Principle of Prime Number Inherency and Constant Relative
Abundance of the Topology of Symmetrical Structural
Systems
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223.00
Principle of Prime Number Inherency and Constant Relative
Abundance of the Topology of Symmetrical Structural
Systems
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223.01
Definition: The number of vertexes of every omnitriangulated
structural
system is rationally and differentially accountable,
first, by selecting and separating out the
always additive two polar vertexes that must accommodate
the neutral axis of spin
inherent in all individual structural systems to permit
and account for their independent
motional freedom relationship from the rest of Universe.
The number of nonpolar vertexes
is called the base number. Second, we identify the always
multiplicative duality factor of
two characterizing the always coexistent insideness-outsideness
of systems and their
inherently positively and negatively congruent disparity
of convexity and concavity. Third,
we find the multiplicative duality factor of two to
be multiplied by one of the first four
prime numbers, 1, 2, 3, or 5 (multiplied by 1 if the
structural system is tetrahedral, by 2 if it
is octahedral, by 3 if it is the triangularly structured
cube, or by 5 if it is the icosahedron or
the triangularly stabilized vector equilibrium), or
factored by a variety of multiples
comprised of combinations of only those first four prime
numbers, whether the polyhedra
are, in the Platonic, Archimedean, or any other progression
of symmetrical structural
systems. When the vector edges of the symmetrical systems
are modularly subdivided, all
of the foregoing products are found to be multiplied
again to the second power by the
frequency of uniform modular subdivisions of the vector
edges of the symmetrical
structural system. In respect to the original base number
of nonpolar vertexes, there will
always be twice as many openings ("faces") and three
times as many vector edges of the
symmetrical structural system, always remembering that
the two polar vertexes were first
extracted from the inventory of topological characteristics
before multiplying the
remaining number of vertexes in the manner described
and in relation to which the number
of nonpolar vertexes and the relative abundance of the
other topological characteristics are
accurately derived and operationally described.
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223.02
Axis of Spin: Any two vertexes may be selected as
the axis of spin, whether
or not the axis described by them is immediately conceivable
as the logical axis of
spinnability, i.e., the axis need not be statically
symmetrical. (You can take hold of a boy
by his two hands and, holding one above the other and
leaning backward spin him
centrifugally around you. Although his two hands do
not represent the symmetrical static
axis of the boy’s body, their dynamic positions defined
the axis of your mutual spinning.)
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223.03
Equation of Prime Number Inherency of All Symmetrical
Structural
Omnitriangulated Systems:
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223.04
Equation of Constant Relative Abundance of Topological
Aspects of All
Symmetrical Structural Systems: Multiplication of one
of the first four prime numbers
or their powers or multiples by the constant of relative
topological characteristics
abundance:
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223.05
Two Kinds of Twoness: There are two kinds of twoness:
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223.06
There are four kinds of positive and negative:
| (1) the eternal, equilibrium-disturbing plurality of
differentially unique, only-
positively-and-negatively-balanced aberratings; |
(2) the north and south poles; |
(3) the concave and convex; and |
(4) the inside (microcosm) and outside (macrocosm),
always cosmically
complementing the local system's inside-concave and
outside-convex limits. |
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223.07
There is a fourfold twoness: one of the exterior,
cosmic, finite
(“nothingness”) tetrahedron__i.e., the macrocosm outwardly
complementing all
(“something”) systems__and one of the interior microcosmic
tetrahedron of nothingness
complementing all conceptually thinkable and cosmically
isolatable "something" systems.
(See Sec. 1070.)
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223.08
A pebble dropped into water precessionally produces
waves that move both
outwardly from the circle's center__i.e., circumferentially
of the Earth sphere__and
reprecessionally outwardly and inwardly from the center
of the Earth__i.e., radially in
respect to the Earth sphere. Altogether, this interregeneratively
demonstrates (1) the
twoness of local precessional system effects at 90 degrees,
and (2) the Universe-cohering
gravitational effects at 180 degrees. These are the
two kinds of interacting forces
constituting the regenerative structural integrity of
both subsystem local twonesses and
nonunitarily conceptual Scenario Universe. The four
cosmically complementary twonesses
and the four local system twonesses altogether eternally
regenerate the scientific
generalization known as complementarity. Complementarity
is sum-totally eightfoldedly
operative: four definitive local system complementations
and four cosmically synergetic
finitive accountabilities.
