430.01 Definition
430.011
The geometric form most compactly developed from the closest packing of spheres around one nuclear sphere is not that of a composite sphere, but is always a polyhedron of 14 faces composed of six squares and eight triangles, with 12 vertexes extending in tangential radius from the original 12 spheres surrounding the nucleus sphere. (See illustration 413.01.)
430.02
It is called the vector equilibrium because the radials and the circumferentials are all of the same dimension and the tendencies to both explode and implode are symmetrical. That the explosive and implosive forces are equal is shown by the four- dimensional hexagonal cross sections whose radial and circumferential vectoM balance. The eight triangular faces reveal four opposite pairs of single-bonded tetrahedra in a positive and negative tetrahedral system array with a common central vertex and with coinciding radial edges. The four hexagonal planes that cross each other at the center of the vector-equilibrium system are parallel to the four faces of each of its eight tetrahedra. Six square faces occur where the six half-octahedra converge around the common vector- equilibrium nuclear vertex.
430.03
In terms of vectorial dynamics, the outward radial thrust of the vector equilibrium is exactly balanced by the circumferentially restraining chordal forces: hence the figure is an equilibrium of vectors. All the edges of the figure are of equal length, and this length is always the same as the distance of any of its vertexes from the center of the figure. The lines of force radiating from its center are restrainingly contained by those binding inward arrayed in finite closure circumferentially around its periphery—barrel- hooping. The vector equilibrium is an omnidirectional equilibrium of forces in which the magnitude of its explosive potentials is exactly matched by the strength of its external cohering bonds. If its forces are reversed, the magnitude of its contractive shrinkage is exactly matched by its external compressive archwork’s refusal to shrink.
430.04
The vector equilibrium is a truncated cube made by bisecting the edges and truncating the eight corners of the cube to make the four axes of the four planes of the vector equilibrium. The vector equilibrium has been called the “cuboctahedron” or ” cubo- octahedron” by crystallographers and geometers of the non-experimentally-informed and non-energy-concerned past. As such, it was one of the original 13 Archimedean “solids.”
430.05
The vector equilibrium is the common denominator of the tetrahedron, octahedron, and cube. It is the decimal unit within the octave system. Double its radius for octave expansion.
430.06
The vector equilibrium is a system. It is not a structure. Nor is it a prime volume, because it has a nucleus. It is the prime nucleated system. The eight tetrahedra and the six half-octahedra into which the vector equilibrium may be vectorially subdivided are the volumes that are relevantly involved.
431.00 Volume
431.01
The vector equilibrium consists of six one-half octahedra, each with a volume of two (6 × 2 = 12), and eight tetrahedra each with a volume of one, so 8 + 12 = 20, which is its exact volume. (See illustration 222.30.)
431.02
The volume of a series of vector equilibria of progressively higher frequencies is always frequency to the third power times 20, or 20F³ , where F=frequency. When the vector equilibrium’s frequency is one (or radiationally inactive), its volume is 20 × 1³ = 20.
431.03
But frequency, as a word key to a functional concept, never relates to the word one because frequency obviously involves some plurality of events. As a one- frequency, ergo sub-frequency, system, the vector equilibrium is really subsize, or a size- independent, conceptual integrity. Therefore, frequency begins with two—where all the radials would have two increments. When the edge module of a cube is one, its volume is one; when the edge module of a cube is two, its volume is eight. But when the edge module of a vector equilibrium is one, its volume is 20. A nuclear system is subsize, subfrequency. Equilibrious unity is 20; its minimum frequency state is 160 = 2⁵ × 5. This is one of the properties of 60-degree coordination.
431.04
Looking at a two-frequency vector equilibrium (with all the radials and edge units divided into two) and considering it as the domain of a point, we find that it has a volume of 480 A and B Modules. The formula of the third power of the frequency tells us the exact number of quanta in these symmetrical systems, in .terms of quantum accounting and in terms of the A and B Modules (see Chapter 9, Modelability).
432.00 Powering
432.01
The vector equilibrium makes it possible to make conceptual models of fourth-, fifth-, and sixth-dimensional omniexpeAence accounting by using tetrahedroning. If we have a volume of 20 around a point, then two to the fourth power (16) plus two to the second power (4) equals 20. We can then accommodate these powerings around a single point.
