BUCKYVERSE

223.00 Principle of Prime Number Inherency and Constant Relative Abundance of the Topology of Symmetrical Structural Systems

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.

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.)

223.03 Equation of Prime Number Inherency of All Symmetrical Structural Omnitriangulated Systems:

X = 2NF^{2}+ 2

Where:

X = number of vertexes (crossings) or spheres in the outer layer or shell of any symmetrical system;

N = one of the first four prime numbers: 1, 2, 3, or 5; and F = edge frequency, i.e., the number of outer layer edge modules.

223.04 Equation of Constant Relative Abundance of Topological Aspects of All Symmetrical Structural Systems: Multiplication of one of the first four prime numbers of their powers or multiples by the constant of relative topological characteristics abundance:

1 + 2 = 3

  1. Nonpolar vertex

  2. Faces

  3. Edges

In addition to the product of such multiplication of the constant relative abundance equation by one of the first four prime numbers$1, 2, 3, or
5$or their powers or multiples, there will always be two additional vertexes assigned as the poles of the axial spinnability of the system.

223.05 Two Kinds of Twoness: There are two kinds of twoness:

  1. the numerical, or morphationally unbalanced twoness; and

  2. the balanced twoness.

The vector equilibrium is the central symmetry through which both balanced and unbalanced asymmetries pulsatingly and complexedly intercompensate and synchronize. The vector equilibrium’s frequency modulatability accommodates the numerically differentiated twonesses.

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.

223.07 There is a fourfold twoness: one of the exterior, cosmic, finite (“nothingness”) tetrahedroni.e., the macrocosm outwardly complementing all (“something”) systemsand one of the interior microcosmic tetrahedron of nothingness complementing all conceptually thinkable and cosmically isolatable “something” systems. (See Sec. 1070.)

223.08 A pebble dropped into water precessionally produces waves that move both outwardly from the circle’s centeri.e., circumferentially of the Earth sphereand reprecessionally outwardly and inwardly from the center of the Earthi.e., radially in respect to the Earth sphere. Altogether, this inter-regeneratively 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.

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.)

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.

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.

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.

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 sphere-like 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.

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 inside-outing tetrahedron is the minimum compound curveergo, minimum sphere.

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.

223.16 The number of openings (or “faces”) will be twice that of the vertexes, minus two.

223.17 The number of vector edges will be three times the number of vertexes, minus two.

223.18 When we reduce the topological inventory of basic vertexes, areas, and edges of all omnitriangulated structural systems in Universe whether symmetrical or asymmetricalby 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.

223.19 In an omni-triangulated structural system:

  1. the number of vertexes (“crossings” or “points”) is always evenly divisible by two;

  2. the number of faces (“areas” or “openings”) is always evenly divisible by four; and

  3. the number of edges (“lines,” “vectors,” or “trajectories”) is always evenly divisible by six.

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:

  1. Tetrahedron (four sides):
2 +{[}(2 \times 1) \times F^{2}{]}= 4 \text{vertexes (crossings)}
  1. Octahedron (eight sides):
2 +{[}(2 \times 2) \times F^{2}{]}= 6 \text{vertexes (crossings)}
  1. Cube (six Sides):
2+{[}(2 \times 3) \times F^{2}{]}= 8 \text{vertexes (crossings)}
  1. Vector Equilibrium (fourteen sides):
2+{[}(2x 5) xF\textsuperscript{2}{]}= 12 vertexes (crossings)
``` *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:

1.  *Tetrahedron:*

*X = 2F^2^ + 2*

1.  *Octahedron:*

X = 4F^2^ + 2

1.  *Cube:*

*X = 6F^2^ + 2*

1.  *Vector Equilibrium (Icosahedron)-.*

*X = 10F^2^+2*

Where:

*X =* the number of spheres in the outer layer or shell of the primary system;

F = edge frequency, i.e., the number of outer-layer edge modules.

1.  Symmetrical Analysis of Topological Hierarchies: Symmetrical means having no local asymmetries. Omnisymmetrical permits local asymmetries.

2.  The following omnitriangulated systems are symmetrical:

Tetrahedron

Octahedron

Icosahedron

*223.30* The following omnitriangulated systems are omnisymmetrical: Cube

Diagonal Rhombic Dodecahedron

Rhombic Dodecahedron

Dodecahedron

Tetraxidecahedron

Triacontahedron

Enenicontrahedron

*223.33* The vector equilibrium is locally mixed symmetrical and asymmetrical.

*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 omni2triangulated, vector equilibrium may function as a system but not as a structure.

*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 non-demonstrable 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.

*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.

*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).

*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.

*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](#sec:920.00-functions-of-a-and-b-modules) through [940](#sec:940.00-hierarchy-of-quanta-module-orientations)). They each have a volume of one-twenty-fourth of a tetrahedron.

*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.

*223.61* Only triangles are structures, as will be shown in Sec. 610. Systems have insideness and outsidenessergo, 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.

*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.”](#subsec:608.00-stability:-necklace))

*223.64* Table: Synergetics Hierarchy of Topological Characteristics of Omnitriangulated Polyhedral Systems ([See pp. 46–47.](#tab:223.64))

*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.

*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 twothe 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.*