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223.09
Topologically the additive twoness identifies the
opposite poles of
spinnability of all systems; the multiplicative twoness
identifies the concave-insideness and
convex-outsideness of all systems: these four are the
four unique twonesses of the
eternally regenerative, nonunitarily conceptual Scenario
Universe whose conceptual think-
aboutedness is differentially confined to local "something"
systems whose insideness-and-
outsideness-differentiating foci consist at minimum
of four event "stars."
(See Secs. 510.04 and
510.09.)
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223.10
Constant Relative Abundance: Topological systems that
are structurally
stabilized by omnitriangulation reveal a constant relative
abundance of certain fundamental
characteristics deriving from the additive twoness and
the multiplicative twoness of all
finite systems.
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223.11
The additive twoness derives from the polar vertexes
of the neutral axis of
spin of all systems. This twoness is the beginning and
essence of consciousness, with
which human awareness begins: consciousness of the other,
the other experience, the
other being, the child's mother. To describe that of
which we are aware, we employ
comparison to previous experience. That which we are
aware of is hotter, or bigger, or
sharper than the other experience or experiences. The
a priori otherness of comparative
awareness inherently requires time. Early humanity's
concept of the minimum increment of
time was the second, because time and awareness begin
with the second experience, the
prime other. If there is only one think, one think is
naught. Life and Universe that goes
with it begins with two spheres: you and me . . . and
you are always prior to me. I have
just become by my awareness of you.
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223.12
The multiplicative twoness is inherent in the disparity
of the congruent
convexity and concavity of the system. The multiplicative
twoness is because both you and
I have insideness and outsideness, and they are not
the same: one is convex and one is
concave.
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223.13
Conceptual systems having inherent insideness and
outsideness are defined at
minimum by four event foci and are, ergo, tetrahedral;
at maximum symmetrical
complexity, they are superficially "spherical"__that
is, they are a spherelike array of event
foci too minute for casual resolution into the plurality
of individual event foci of which, in
experiential fact, they must consist, each being approximately
equidistant from one
approximately identifiable event focus at the spherical
array's center. Since all the
"surface" event foci may be triangularly interconnected
with one another by chords that
are shorter than arcs, all spherical experience arrays
are, in fact, polyhedra. And all spheres
are polyhedra. Spherical polyhedra may at minimum consist
of the four vertexes of the
regular tetrahedron.
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223.14
We discover that the additive twoness of the two polar
(and a priori
awareness) spheres at most economical minimum definition
of event foci are two
congruent tetrahedra, and that the insideness and outsideness
of complementary tetrahedra
altogether represent the two invisible complementary
twoness that balances the visible
twoness of the polar pair. This insideouting tetrahedron
is the minimum compound
curve__ergo, minimum sphere.
(See Sec. 624.)
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223.15
When the additive twoness and the multiplicative twoness
are extracted from
any symmetrical and omnitriangulated system, the number
of vertexes will always be a
rational product of one or more of the first four prime
numbers, 1, 2, 3, or 5, or their
powers or multiples.
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223.16
The number of openings (or "faces") will be twice
that of the vertexes,
minus two.
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223.17
The number of vector edges will be three times the
number of vertexes,
minus two.
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223.18
When we reduce the topological inventory of basic
vertexes, areas, and
edges of all omnitriangulated structural systems in
Universe__whether symmetrical or
asymmetrical__by taking away the two poles and dividing
the remaining inventory by two,
we discover a constant relative abundance of two faces
and three lines for every one
vertex. This is to say that there is a constant topological
abundance characterizing all
systems in Universe in which for every nonpolar vertex
there are always two faces and
three (vectorial) edges.
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223.19
In an omnitriangulated structural system:
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223.20
Primary Systems: Only four primary systems or contours
can be developed
by closest packing of spheres in omnisymmetrical concentric
layers. The exterior contours
of these points are in the chart:
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223.21
Primary Systems: Equations: The formulas for the number
of spheres in the
outer layer of closest packed spheres in primary systems
is as follows:
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223.30
Symmetrical Analysis of Topological Hierarchies: Symmetrical
means
having no local asymmetries. Omnisymmetrical permits
local asymmetries .
