41500-Concentric-Shell-Growth-Rates

415.00 Concentric Shell Growth Rates

415.01 Minimal Most Primitive Concentric Shell Growth Rates of Equiradius, Closest-Packed, Symmetrical Nucleated Structures: Out of all possible symmetrical polyhedra produceable by closest-packed spheres agglomerating, only the vector equilibrium accommodates a one-to-one arithmetical progression growth of frequency number and shell number developed by closest-packed, equiradius spheres around one nuclear sphere. Only the vector equilibrium“equanimity” accommodates the symmetrical growth or contraction of a nucleus-containing aggregate of closest-packed, equiradius spheres characterized by either even or odd numbers of concentric shells.

415.02 Odd or Even Shell Growth: The hierarchy of progressive shell embracements of symmetrically closest-packed spheres of the vector equilibrium is generated by a smooth arithmetic progression of both even and odd frequencies. That is, each successively embracing layer of closest-packed spheres is in exact frequency and shell number atunement. Furthermore, additional embracing layers are accomplished with the least number of spheres per exact arithmetic progression of higher frequencies.

415.03 Even-Number Shell Growth: The tetrahedron, octahedron, cube, and rhombic dodecahedron are nuclear agglomerations generated only by even-numbered frequencies:

image

Chart 415.03 Rate of Occurrence of Symmetrically Nucleated Polyhedra of Closest Packing

415.10 Yin-Yang As Two (Note to Chart 415.03): Even at zero frequency of the vector equilibrium, there is a fundamental twoness that is not just that of opposite polarity, but the twoness of the concave and the convex, i.e., of the inwardness and outwardness, i.e., of the microcosm and of the macrocosm. We find that the nucleus is really two layers because its inwardness turns around at its own center and becomes outwardness. So we have the congruence of the inbound layer and the outbound layer of the center ball.

10F^{2}+ 2

F = 0

10x0 = 0

0 + 2 = 2 (at zero frequency)

Because people thought of the nucleus only as oneness, they for long missed the significant twoness of spherical unity as manifest in the atomic weights in the Periodic Table of the Elements.

415.11 When they finally learned that the inventory of data required the isolation of the neutron, they were isolating the concave. When they isolated the proton, they isolated the convex.

415.12 As is shown in the comparative table of closest-packed, equiradius nucleated polyhedra, the vector equilibrium not only provides an orderly shell for each frequency, which is not provided by any other polyhedra, but also gives the nuclear sphere the first, or earliest possible, polyhedral symmetrical enclosure, and it does so with the least number spheres; whereas the octahedron closest packed requires 18 spheres; the tetrahedron, 34; the rhombic dodecahedron, 92; the cube, 364; and the other two symmetric Platonic solids, the icosahedron and the dodecahedron, are inherently, ergo forever, devoid of equiradius nuclear spheres, having insufficient radius space within the triangulated inner void to accommodate an additional equiradius sphere. This inherent disassociation from nucleated systems suggests both electron and neutron behavior identification relationships for the icosahedron’s and the dodecahedron’s requisite noncontiguous symmetrical positioning outwardly from the symmetrically nucleated aggregates. The nucleation of the octahedron, tetrahedron, rhombic dodecahedron, and cube very probably plays an important part in the atomic structuring as well as in the chemical compounding and in crystallography. They interplay to produce the isotopal Magic Number high point abundance occurrences. (See Sec. 984.10 hangeadd ref)

415.13 The formula for the nucleated rhombic dodecahedron is the formula for the octahedron with frequency plus four (because it expands outwardly in four-wavelength leaps) plus eight times the closest-packed central angles of a tetrahedron. The progression of layers at frequency plus four is made only when we have one ball in the middle of a five-ball edge triangle, which always occurs again four frequencies later.

415.14 The number of balls in a single-layer, closest-packed, equiradius triangular assemblage is always 2.

N²-N

415.15 To arrive at the cumulative number of spheres in the rhombic dodecahedron, you have to solve the formula for the octahedron at progressive frequencies plus four, plus the solutions for the balls in the eight triangles.

