410.00 Closest Packing of Spheres
410.01 Natures Coordination
410.011 About 1917, I decided that nature did not have separate, independently operating departments of physics, chemistry, biology, mathematics, ethics, etc. Nature did not call a department heads’ meeting when I threw a green apple into the pond, with the department heads having to make a decision about how to handle this biological encounter with chemistry’s water and the unauthorized use of the physics department’s waves. I decided that it didn’t require a Ph.D. to discern that nature probably had only one department and only one coordinate, omnirational, mensuration system.
410.02 I determined then and there to seek out the comprehensive coordinate system employed by nature. The omnirational associating and disassociating of chemistryalways joining in whole low-order numbers, as for instance
410.03 I was dissatisfied with abstract, weightless, unstable, ageless, temperatureless, straight-line-defined squares and cubes as models for calculating our omnicurvilinear experience. I was an early rebel against blockheads and squares. Reviewing the history of chemical science, I became intuitively aware that the clue to vectorial, volumetric, geometrical coordination with physical reality and all the fundamental energetic experiences of reality, such as temperature, time, and force, might be found in Avogadro’s experimental proof of his earlier hypothesis, which stated that all gases under identical conditions of heat and pressure will always disclose the same number of molecules per given volume. Here was disclosed a “Grand Central Station” accommodating all corners; despite “fundamental” or elementarily unique differences of identity, all accommodated on a common volume (space)-to-number basis. One molecule of any element: One space. A cosmic democracy.
410.04 I felt intuitively that inasmuch as the variety of gases experimented with often consisted of only one unique chemical element, such as hydrogen or oxygen, and that inasmuch as these gases also could be liquefied, and also inasmuch as most of the elements are susceptible to some heat- or pressure-produced transformation between their liquid, crystalline, and vapor, or incandescent, states, it might also be hypothetically reasonable to further generalize Avogadro’s hypothesis by assuming that, under identical energy conditions, all elements may disclose the same number of “somethings” per given volume. Such a generalized concept is not limited to pressure and heat: we wanted to be much more inclusive, so we said we assume that the conditions of energy are identical; this includes not only the pressure and heat conditions of thermodynamics, which developed before electromagnetics became an applied realization, but also the conditions of electromagnetics as constant.
410.05 I went on from there to reason that vectors, being the product of physical energy constituents, are “real,” having velocity multiplied by mass operating in a specific direction; velocity being a product of time and size; and mass being a volume-weight relationship. On impact, mass at velocity transforms into heat and work. These energy factors can be translated not only into work, but into heat or into time as well. Furthermore, electromagnetic scientists had found that all their EMF (electromotive force) problems could be graphed vectorially; the fact that “graphable” or “modelable” vectors can interact modelably in real Universe space seemed to promise that the equations of nature’s omnicoordinate transactions, expressed in omni-space-intruding vectorial models, might produce real models of reality of nature’s Grand Central Station of omnicoordination.
410.06 So I then went on to say that, if all the energy conditions were everywhere the same, then all the vectors would be the same length and all of them would interact at the same angle. I then explored experimentally to discover whether this “isotropic vector matrix,” as so employed in matrix calculus, played with empty sets of symbols on flat sheets of paper, could be realized in actual modeling. Employing equilength toothpicks and semi-dried peas, as I had been encouraged to do in kindergarten at the age of four (before receiving powerful eyeglasses and when I was unfamiliar with the right-angled structuring of buildings as were the children with normal vision), I fumbled tactilely with the toothpicks and peas until I could feel a stable structure, and thus assembled an omnitriangulated complex and so surprised the teachers that their exclamations made me remember the event in detail. I thus rediscovered the octet truss whose vertexes, or convergent foci, were all sixty-degree-angle interconnections, ergo omniequilateral, omniequiangled, and omni-intertriangulated; ergo, omni structured. Being omnidirectionally equally interspaced from one another, this omni-intertriangulation produced the isotropic matrix of foci for omni-closest-packed sphere centers. This opened the way to a combinatorial geometry of closest-packed spheres and equilength vectors.
410.07 Over and over again, we are confronted by nature obviously formulating her structures with beautiful spherical agglomerations. The piling of oranges, coconuts, and cannonballs in tetrahedral or half-octahedral pyramids has been used for centuries and possibly for ages. Almost a half-century ago, F. W. Aston, the British scientist first identified for physics the most economical uniradius spherical interagglomerations as the “closest packing of spheres,” which had fresh interest for the physicist and crystallographer because of the then recently discovered microscopic realization that nature frequently employed omni-intertriangulated systems, which hold mathematical clues to the principles of symmetrical coordination governing natural structure, the dynamic vectorial geometry of the atomic nucleus, as well as of the atoms themselves.