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223.31
The following omnitriangulated systems are symmetrical:
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223.32
The following omnitriangulated systems are omnisymmetrical:
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223.33
The vector equilibrium is locally mixed symmetrical
and asymmetrical.
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223.34
Symmetrical Analysis of Topological Hierarchies: Whenever
we refer to
an entity, it has to be structurally valid, and therefore
it has to be triangulated. Being
locally mixed, vectorially symmetrical but facially
asymmetrical, being triangulated but not
omnitriangulated, vector equilibrium may function as
a system but not as a structure.
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223.40
Powering: Second powering in the topology of synergetics
is identifiable
only with the vertexes of the system and not with something
called the "surface area."
Surfaces imply experimentally nondemonstrable continuums.
There are no topologically
indicated or implied surfaces or solids. The vertexes
are the external points of the system.
The higher the frequency of the system, the denser the
number of external points. We
discover then that second powering does not refer to
"squaring" or to surface
amplification. Second powering refers to the number
of the system's external vertexes in
which equating the second power and the radial or circumferential
modular subdivisions of
the system multiplied by the prime number one if a tetrahedral
system; by the prime
number two if an octahedral system; by the prime number
three if a triangulated cubical
system; and by the prime number five if an icosahedral
system; each, multiplied by two,
and added to by two, will accurately predict the number
of superficial points of the system.
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223.41
This principle eliminates our dilemma of having to
think of second and third
powers of systems as referring exclusively to continuum
surfaces or solids of the systems,
neither of which states have been evidenced by experimental
science. The frequencies of
systems modify their prime rational integer characteristics.
The second power and third
power point aggregations identify the energy quanta
of systems and their radiational
growth or their gravitational contraction. They eliminate
the dilemma in which physics
failed to identify simultaneously the wave and the particle.
The dilemma grew from the
misconceived necessity to identify omnidirectional wave
growth exclusively with the rate
of a nonexperimentally existent spherical surface continuum
growth, the second power of
radiational growth being in fact the exterior quanta
and not the spherical surface being
considered as a continuum.
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223.50
Prime Number Inherency: All structurally stabilized
polyhedra are
characterized by a constant relative abundance of Euler's
topological aspects in which
there will always be twice as many areas and three times
as many lines as the number of
points in the system, minus two (which is assigned to
the polar axis of spin of the system).
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223.51
The number of the topological aspects of the Eulerian
system will always be
an even number, and when the frequency of the edge modulation
of the system is reduced
to its second root and the number of vertexes is divided
by two, the remainder will be
found to consist exclusively of a prime number or a
number that is a product exclusively of
two or more intermultiplied prime numbers, which identify
the prime inherency
characteristics of that system in the synergetic topological
hierarchy of cosmically simplest
systems.
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223.52
All other known regular symmetrical polyhedra (other
than the tetrahedron
and the octahedron) are described quantitatively by
compounding rational fraction
elements of the tetrahedron and the octahedron. These
elements are known as the A and B
Quanta Modules
(see Sec.920 through 940).
They each have a volume of one-twenty-fourth of a tetrahedron.
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223.60
Analysis of Topological Hierarchies: Omnitriangulation:
The areas and
lines produced by omnitriangularly and circumferentially
interconnecting the points of the
system will always follow the rule of constant relative
abundance of points, faces, and
lines.
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223.61
Only triangles are structures, as will be shown in
Sec. 610.
Systems have
insideness and outsideness ergo, structural systems
must have omnitriangulated isolation
of the outsideness from the insideness. Flexibly jointed
cubes collapse because they are not
structures. To structure a cubical form, the cube's
six square faces must be diagonally
divided at minimum into 12 triangles by one of the two
inscribable tetrahedra, or at
maximum into 24 triangles by both the inherently inscribable
positive-negative tetrahedra
of the cube’s six faces.
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223.62
Lacking triangulation, there is no structural integrity.
Therefore, all the
polyhedra must become omnitriangulated to be considered
in the Table. Without
triangulation, they have no validity of consideration.
(See Sec. 608, "Necklace.")