415.16 The first cube with 14 balls has no nucleus. The first cube with a nucleus occurs by the addition of 87-ball corners to the eight triangular facets of a four-frequency vector equilibrium.

415.17 Nucleated Cube: The “External” Octahedron: The minimum allspace-filling nuclear cube is formed by adding eight Eighth-Octahedra to the eight triangular facets of the nucleated vector equilibrium of tetravolume-20, with a total tetravolume involvement of 4 + 20 = 24 quanta modules. This produces a cubical nuclear involvement domain (see Sec. 1006.30) of tetravolume-24: $24 \times 24 = 5
76$ quanta modules. (See Sec. 463.05 and Figs. 415.17A-F.)

415.171 The nuclear cube and its six neighboring counterparts are the volumetrically maximum members of the primitive hierarchy of concentric, symmetric, pre-time-size, subfrequency-generalized, polyhedral nuclear domains of synergetic-energetic geometry.

415.172 The construction of the first nuclear cube in effect restores the vector-equilibrium truncations. The minimum to be composited from closest- packed unit radius squares has 55 balls in the vector equilibrium. The first nucleated cube has 63 balls in the total aggregation.

image

Fig. 415.17 Nucleated Cube: The “External" Octahedron: ABC show that eight additional closest-packed spheres are required to form the minimum allspace-filling nuclear cube to augment the nuclear vector equilibrium. DEF show the eight Eighth-Octa required to complete the polyhedral transformation. (Compare Fig. 1006.32.)

415.20 Organics: It could be that organic chemistries do not require nuclei.

415.21 The first closest-packed, omnitriangulated, ergo structurally stabilized, but non-nuclear, equiradius-sphered, cubical agglomeration has 14 spheres. This may be Carbon 14, which is the initially closest-packed, omnisymmetrical, polyhedral fourteenness, providing further closest-packability surface nests suitable for structurally mounting hydrogen atoms to produce all organic matter.

415.22 The cube is the prime minimum omnisymmetrical allspace filler. But the cube is nonstructural until its six square faces are triangularly diagonaled. When thus triangularly diagonaled, it consists of one tetrahedron with four one-eighth octahedra, of three isosceles and one equilateral-faced tetrahedron, outwardly applied to the nuclear equilateral tetrahedron’s four triangular faces. Thus structurally constituted, the superficially faced cube is prone to closest-packing self-associability. In order to serve as the carbon ring (with its six-sidedness), the cube of 14 spheres (with its six faces) could be joined with six other cubes by single atoms nestable in its six square face centers, which singleness of sphericity linkage potential is providable by Hydrogen 1.

415.23 In the atoms, we are always dealing in equiradius spheres. Chemical compounds may, and often do, consist of atomic spheres with a variety of radial dimensions. Since each chemical element’s atoms are characterized by unique frequencies, and unique frequencies impose unique radial symmetries, this variety of radial dimensionality constitutes one prime difference between nuclear physics and chemistry.

415.30 Eight New Nuclei at Fifth Frequency of Vector Equilibrium: Frequency five embraces nine nuclei: the original central nucleus plus eight new nuclei occurring at the centers of volume of the eight tetrahedra symmetrically surrounding the nucleus, with each of the nine enclosed with a minimum of two layers of spheres.

image image image image

Fig. 415.22 Rational Volumes of Tetrahedroning:

The cube may be formed by placing four 1/s-octahedra with their equilateral faces on the faces of a tetrahedron. Since tetrahedron volume equals one, and Vs- octahedron equals V2, the volume of the cube will be: 1 + 4(¹/2) = 3.
The rhombic dodecahedron may be formed by placing eight V-i-tetrahedra with their equilateral faces on the faces of an octahedron. Since the octahedron volume equals four and V4-tetrahedron equals V4, the volume of the rhombic dodecahedron will be: 4+8(V4)=6.

415.31 The vector equilibrium at /= 12; at/² = 42; /³ = 93; f⁴= 162 spheres in the outer shell; and atf⁵ = 252 we get eight new nuclei. Therefore, their eightness of “begetness” relates to the eight triangles of the vector equilibrium.

