Multiplication Only by Division
987.00
Multiplication Only by Division
[987.00-987.416 Multiplication by Division Scenario]
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987.010
Operational Scenario in Proof of Multiplication Only
by
Progressive Divisioning of Simplest Vectorially Structured
Polyhedra
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987.011
Six equi-zero-magnitude, mass-times-velocity-produced
vectors representing
the six equi-energetic, differently angled (i.e., differently
directioned) cosmic forces that
always cointeract to freshly reinitiate minimum local
structuring in Universe, constitute the
minimum-maximum 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.)
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987.012
A system is a polyhedral pattern__regular or irregular__that
definitively
closes back upon itself topologically to subdivide Universe
locally into four parts: (1) all
the Universe outside the system, the macrocosm; (2)
all the Universe within the system,
the microcosm; (3) the convex-outside little bit of
the Universe of which the system itself
is constituted; and (4) the concave-inside little bit
of the Universe of which the system is
constituted.
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987.013
The tetrahedron, with its six equi-lengthed vector
edges and four vertexes
and with its four triangular windows, is experimentally
demonstrable to be the
topologically simplest structural system of Universe.
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987.020
Topological Uniqueness
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987.021
Recognizing that angles are conceptual independent
of the lengths of the
lines converging to form them, it follows that a triangle
or a tetrahedron or any polygons
or polyhedra are conceptual__and conceptually
different__quite
independent of the time-
size lengths of the lines defining the polyhedra. All
primitive non-space-time differences
are exclusively angular and topological.
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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.
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987.030
Finite Synergetics
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987.031
Starting with mass = zero and velocity = zero (i.e.,
MV = 0), as the energy-
quantum product of the six vectors of the minimum structural
system in Universe (that is,
each of the tetrahedron's six equi-lengthed edges individually
= 0), the mathematical art
and science known as Synergetics provides a cosmically
comprehensive mathematical
strategy of employing always and only physically demonstrable,
omnidimensional,
quantum-compatible multiplication only by division of
a no-gain-no-loss, no-beginning-
no-ending, omnicomplexedly and nonsimultaneously overlapping,
ceaselessly and
differently intertransforming, eternally self-regenerative,
100-percent-efficient, energetic
Universe.
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987.032
The omnidirectionally multiplying amplification of
information in Universe is
arrived at only by discretely progressive subdivision
of the structural system that has been
already experimentally and operationally demonstrated
to be the simplest-the regular
tetrahedron.
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987.033
Synergetics progressively divides and progressively
discovers the
omnirational tetrahedral-related volumes (see Table
1033.192
for table of tetravolume
values) and the other topological and angular characteristics
of the great-circle-spun,
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.) |
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987.040
Macro-medio-micro Mensuration Limits
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987.041
Primitive unity is at minimum a union of two uniquely
individual quantum
vectorizations of each member of the primitive minimum
polyhedral system hierarchy,
each of whose polyhedra described by the quantum vectorizations
are complementarily
intravolumed and intra-energetic. The quantum-vectored
polyhedra of the primitive
hierarchy are always relative volumetrically, topologically,
and vectorially__
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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 rational-but
proportionately constant and interequatable__mensuration
reference limits in all
geometrical, topological, chemical, and quantum-coordinate
scientific interconsiderability.
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987.043
The medio octahedron serves as the average, between-limits,
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: 22 = 4.
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987.044
The self-regeneration of the nonsimultaneously and
only-partially-
overlappingly-episoded, 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.
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987.050
Intercommensurable Functions of Jitterbug
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987.051
The vector equilibrium of tetravolume-20 = prime 5
× prime 22, is rationally
coordinate with the tetrahedron representing the prime
number 1 and with the octahedron
representing the prime number 2. But the 20-tetravolume
(5 × 22) 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.
461.02-06).
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987.052
The mathematical span between the second power of 2,
(22 = 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.
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987.053
The 2 as constant proportional equity median is clearly
evidenced as each of
the VE's six square unit-length-vector-edged faces jitterbuggingly
transform into the two
unit-vector-edged equilateral triangles. We recall that
the diagonal of each square was the
hypotenuse of a right-angle-apexed, unit-vector-edged
isosceles triangle whose
hypotenuse jitterbuggingly contracted in length to the
length of each of the unit-vector
edges. We have the well-known formula for the second
power of the hypotenuse equaling
the sum of the second powers of the right triangle's
right-angle sides, and since the right-
angle-apexed isosceles triangle's sides were of unit-vector
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
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987.054
Proportionately expressed this equation reads:
VE:icosa = 2:sqrt(2) = sqrt(2):1
Fractionally expressed the equation reads:
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987.055
In the jitterbug, as the 20-tetravolume VE contracts
symmetrically through
the icosahedral phase with a tetravolume of 18.51229587,
and then ever symmetrically
contracts to the bivalent octahedral phase of tetravolume-4,
the six-membered axis of the
concentric system does not rotate while the other 18
nonaxis "equatorial" members rotate
around the axis. (Fig.
