222.00 Omnidirectional Closest Packing of Spheres
222.01 Definition: The omnidirectional concentric closest packing of equal radius spheres about a nuclear sphere forms a matrix of vector equilibria of progressively higher frequencies. The number of vertexes or spheres in any given shell or layer is edge frequency (
222.02 Equation:
Where,
222.03 The frequency can be considered as the number of layers (concentric shells or radius) or the number of edge modules of the vector equilibrium. The number of layers and the number of edge modules is the same. The frequency, that is the number of edge modules, is the number of spaces between the spheres, and not the number of spheres, in the outer layer edge.
222.10 Equation for Cumulative Number of Spheres: The equation for the total number of vertexes, or sphere centers, in all symmetrically concentric vector equilibria shells is:
$10(F^\textsuperscript4 -F{2}\textsuperscript{2}+F\textsuperscript
- F^) + 2F+1$
222.20 Characteristics of Closest Packing of Spheres: The closest packing of spheres begins with two spheres tangent to each other, rather than omnidirectionally. A third sphere may become closest packed by becoming tangent to both of the first two, while causing each of the first two also to be tangent to the two others: this is inherently a triangle.
One-Frequency
A.
Two-Frequency
B.
Fig. 222.01 Equation for Omnidirectional Closest Packing of Spheres: Omnidirectional concentric closest packings of equal spheres about a nuclear sphere form series of vector equilibria of progressively higher frequencies. The number of spheres or vertexes on any symmetrically concentric shell or layer is given by the equation
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222.21 A fourth sphere may become closest packed by becoming tangent to all three of the first three, while causing each of the others to be tangent to all three others of the four-sphere group: this is inherently a tetrahedron.
222.22 Further closest packing of spheres is accomplished by the omniequiangular, intertriangulating, and omnitangential aggregating of identical radius spheres. In omnidirectional closest-packing arrays, each single sphere finds itself surrounded by, and tangent to, at most, 12 other spheres. Any center sphere and the surrounding 12 spheres altogether describe four planar hexagons, symmetrically surrounding the center sphere.
222.23 Excess of Two in Each Layer: The first layer consists of 12 spheres tangentially surrounding a nuclear sphere; the second omnisurrounding tangential layer consists of 42 spheres; the third 92, and the order of successively enclosing layers will be 162 spheres, 252 spheres, and so forth. Each layer has an excess of two diametrically positioned spheres which describe the successive poles of the 25 alternative neutral axes of spin of the nuclear group. (See illustrations 450.11a and 450.11b.)
222.24 Three Layers Unique to Each Nucleus: In closest packing of spheres, the third layer of 92 spheres contains eight new potential nuclei which do not, however, become active nuclei until each has three more layers surrounding itthree layers being unique to each nucleus.
222.25 Isotropic Vector Matrix: The closest packing of spheres characterizes all crystalline assemblages of atoms. All the crystals coincide with the set of all the polyhedra permitted by the complex configurations of the isotropic vector matrix (see Sec. 420), a multidimensional matrix in which the vertexes are everywhere the same and equidistant from one another. Each vertex can be the center of an identical-diameter sphere whose diameter is equal to the uniform vector’s length. Each sphere will be tangent to the spheres surrounding it. The points of tangency are always at the mid-vectors.
222.26 The polyhedral shape of these nuclear assemblages of closest-packed spheresreliably interdefined by the isotropic vector matrix’s vertexesis always that of the vector equilibrium, having always six square openings (“faces”) and eight triangular openings (“faces”).
222.30 Volume of Vector Equilibrium: If the geometric volume of one of the uniform tetrahedra, as delineated internally by the lines of the isotropic vector matrix system, is taken as volumetric unity, then the volume of the vector equilibrium will be 20.
Fig. 222.30 Volume of Vector Equilibrium: The volume of the vector equilibrium consists of eight tetrahedra and six half-octahedra. Therefore, the volume of the vector equilibrium is exactly twenty.
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222.31 The volume of any series of vector equilibria of progressively higher frequencies is always frequency to the third power times 20.
222.32 Equation for Volume of Vector Equilibrium:
Volume of vector equilibrium
Where
222.40 Mathematical Evolution of Formula for Omnidirectional Closest Packing of Spheres: If we take an inventory of the number of balls in successive vector equilibria layers in omnidirectional closest packing of spheres, we find that there are 12 balls in the first layer, 42 balls in the second layer, and 92 balls in the third. If we add a fourth layer, we will need 162 balls, and a fifth layer will require 252 balls. The number of balls in each layer always comes out with the number two as a suffix. We know that this system is a decimal system of notation. Therefore, we are counting in what the mathematician calls congruence in modulo ten a modulus of ten unitsand there is a constant excess of two.
222.41 In algebraic work, if you use a constant suffixwhere you always have, say, 33 and 53you could treat them as 30 and 50 and come out with the same algebraic conditions. Therefore, if all these terminate with the number two, we can drop off the two and not affect the algebraic relationships. If we drop off the number two in the last column, they will all be zeros. So in the case of omnidirectional closest packing of spheres, the sequence will read; 10, 40, 90, 160, 250, 360, and so forth. Since each one of these is a multiple of 10, we may divide each of them by 10, and then we have 1, 4, 9, 16, 25, and 36, which we recognize as a progression of second poweringtwo to the second power, three to the second power, and so forth.
222.42 In describing the number of balls in any one layer, we can use the term frequency of modular subdivisions of the radii or chords as defined by the number of layers around the nuclear ball. In the vector equilibrium, the number of modular subdivisions of the radii is exactly the same as the number of modular subdivisions of the chords (the “edge units”), so we can say that frequency to the second power times ten plus two is the number of balls in any given layer.
222.43 This simple formula governing the rate at which balls are agglomerated around other balls or shells in closest packing is an elegant manifest of the reliably incisive transactions, formings, and transformings of Universe. I made that discovery in the late 1930s and published it in 1944. The molecular biologists have confirmed and developed my formula by virtue of which we can predict the number of nodes in the external protein shells of all the viruses, within which shells are housed the DNA-RNA-programmed design controls of all the biological species and of all the individuals within those species. Although the polio virus is quite different from the common cold virus, and both are different from other viruses, all of them employ frequency to the second power times ten plus two in producing those most powerful structural enclosures of all the biological regeneration of life. It is the structural power of these geodesic-sphere shells that makes so lethal those viruses unfriendly to man. They are almost indestructible.
222.50 Classes of Closest Packing: There are three classes of closest packing of unit-radius spheres:
222.51 SYSTEMATIC Symmetrical Omnidirectional Closest Packing: Twelve spheres closest pack omnitangentially around one central nuclear sphere. Further symmetrical enclosure by closest-packed sphere layers agglomerate in successive vector equilibria. The nucleus is inherent.
222.52 ASYMMETRICAL Closest Packed Conglomerates: Closest-packed conglomerates may be linear, planar, or “crocodile.” Closest packed spheres without nuclear organization tend to arrange themselves as the octet truss or the isotropic vector matrix. The nuclei are incidental.
222.53 VOLUMETRIC Symmetrical Closest Packing: These are nonnuclear symmetrical embracements by an outer layer. The outer layer may be any frequency, but it may not be expanded or contracted by the addition inwardly or outwardly of complete closest-packed layers. Each single-layer frequency embracement must be individually constituted. Volumetric symmetrical closest packing aggregates in most economical forms as an icosahedron geodesic network. The nucleus is excluded.