415.32 Six nucleated octahedra with two layer omni-enclosure of their nuclei does not occur until /⁶ = 362 in the outer shell of the vector equilibrium. At this stage we have six new nuclei, with 14 nuclei surrounding the 15th, or original, nucleus.

415.40 Begetted Eightness: The “begetted” eightness as the systemlimit number of nuclear uniqueness of self-regenerative symmetrical growth may well account for the fundamental octave of unique interpermutative integer effects identified as plus one, plus two, plus three, plus four, as the inter- permutated effects of the integers one, two, three, and four, respectively; and as minus four, minus three, minus two, minus one, characterizing the integers five, six, seven, and eight, respectively. The integer nine always has a neutral, or zero, intermutative effect on the other integers. This permutative, synergetic or interamplifying or dimensioning effect of integers upon integers, together with the octave interinsulative accommodation produced by the zero effect of the nineness, is discussed experientially in our section on Indigs in Chapter 12, Numerology.

415.41 The regenerative initial eightness of first-occurring potential nuclei at the frequency-four layer and its frequency-five confirmation of those eight as constituting true nuclei, suggest identity with the third and fourth periods of the Periodic Table of Chemical Elements, which occur as

1st period = 2 elements

2nd period = 8 elements

3rd period = 8 elements

415.42 Starting with the center of the nucleus: plus one, plus two, plus three, plus four, outwardly into the last layer of nuclear uniqueness, whereafter the next pulsation becomes the minus fourness of the outer layer (fifth action); the sixth event is the minus threeness of canceling out the third layer; the seventh event is the minus twoness canceling out the second layer; the eighth event is the minus oneness returning to the center of the nucleusall of which may be identified with the frequency pulsations of nuclear systems.

415.43 The None or Nineness/Noneness permits wave frequency propagation cessation. The Nineness/Zeroness becomes a shutoff valve. The Zero/Nineness provides the number logic to account for the differential between potential and kinetic energy. The NinenessIZeroness becomes the number identity of vector equilibrium, that is, energy differentiation at zero. (See Secs. 1230 et seq. and the Scheherazade Number.)

415.44 The eightness being nucleic may also relate to the relative abundance of isotopal magic numbers, which read 2, 8, 20, 50, 82, 126…

415.45 The inherent zero-disconnectedness accounts for the finite energy packaging and discontinuity of Universe. The vector equilibria are the empty set tetrahedra of Universe, i.e., the tetrahedron, being the minimum structural system of Universe independent of size, its four facet planes are at maximum remoteness from their opposite vertexes and may have volume content of the third power of the linear frequency. Whereas in the vector equilibrium all four planes of the tetrahedra pass through the same opposite vertexwhich is the nuclear vertex and have no volume, frequency being zero: F^{0}.

415.50 Vector-Equilibrium Closest-Packing Configurations: The vector equilibrium has four unique sets of axes of symmetry:

The three intersymmetrical axes perpendicular to, i.e., normal to, i.e., joining, the hemispherically opposite six square faces;
The four axes normal to its eight triangular faces;
The six axes normal to its 12 vertexes; and
the 12 axes normal to its 24 edges.

The tetrahedron, vector equilibrium, and octahedron, with all their planes parallel to those of the tetrahedron, and therefore derived from the tetrahedron, as the first and simplest closest-packed, ergo omnitriangulated, symmetrical structural system, accept further omnidirectional closest packing of spheres. Because only eight of its 20 planar facets are ever parallel to the four planes of the icosahedron, the icosahedron refuses angularly to accommodate anywhere about its surface further omnidirectional closest packing of spheres, as does the tetrahedron.

415.51 Consequently, the (no-nucleus-accommodating) icosahedron formed of equiradius, triangularly closest-packed spheres occurs only as a one-sphere-thick shell of any frequency only. While the icosahedron cannot accommodate omnidirectionally closest-packed multishell growth, it can be extended from any one of its triangular faces by closest-packed sphere agglomerations. Two icosahedra can be face-bonded.

