the negative phases of the octa's duotwoness octa icosa alternate VE. 
987.064 VE is potentially patterndivisible both positively and negatively and both internally and surfacewise. Icosahedron is potentially patterndivisible both positively and negatively and both internally and surfacewise. The octahedron has internal comprehensive (duotet) twoness of 2^{2}, 2, sqrt(2). Tetrahedron is likewise both positively and negatively integrally intertransformable. 
987.065 The incommensurability of the icosa derives from its lack of a nucleus. The VE is inherently nucleated. The primitive tetra is nonnuclear but acquires a nucleus with frequency. The icosa cannot acquire a nucleus whatever the frequency. (See Sec. 466 and Fig. 466.01 for jitterbug transformation pumping out of nuclear sphere.) 
987.066 Since multiplication is accomplished only by division, we observe that the macroisotropicity of seemingly Expanding Universe is equally explicable as the shrinking relative magnitude of the system viewpoint of the observer. (See Secs. 986.75657 and 1052.62.) 
987.067 Octaphase: The eternally inherent incommensurability of the regenerative turmoil of eternally selfregenerative Universe occurs always at its mediophase of intertransforming between VE and icosa and between icosa and tetra: at these mediophases the neverrepeating maxiasymmetry patterns are generated. 
987.070 Topological Minima 
987.071 In synergetics all topological characteristics are interconformationally conceptual independent of size; for instance, a vertex is one of the convergence loci of a system's inherent plurality of conceptual interrelationships. 
987.072 Since vertexes are omnidimensional, system topology deals with the loci of interrelationship convergences at any one of the system's set of defining loci^{__}with a closestpackingofspheresimposed maximum of 12 unitradius convergences around any one unitradius locus sphere. In the latter case vertexes may be predominantly identified as spheres of unit radius and may identify a prime nucleated system. 
987.073 The minimum conceptual system in Universe is the regular tetrahedron, which consists of a minimum of four vertexes that can be represented as four approximately intertangent, equiradius spheres. Vertexrepresenting spheres do not occur in Universe or become conceptually considerable in sets of less than four. (This process is described at Secs. 100.331 and 411. A minimum of four successive events and three intervals is required to define a frequency cycle; see Sec. 526.23.) In the same way lines^{__}or edges, as they are spoken of in topology^{__}occur only in sets of six, as the most economical interrelationships of vertexes of polyhedral systems. 
987.074
The minimum system in Universe is the tetrahedron;
its unit radius spheres at
each of the four vertexes have a minimum of six intersystem
vertexial relationships. We
have learned that topological system vertex interrelationships
always occur in sets of six.
The formula for the number of system interrelationships
is

987.075 Although Alfred North Whitehead and Bertrand Russell did not recognize the full conceptual implications, their "new mathematics" of set theory and empty sets were tour de force attempts by the leading abstract nonconceptual mathematicians of their day to anticipate the inevitable historical convergence of their mathematics with the inherently conceptual topology of Euler, as well as with the phase rule of Gibbs in chemistry, the simplified quantum mechanics of Dirac in physics, and the homogenizing biochemistry and physics of virology's DNARNA design programming^{__}all remotely but inexorably rendezvousing with Boltzmann's, Einstein's, and Hubble's astrophysics and cosmology to constitute unitary science's unitary selfregenerative, untenably equilibrious, cosmiccoordinate system to be embraced and accommodated by the epistemography of synergetics. 
987.076 What are known in the terminology of topology as faces^{__}the polyhedron's hedra sides or facets^{__}are known in synergetics as windows, being the consequences of systemvertex interrelationship lines framing or viewing "windows of nothingness"^{__} windows opening to a nonconvergence, to nonrelatedness, to the untunedin. Nothingness is the atpresentuntunedin information of each special case individual's special localin Universe, momentary, tunedin, preoccupying consideration. Vertexes are tunedin; hedra are untunedin, ergo out. Hedra faces are system outs. 
987.077 Unit radius spheres are unitwavelength, tunedin, event loci; topological faces (Greek hedra) are all the windows looking out upon all the rest of the Universe's presentlyuntunedin information in respect to the considered or tunedin system. 
987.078 Since two systeminterrelationship lines (vectorially energetic in pure principle) cannot pass through the same point at the same time, the windows' "corners" are always superimposed timecrossing aspects^{__}one crossing behind or in front of, but not touching, the other. The topological windows of synergetics are polygonal aspects of the system's interrelationships and not of physical lines. 
987.079 Synergetics' experimentally produced, minimumstructuralsystem subdivisions of Universe have four tunedin vertexial loci, four windows looking toward all the untunedin complementary balance of Universe. and six vertexial interrelationship vector lines, with all the latter occurring as outermost system features. The minimum nonnucleated structural system does not require internal vertexes. 
987.080 Vertexial Spheres Form Rigids 
Fig. 987.081 
987.081 In addition to the four vertexial spheres of the precleavage primitive tetrahedron, we find after the cleavage all six vertexial unitradiusvertex spheres of the octahedron also occurring in the outermost structure of the nonnucleated system. Neither the primitive tetrahedron nor the primitive octahedron have internal or nuclear unitradius vertex spheres. For an illustration of these structural "rigids" see Fig. 987.081. 
987.100 Greatcirclespun Symmetries and Cleavagings 
987.110 Intercommensurability Functions 
987.111
In the greatcirclespun cleavaging of synergetics'
multiplication only by
division there are seven primitive symmetries of spinnability.
(See Sec.
1040.) Four
symmetries belong directly to the separate tetrahedral
commensurability, and three
symmetries belong to the separate icosahedral commensurability,
with the integrity of
eternal interrelationship being provided by the symmetrically
contractive, concentric
intertransformability of the two sets of symmetry at
the jitterbug VEricosa stage. This
symmetrically embraced intertransformable stage corresponds
to the constant
interproportionality stage of the VE and icosa manifest
as
2:sqrt(2) = sqrt(2):1.

987.120 Sequence of Symmetries and Cleavagings 
987.121
Table

987.122 Starting with the regular tetrahedron the progressive primitive subdividing of synergetics is initially accomplished only by the successive equatorial halvings of the progressively halvedout parts of the first four of the only seven cosmic symmetries of axial spin of the primitive structural systems . 
Next Section: 987.130 