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223.64
Table: Synergetics Hierarchy of Topological Characteristics
of
Omnitriangulated Polyhedral Systems (See pp. 46-47.)
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223.65
The systems as described in Columns 1 through 5 are
in the prime state of
conceptuality independent of size: metaphysical. Size
is physical and is manifest by
frequency of "points-defined" modular subdivisions of
lengths, areas, and volumes. Size is
manifest in the three variables of relative length,
area, and volume; these are all expressible
in terms of frequency. Frequency is operationally realized
by modular subdivision of the
system.
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223.66
Column 1 provides a statement of the true rational
volume of the figure
when the A and B Quanta Modules are taken as unity.
Column 2 provides a statement of the true rational volume of the figure when the tetrahedron is taken as unity. Columns 1 and 2 describe the rationality by complementation of two selected pairs of polyhedra considered together. These are (a) the vector-edged icosahedron and the vector-edged cube; and (b) the vector-edged rhombic dodecahedron and the vector-edged dodecahedron. Column 3 provides the ratio of area-to-volume for selected polyhedra. Column 4 denotes self-packing, allspace-filling polyhedra. Column 5 identifies complementary allspace-filling polyhedra. These are: (a) the A and B Quanta Modules in combination with each other; (b) the tetrahedron and octahedron in combination with each other; and (c) the octahedron and vector equilibrium in combination with each other. Column 6 presents the topological analysis in terms of Euler. Columns 7 through 15 present the topological analysis in terms of synergetics, that is, with the polar vertexes extracted from the system and with the remainder divided by two. Column 7 accounts the extraction of the polar vertexes. All systems have axes of spin. The axes have two poles. Synergetics extracts two vertexes from all Euler topological formulas to function as the poles of the axis of spin. Synergetics speaks of these two polar vertexes as the additive two. It also permits polar coupling with other rotative systems. Therefore a motion system can have associability. Column 9 recapitulates Columns 7 and 8 in terms of the equation of constant relative abundance. Column 10 accounts synergetics multiplicative two. Column 11. The synergetics constants of all systems of Universe are the additive two and the multiplicative two__the Holy Ghost; the Heavenly Twins; a pair of twins. Columns 12 and 15 identify which of the first four prime numbers are applicable to the system considered. Column 13 recapitulates Columns 11 and 12. |
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223.67
Synergetics Hierarchy: The Table of Synergetics Hierarchy
(223.64)
makes it possible for us to dispense with the areas
and lines of Euler's topological
accounting; the hierarchy provides a definitive description
of all omnitriangulated
polyhedral systems exclusively in terms of points and
prime numbers.
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223.70
Planck's Constant
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223.71
Planck's constant: symbol = h. h = 6.6__multiplied
by l0-27 grams per
square centimeters per each second of time. The constant
h is the invariable number found
empirically by Planck by which the experimentally discovered,
uniformly energized,
minimum increment of all radiation, the photon, must
be multiplied to equate the
photon’s energy value as rated by humans' energy-rating
technique, with the effort
expended in lifting weights vertically against gravity
given distances in given times. Thus
automotive horsepower or electromagnetic kilowatts per
hour performance of stationary
prime movers, engines, and mobile motors are rated.
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223.72
Max Planck's photons of light are separately packaged
at the radiation
source and travel in a group-coordinated flight formation
spherical surface pattern which
is ever expanding outwardly as they gradually separate
from one another. Every photon
always travels radially away from the common origin.
This group-developed pattern
produces a sum-totally expanding spherical wave-surface
determined by the plurality of
outwardly traveling photons, although any single photon
travels linearly outwardly in only
one radial direction. This total energy effort is exactly
expressed in terms of the
exponential second-power, or areal "squaring," rate
of surface growth of the overall
spherical wave; i.e., as the second power of the energy
effort expended in lifting one gram
in each second of time a distance of one "vertical"
centimeter radially outward away from
the origin center.