415.52 The icosahedron has three unique sets of axes of symmetry:

The 15 intersymmetric axes perpendicular to and joining the hemispherically opposite mid-edges of the icosahedron’s 30 identical, symmetrically interpatterned edges;
The 10 intersymmetric axes perpendicular to the triangular face centers of the hemispherically opposite 20 triangular faces of the icosahedron; and
The six intersymmetric axes perpendicularly interconnecting the hemispheric opposites of the icosahedron’s 12 vertexes, or vertexial corner spheres of triangular closest packing.

415.53 While the 15-axes set and the 6-axes set of the icosahedron are always angularly askew from the vector equilibrium’s, four out of its 10 axes of symmetry are parallel to the set of four axes of symmetry of the vector equilibrium. Therefore, the icosahedron may be face-extended to produce chain patterns conforming to the tetrahedron, octahedron, vector equilibrium, and rhombic dodecahedron in omnidirectional, closest-packing coordinationbut only as chains; for instance, as open linear models of the octahedron’s edges, etc.

415.55 Nucleus and Nestable Configurations in Tetrahedra: In any number of successive planar layers of tetrahedrally organized sphere packings, every third triangular layer has a sphere at its centroid (nucleus). The dark ball rests in the valley between three balls, where it naturally falls most compactly and comfortably. The next layer is three balls to the edge, which means two-frequency. There are six balls in the third layer, and there very clearly is a nest right in the middle. There are ten balls in the fourth layer: but we cannot nest a ball in the middle because it is already occupied by a dark centroid ball. Suddenly the pattern changes, and it is no longer nestable.

415.56 At first, we have a dark ball at the top; then a second layer of three balls with a nest but no nucleus. The third layer with six balls has a nest but no nucleus. The fourth layer with ten balls has a dark centroid ball at the nucleus but no nestable position in the middle. The fifth layer (five balls to the edge; four frequency) has 15 balls with a nest again, but no nucleus. This 35- sphere tetrahedron with five spheres on each edge is the lowest frequency tetrahedron system that has a central sphere or nucleus. (See Fig. A, illustration 415.55. )

415.57 The three-frequency tetrahedron is the highest frequency single-layer, closest-packed sphere shell without a nuclear sphere. This three-frequency, 20-sphere, empty, or nonsphere nucleated, tetrahedron may be enclosed by an additional shell of 100 balls; and a next layer of 244 balls totaling 364, and so on. (See Fig. B, illustration 415.55.)

415.58 Basic Nestable Configurations: There are three basic nestable possibilities shown in Fig. C. They are (1) the regular tetrahedron of four spheres; (2) the one-eighth octahedron of seven spheres; and (3) the quarter tetrahedron, with a 16th sphere nesting on a planar layer of 15 spheres. Note that this “nesting” is only possible on triangular arrays that have no sphere at their respective centroids. This series is a prime hierarchy. One sphere on three is the first possibility with a central nest available. One sphere on six is the next possibility with an empty central nest available. One sphere on 10 is impossible as a ball is already occupying the geometrical center. The next possibility is one on 15 with a central empty nest available.

image image image

Fig. 415.55 Tetrahedral Closest Packing of Spheres: Nucleus and Nestable Configurations:

In any number of successive planar layers of tetrahedrally organized sphere packings, every third triangular layer has a sphere at its centroid (a nucleus). The 36- sphere tetrahedron with five spheres on an edge (four-frequency tetrahedron) is the lowest frequency tetrahedron system which has a central sphere or nucleus.
The three-frequency tetrahedron is the highest frequency without a nucleus sphere.
Basic “nestable” possibilities show how the regular tetrahedron, the V-i- tetrahedron and the 1/s-octahedron may be defined with sets of closest packed spheres. Note that this “nesting” is only possible on triangular arrays which have no sphere at their respective centroids.

415.59 Note that the 20-ball empty set (see Fig. B, illustration 415.55 ) consists of five sets of four-ball simplest tetrahedra and can be assembled from five separate tetrahedra. The illustration shows four four-ball tetrahedra at the vertexes colored “white.” The fifth four-ball tetrahedron is dark colored and occupies the central octahedral space in an inverted position. In this arrangement, the four dark balls of the inverted central tetrahedron appear as center balls in each of the four 10-ball tetrahedral faces.