Vector Equilibrium: Potential and Primitive Tetravolumes
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1033.180
Vector Equilibrium: Potential and Primitive Tetravolumes
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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.
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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 innermost-economically-interconnecting 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.
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1033.183
The tetravolume-6 rhombic dodecahedron is the domain
of each closest-
packed, unit-radius sphere, for it tangentially embraces
not only each sphere, but that
sphere's proportional share of the intervening space
produced by such unit-radius-sphere
closest packing.
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1033.184
When the time-sizing is initiated with frequency2,
the rhombic
dodecahedron's volume of 6 is eightfolded to become
48. In the plurality of closest-
packed-sphere domains, the sphere-into-space, space-into-sphere
dual rhombic
dodecahedron domain has a tetravolume of 48. The total
space is 24__with the vector
equilibrium's Eighth-Octahedra extroverted to form the
rhombic dodecahedron. For every
space there is always an alternate space: This is where
we get the 48-ness of the rhombic
dodecahedron as the macrodomain of a sphere:
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1033.185
The 12 spheric domains around one nuclear sphere domain
equal 13 rhombic
dodecahedra__nuclear 6 + (12 × 6) = tetravolume 78.
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1033.192
Table: Prime Number Consequences of Spin-halving of
Tetrahedron's
Volumetric Domain Unity
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1033.20
Table: Cosmic Hierarchy of Primitive Polyhedral Systems:
The constant
octave system interrelationship is tunable to an infinity
of different frequency keys:
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1033.30
Symmetrical Contraction of Vector Equilibrium: Quantum
Loss
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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 half-octahedra's four internal, equiangular,
triangular faces progressively
paired into congruence, at which point each of the six
half-octahedra__ergo, three
quanta__has been annihilated.
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1033.32
In the always-omnisymmetrical 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 half-octahedra become mildly
folded diamonds ridge-poled along the diamond's shorter
axis and thereby bent into six
ridge-pole diamond facets, thus producing 12 primitively
equilateral triangles. Not until
the six squares are diagonally vectored is the vector
equilibrium stabilized into an
omnitriangulated, 20-triangled, 20-tetrahedral 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.
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1033.33
See Sec.
611.02
for the tetravolumes per vector quanta
structurally invested
in the tetra, octa, and icosa, in which we accomplish__
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1033.35
The six new vector diagonals of the three pairs of
opposing half-octahedra
become available to provide for the precession of any
one of the equatorial quadrangular
vectors of the half-octahedra 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 mass-interattractively 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.
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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.)
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1033.40
Asymmetrical Contraction of Vector Equilibrium: Quantum
Loss
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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 half-octahedra
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 half-octahedra, 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 prime-vector-edge
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 right-angle-
apexed and prime-vector-base-edged internal isosceles-triangle
faces, each of whose
interior apexes occurs congruently at the center of
volume of the symmetrical
octahedron__ergo, each of which eight regular-to-asymmetric-transformed
tetrahedra are
now seen to be our familiar Eighth-Octahedra, 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 tetravolume-4.
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1033.42
This transformation may also have been accomplished
in an alternate
manner. We recall how the jitterbug vector equilibrium
demonstrated the four-dimensional
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 as-yet-12-vectored 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 two-vertex-to-two-vertex-bonded
"pair" of regular tetrahedra
whose most-remotely-opposite, 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 southeastward-slantwise rotating vertices originally
situated at the VE jitterbug
equator spiral into congruence with the three corner
vertices of the nonrotating south pole
triangle.
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 Fig. 1033.43 |
1033.43
As part of the comprehensively symmetrical contraction
of the whole
primitive VE system, we may consider the concurrent
north-to-south polar-axis
contraction (accomplished as the axis remained motionless
with respect to the equatorial
motions) to have caused the two original vertex-to-vertex
regular polar tetrahedra to
penetrate one another vertexially as their original
two congruent center-of-VE-volume
vertices each slid in opposite directions along their
common polar-axis 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.)