460.08.)
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987.056
As the system contraction continues beyond the octahedron
stage of
tetravolume-4, the axis also torques and contracts as
the octahedron either (1) contracts
symmetrically and rotationally into the regular tetrahedron
of tetravolume-1 (or
counterrotates into the alternate regular tetrahedron
of tetravolume-1), or (2) flattens by
contraction of its axis to form zerovolume, edge-congruent
pair of triangular patterns;
thereafter the triangle's three corners are foldable
alternately into the quadrivalent positive
or the quadrivalent negative regular tetrahedron of
tetravolume-1.
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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 only-proportionally-equatable, separately
rational, geometrical membership sets.
Though the tetra and icosa are incommensurable with
each other, the octahedron is
transformatively commensurable with either.
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987.058
The inherent volumetric incommensurability of VE and
icosa (and their
respective four- and three-unique-symmetrical-great-circle-system
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 tetravolume-5 (as a product of the
icosa's 15-great-circle 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.)
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987.060
Isotropic Limits
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987.061
Cosmic regeneration, metaphysical and physical, involves
phases of
maximum asymmetry or of random pattern uniqueness. The
self-regeneration 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 maxi-entropic 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
time-size 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.
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987.062
The median turbulence and kaleidoscopically nonrepetitive,
random,
individually unique, local patterning events occur between
the four successive,
symmetrically orderly, "click-stop" 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.
Time-size infinity is embraced by
primitive finity.
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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 duo-twoness octa
icosa alternate VE.
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987.064
VE is potentially pattern-divisible both positively
and negatively and both
internally and surfacewise. Icosahedron is potentially
pattern-divisible both positively and
negatively and both internally and surfacewise. The
octahedron has internal comprehensive
(duo-tet) twoness of 22, 2, sqrt(2). Tetrahedron is likewise
both positively and
negatively integrally intertransformable.
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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.)
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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.
986.756-57 and
1052.62.)
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987.067
Octaphase: The eternally inherent incommensurability
of the regenerative
turmoil of eternally self-regenerative Universe occurs
always at its mediophase of
intertransforming between VE and icosa and between icosa
and tetra: at these
mediophases the never-repeating maxi-asymmetry patterns
are generated.
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987.070
Topological Minima
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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.
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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
closest-packing-of-spheres-imposed maximum of 12 unit-radius
convergences around any
one unit-radius locus sphere. In the latter case vertexes
may be predominantly identified as
spheres of unit radius and may identify a prime nucleated
system.
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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. Vertex-representing
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.
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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
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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 DNA-RNA design
programming__all remotely but
inexorably rendezvousing with Boltzmann's, Einstein's,
and Hubble's astrophysics and
cosmology to constitute unitary science's unitary self-regenerative,
untenably equilibrious,
cosmic-coordinate system to be embraced and accommodated
by the epistemography of
synergetics.
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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
system-vertex interrelationship lines framing or viewing
"windows of nothingness"__
windows opening to a nonconvergence, to nonrelatedness,
to the untuned-in. Nothingness
is the at-present-untuned-in information of each special
case individual's special local-in-
Universe, momentary, tuned-in, preoccupying consideration.
Vertexes are tuned-in; hedra
are untuned-in, ergo out. Hedra faces are system outs.
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987.077
Unit radius spheres are unit-wavelength, tuned-in,
event loci; topological
faces (Greek hedra) are all the windows looking out
upon all the rest of the Universe's
presently-untuned-in information in respect to the considered
or tuned-in system.
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987.078
Since two system-interrelationship lines (vectorially
energetic in pure
principle) cannot pass through the same point at the
same time, the windows' "corners"
are always superimposed time-crossing 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.
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987.079
Synergetics' experimentally produced, minimum-structural-system
subdivisions of Universe have four tuned-in vertexial
loci, four windows looking toward
all the untuned-in 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.
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987.080
Vertexial Spheres Form Rigids
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 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
unit-radius-vertex 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 unit-radius-
vertex spheres. For an illustration of these structural
"rigids" see Fig.
987.081.
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987.100
Great-circle-spun Symmetries and Cleavagings
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987.110
Intercommensurability Functions
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987.111
In the great-circle-spun 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.
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987.120
Sequence of Symmetries and Cleavagings
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987.121
Table
icosa and the further
progressive halvings of:
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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 halved-out parts of the first four of
the only seven cosmic symmetries of
axial spin of the primitive structural systems .
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