413.00 Omnidirectional Closest Packing
413.01 In omnidirectional closest packing of equiradius spheres around a nuclear sphere, 12 spheres will always symmetrically and intertangentially surround one sphere with each sphere tangent to its immediate neighbors. We may then close-pack another symmetrical layer of identical spheres surrounding the original 13. The spheres of this outer layer are also tangent to all of their immediate neighbors. This second layer totals 42 spheres. If we apply a third layer of equiradius spheres, we find that they, too, compact symmetrically and tangentially. The number of spheres in the third layer is 92.
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NO. RODS PER CLUSTER 1 7 19 37 61 91 127
NO. RODS OUTER LAYER 6 12 18 24 30 36
TOTAL ROD AREAS 314.159 314.159 314.159 314.159 314.159 314.159 314.159
DIAM. OF SINGLE ROD 20.000 7.556 4.588 3.288 2.561 2.099 1.775
AREA OF SINGLE ROD 314.159 44.880 16.535 8.491 5.150 3.452 2.474
PERIM. OF SINGLE ROD 62.832 23.748 14.415 10.330 8.045 6.594 5.575
TOTAL PERIMETER OF RODS IN BUNDLE 62.832 166.24 273.88 382.19 489.88 600.07 708.08
TOTAL PERIMETER OF RODS IN OUTER LAYER 62.832 142.49 172.98 185.93 193.08* 197.83* 205.39*
MIN. DIAMETER OF HEXAGONAL BUNDLE 20.649 20.483 20.373 20.302 20.277 20.218
MAX. DIAMETER OF HEXAGONAL BUNDLE 22.668 22.940 23.016 23.046 23.089 23.095
TOTAL AREA OF RODS OUTER LAYER ONLY 269.279 198.416 152.834 123.604 103.569 89.053
NOTE THAT PERIMETERS OF OUTER LAYER RODS ALONE EXCEEDS THREE TIMES PERIMETER OF LARGE ROD.
Fig. 412.01 Closest Packing of Rods.
413.02 Equi-radius spheres closest packed around a nuclear sphere do not form a supersphere, as might be expected. They form a symmetrical polyhedron of 14 faces: the vector equilibrium.
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3 Frequency Triangle A.
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Fig. 413.01 Vector Equilibrium: Omnidirectional Closest Packing Around a Nucleus: Triangles can be subdivided into greater and greater numbers of similar units. The number of modular subdivisions along any edge can be referred to as the frequency of a given triangle. In triangular grids each vertex may be expanded to become a circle or sphere showing the inherent relationship between closest packed spheres and triangulation. The frequency of triangular arrays of spheres in the plane is determined by counting the number of intervals (A) rather than the number of spheres on a given edge. In the case of concentric packings or spheres around a nucleus the frequency of a given system can either be the edge subdivision or the number of concentric shells or layers. Concentric packings in the plane give rise to hexagonal arrays (B) and omnidirectional closest packing of equal spheres around a nucleus (C) gives rise to the vector equilibrium (D).
413.03 If we add on more layers of equiradius spheres to the symmetrical polyhedron of 14 faces close-packed around one sphere, we find that they always compact symmetrically and tangentially, and that this process of enclosure may seemingly be repeated indefinitely. Each layer, however, is in itself a finite or complete and symmetrical embracement of spheres. Each of these embracing layers of spheres constitutes a finite system. Each layer always takes the 14-face conformation and consists of eight triangular and six square faces. Together with the layers they enclose and the original sphere center, or nucleus, these symmetrically encompassing layers constitute a concentric finite system.
413.04 As additional layers are added, it is found that a symmetrical pattern of concentric systems repeats itself. That is, the system of three layers around one sphere, with 92 spheres in the outer layer, begins all over again and repeats itself indefinitely with successively enclosing layers in such a way that the successive layers outside of the 92-sphere layer begin to penetrate the adjacent new nuclear systems. We find then that only the concentric system of spheres within and including the layer of 92 are unique and individual systems. We will pursue this concept of a finite system in universal geometry still further (see Sec. 418, et seq.) in order to relate it to the significance of the 92 self-regenerative chemical elements.