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223.73
Whereas: All the volumes of all the equi-edged regular
polyhedra are
irrational numbers when expressed in the terms of the
volume of a cube = 1;
Whereas: The volume of the cube and the volumes of the other regular polyhedra, taken singly or in simple groups, are entirely rational; Whereas: Planck's constant was evaluated in terms of the cube as volumetric unity and upon the second-power rate of surface expansion of a cube per each second of time; Whereas: Exploring experimentally, synergetics finds the tetrahedron, whose volume is one-third that of the cube, to be the prime structural system of Universe: prime structure because stabilized exclusively by triangles that are experimentally demonstrable as being the only self-stabilizing polygons; and prime system because accomplishing the subdivision of all Universe into an interior microcosm and an external macrocosm; and doing so structurally with only the minimum four vertexes topologically defining insideness and outsideness; Whereas: Structuring stability is accomplished by triangularly balanced energy investments; Whereas: Cubes are unstable; Whereas: The radial arrangement of unit tetrahedral volumes around an absolute radiation center (the vector equilibrium) constitutes a prime radiational-gravitational energy proclivity model with a containment value of 20 tetrahedra (where cube is 3 and tetrahedron 1); Whereas: Max Planck wished to express the empirically emerged value of the photon, which constantly remanifested itself as a unit-value energy entity in the energy-measuring terms of his contemporary scientists; Wherefore: Planck employed the XYZ rectilinear frame of shape, weight, volume, surface, time, distance, antigravity effort, and metric enumeration, mensuration tools adopted prior to the discovery of the photon value. |
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223.74
Planck's constant emerged empirically, and to reconvert
it to conformity with
synergetics the 6.6ness is canceled out:
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223.75
In synergetics' topological accounting, surface areas
are always structural
triangles of the systems, which systems, being vectorially
structured, are inherently energy-
investment systems. As synergetics' topology also shows,
the number of triangular surface
areas of the system increases at twice the rate of the
nonpolar surface points, ergo the rate
of energetic system’s surface increase is accounted
in terms of the number of the triangular
areas of the system's surface, which rate of system
surface increase is 20F2, where F =
frequency; while the rate of volumetric increase is
20F3. The vector is inherent in the
synergetics system since it is structured with the vector
as unity. Because vectors = mass ×
velocity, all the factors of time, distance, and energy,
as both mass and effort as well as
angular direction, are inherent; and E as energy quantum
of one photon = 20F2.
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223.80
Energy Has Shape
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223.81
I recognize the experimentally derived validity of
the coordinate invariant
the result does not depend on the coordinate system
used. Planck's constant is just what it
says it is: an experimentally ascertained constant cosmic
relationship. Planck's constant as
expressed is inherently an irrational number, and the
irrationality relates to the invariant
quantum of energy being constantly expressed exclusively
in the volume-weight terms of a
special-case shape which, in the geometrical shape-variant
field of weight-strength and
surface-volume ratio limits of local structural science
containment of energy, as mass or
effort, by energy-as-structure, is neither maximum nor
minimum. The special-case
geometrical shape chosen arbitrarily by the engineering-structures-eschewing
pure
scientists for their energy-measurement accommodation,
that of the cube, is structurally
unstable; so much so as to be too unstable to be classified
as a structure. Unwitting of this
mensural shortcoming, Planck's constant inadvertently
refers to the cube, implicit to the
gram, as originally adopted to provide an integrated
unit of weight-to-volume
mensuration, as was the “knot” adopted by navigators
as a velocity unit which integrates
time-space incrementation values. The volume and weight
integrate as a gram. The gram
was arbitrarily assumed to be constituted by a cubic
centimeter of water at a specific
temperature, 4 degrees centigrade.
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223.82
Relationship constants are always predicated on limits.
Only limits are
invariable. (This is the very essence of the calculus.)
Variation is between limits. Though
Planck's constant is indirectly predicated on a limit
condition of physical phenomena, it is
directly expressed numerically only as a prefabricated,
constantly irrational number-
proportionality to that limit, but it is not the inherently
rational unit number of that limit
condition. This is because the cube was nonstructural
as well as occurring structurally
between the specific limit cases of surface-to-volume
ratio between whose limits
of 1 20,
the cube rates as 3.
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223.83
Max Planck found a constant energy-value relationship
emergent in all the
photon-discovery experimental work of others. A great
variety of exploratory work with
measurements of energy behaviors in the field of radiation
disclosed a hitherto unexpected,
but persistent, minimum limit in relation to such energy
phenomena. Planck expressed the
constant, or limit condition, in the scientifically
prevailing numerical terms of the physical
and metaphysical equipment used to make the measuring.