432.02
Using frequency to the third power with a no-frequency nucleus, the vector equilibrium models all of the first four primes. For instance, the number 48 (in 480) is 16 × 3. Three is a prime number, and 16 is two to the fourth power: that is 48, and then times 10. Ten embraces the prime numbers five times the number two; so instead of having 16 times 2, we can call it 32, which is two to the fifth power. The whole 480-moduled vector equilibrium consists of the prime number one times two to the fifth power, times three, times five (1 × 2⁵ × 3 × 5). These are the first four prime numbers.
432.03
Using frequency to the third power with a two-frequency nucleus, we have 2³×2⁵ = 2⁸. If the frequency is two, we have two to the eighth power in the model times three times five (2⁸ × 3 × 5).
432.04
In a three-frequency system, we would have three to the third power times three, which makes three to the fourth power, which we would rewrite as 2⁵ × 3⁴ × 5. We get two kinds of four-dimensionality in here. There is a prime dimensionality of three to the fourth power (3⁴). And there is another kind of four-dimensionality if the frequency is four, which would be written 2⁵ × 3 × 5. But since it is frequency to the third power, and since four is two times two (2 × 2) or two to the second power (2²), we would add two to make two to the seventh power (2⁷), resulting in ⁷ × 3 × 5. If the frequency is five, it would then be two to the fifth power (2⁵) times three, because frequency is to the third power times five, which makes five to the fourth power. Quite obviously, multidimensionality beyond three dimensions is experienceably, i.e., conceptually, modelable in synergetics accounting.
433.00 Outside Layer of Vector Equilibrium
433.01
The unique and constantly remote but-always-and-only co-occurring geometrical “starry” surroundment “outsideness” of the nucleated vector equilibrium is always an icosahedron, but always occurring only as a single layer of vertexes of the same frequency as that of the nuclear vector equilibrium’s outermost vertexial layer.
433.02
There may be multilayer vector equilibria—two-frequency, threefrequency, four-frequency, or whatever frequency. The circumferential vector frequency will always be identical to that of its radial vector frequency contraction of the vector equilibrium’s outer layer of unit radius spheres by local surface rotation of that outer layer’s six square arrays of non-closest-singlelayer packing of tangent spheres inter-rearranging into closest triangular packing as in the vector equilibrium’s eight triangular facets, thus transforming the total outer layer into the icosahedron of equal outer edge length to that of the vector equilibrium, but of lesser interior radius than the vector equilibrium of the same outer edge length, and therefore of lesser interior volume than that of the vector equilibrium, ergo unable to accommodate the same number of interiorally-closest-packed, nuclear-sphere- centered unit radius spheres as that of the vector equilibrium. The icosahedron’s multifrequenced outer layer surface arrays of unit radius, closest-planar-packed spheres cannot accommodate either concentric layers of unit radius closest-packed spheres nor— even at zero frequency—can the icosahedron’s 12-ball, omni-intertangentially triangulated outer shell accommodate one nuclear sphere of the same radius as that of its shell spheres. Icosahedral outer shell arrays of identical frequency to that of the vector equilibria of the same frequency, can therefore only occur as single-layer, symmetrical, enclosure arrays whose individual spheres cannot be tangent to one another but must be remotely equipositioned from one another, thus to form an omni-intertriangulated, icosahedrally conformed starry array, remotely and omnisurroundingly occupying the vector equilibrium’s sky at an omnistar orbit-permitting equidistance remoteness around the vector equilibrium whose outer shell number of spheres exactly corresponds to the number of the icosahedron’s “stars.” This geometrical dynamically interpositioning integrity of relationship strongly suggests the plurality of unique electron shell behaviors of all the chemical elements’ atoms, and the identical number relationships of the atoms’ outer layer protons and its electrons; and the correspondence of the vector equilibrium’s number of concentric closest-packed, nucleus-enclosing layers with the number of quantumjump- spaced electron orbit shells; and finally the relative volume relationship of equi-edged vector equilibria and icosahedra, which is, respectively, as 20 is to 18.51, which suggests the relative masses of the proton and the electron, which is as 1:1/1836.