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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 one-half of the volume of the regular
octahedron of tetravolume-4.
Therefore the remaining space, with the octahedron equatorially
surrounding their axial
core, is also of tetravolume-2__i.e., one-half inside-out
(space) and one-half inside-in
(tetracore).
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1033.45
At this octahedron-forming 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.
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1033.46
In the octahedron-forming state the three polar-base,
corner-to-apex-
connecting-edges of each of the contracting polar-axis
tetrahedra now penetrate the other
tetrahedron's three nonpolar triangle faces at their
exact centers of area.
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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 octahedral-forming positions) at
the respective centers of area of their
opposite equiangular polar triangles. Their two regular-tetrahedra-shaped
apex points
penetrate their former polar-opposite triangles until
the six mid-edges 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.)
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1033.48
The 12 geodesic chords omniinterconnecting these eight
symmetrically
omniarrayed vertices now define the regular cube, one-half
of whose total volume of
exactly 3-tetravolumes is symmetrically cored by the
eight-pointed star core form
produced by the two mutually interpenetrated tetrahedra.
This symmetrical core star
constitutes an inside-in tetravolume of l 1/2, with
the surrounding equatorial remainder of
the cube-defined, insideout space being also exactly
tetravolume 1 1/2. (See Fig.
987.242A.)
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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 mid-edges__which
six mutual tetrahedra's mid-edge points all occur at
the six centers of the six square faces
of the cube.
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1033.491
The rotation around these six points continues until
the six edge-lines 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.
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1033.492
Only during the symmetrical contraction of the tetravolume-3
cube to the
tetravolume- 1 tetrahedron did the original axial contraction
cease, as the two opposing
axis tetrahedra (one inside-out and one outside-out)
rotate simultaneously and
symmetrically on three axes (as permitted only by four-dimensionality
freedoms) to
become unitarily congruent as tetravolume-1__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 inside-out
and six outside-out), while
providing the prime numbers 1,2,3,5 and multiples thereof,
to become available for the
entropic-syntropic, export-import 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 24 1 compaction.
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1033.50
Quanta Loss by Congruence
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1033.51
Euler's Uncored Polyhedral Formula:
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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.)
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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.
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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
(quadri-bonding). 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.
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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.
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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.
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1033.60
Primitive Dimensionality
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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 special-case physical reality
begin with frequency. Pre-time-size conceptuality is
primitive conceptuality.
Unfrequenced angular topology is primitive. (See Sec.
527.70.)
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1033.61
Fifth Dimension Accommodates Physical Size
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1033.611
Dimension begins at four. Four-dimensionality is primitive
and exclusively
within the primitive systems' relative topological abundances
and relative interangular
proportionment. Four-dimensionality is eternal, generalized,
sizeless, unfrequenced.
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1033.612
If the system is frequenced, it is at minimum linearly
five-dimensional,
surfacewise six-dimensional, and volumetrically seven-dimensional.
Size is special case,
temporal, terminal, and more than four-dimensional.
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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 whole-system 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.
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1033.62
Zerovolume Tetrahedron
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1033.621
The primitive tetrahedron is the four-dimensional,
eight-in-one, quadrivalent,
always-and-only-coexisting, inside-out and outside-out
zerovolume whose four great-
circle planes pass through the same nothingness center,
the four-dimensionally
articulatable inflection center of primitive conceptual
reference.
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1033.622
Thus the tetrahedron__and its primitive, inside-out,
outside-out
intertransformability into the prime, whole, rational,
tetravolume-numbered hierarchy of
primitive-structural-system states__expands from zerovolume
to its 24-tetravolume limit
via the maximum-nothingness vector-equilibrium state,
whose domain describes and
embraces the primitive, nucleated, 12-around-one, closest-packed,
unit-radius spheres.
(See cosmic hierarchy at Sec.
982.62.)
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