The measuring system included:
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223.84
Let us assume hypothetically that Ponce de Leon did
find the well of eternal-
youth-sustaining water, and that the well had no "spring"
to replenish it, and that social
demand occasioned its being bailed out and poured into
evaporation-proof containers; and
that the scientists who bailed out that precious well
of water used a cubically-shaped, fine-
tolerance, machined and dimensioned one-inch-thick shelled,
stainless steel bucket to do
their carefully measured bailing and conserving task.
They did so because they knew that
cubes close-pack to fill allspace, and because water
is a constant substance with a given
weight per volume at a given temperature. And having
ten fingers each, they decided to
enumerate in the metric system without any evidence
that meters are whole rational linear
increments of a cosmic nature. Thus organized, the Ponce
de Leon scientists soon
exhausted the well, after taking out only six and two-thirds
cubic bucket loads__with a
little infinitely unaccountable, plus-or-minus, spillage
or overestimate.
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223.85
Planck's constant, h, denotes the minimum energy-as-radiation
increment
known experimentally by humans to be employed by nature,
but the photon's energy value
could and should be expressed in terms of a whole number
as referenced directly by
physical experiment to nature's limit-case transforming
states.
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223.86
Had, for instance, the well-of-youth-measuring scientists
happened to be in a
hurry and had they impatiently used a cubical container
of the same size made of a thin-
wall plastic such as the cubically shaped motel waste
containers, they would have noticed
when they stood their waterfilled plastic cube bucket
on the ground beside the well that its
sides bulged and that the level of the water lowered
perceptibly below the container's rim;
though this clearly was not caused by leaking, nor by
evaporating, but because its shape
was changing, and because its volume-to-container-surface
ratio was changing.
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223.87
Of all regular polyhedra, the sphere (i.e., the high-frequency,
omnitriangulated, geodesic, spheroidal polyhedron) encloses
the most volume with the
least surface. Whereas the tetrahedron encloses the
least volume with the most surface.
The contained energy is at minimum in the tetrahedron.
The structure capability is at
maximum in the tetrahedron.
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223.88
Planck did not deliberately start with the cube. He
found empirically that the
amount of the photon's energy could be expressed in
terms of the CGtS-XYZ decimal-
enumeration coordinate system already employed by science
as the "frame of reference"3
for his photon evaluation which, all inadvertently,
was characterized by awkwardness and
irrationality.
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223.89
Energy has shape. Energy transforms and trans-shapes
in an evoluting way.
Planck's contemporary scientists were not paying any
attention to that. Science has been
thinking shapelessly. The predicament occurred that
way. It's not the size of the
bucket__size is special case__they had the wrong shape.
If they had had the right shape,
they would have found a whole-rational-number constant.
And if the whole number found
was greater than unity, or a rational fraction of unity,
they would simply have had to
divide or multiply to find unity itself.
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223.90
The multiplier 10-27 is required to reduce the centimeter
magnitude of
energy accounting to that of the tuned wavelength of
the photon reception. Frequency and
wave are covariably coupled; detection of one discloses
the other. Since synergetics'
vector equilibrium's energy converging or dispersing
vector is both radially and chordally
subdivided evenly by frequency__whatever that frequency
may be__the frequency
fractionates the unit vector energy involvement by one-to-one
correspondence.
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223.91
If they had taken the same amount of water at the
same temperature in the
form of a regular tetrahedron, they would have come
out with a rational fraction of unity.
They happened to be enumerating with congruence in modulo
10, which does not include
any prime numbers other than 1, 2, and 5. The rational
three-ness of the cube in relation to
the tetrahedron is not accommodated by the decimal system;
nor is the prime 7 inherent in
modulo 10. Therefore, Planck's constant, while identifying
a hitherto undiscovered
invariant limit condition of nature, was described in
the wrong frame of reference in
awkward__albeit in a constantly awkward__term, which works,
because it is the truth; but
at the same time it befogs the otherwise lucid and rational
simplicity covering this
phenomenon of nature, just as does nature's whole number
of utterly rational atoms
exchanging rates in all her chemical combining and separating
transactions accounting.
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