Demass Model — 986.00-986.874

986.00 T and E Quanta Modules

986.00 T and E Quanta Modules: Structural Model of E=mc²: The Discovery that the E Quanta Module Is the True, Experimentally Evidenceable Model of E=mc²

[986.00-986.874 Demass Model Scenario]

986.010 Narrative Recapitulation

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The exposition herein recounts and recapitulates the original surprise and excitement of the progressive, empirically verified conceptionings; the family of relevant experimental-evidence recalls; the modus operandi; the successive, only-evolutionarily- discovered inputs; and the synergetic comprehension of the omniinterresultant cosmic significance of these strategically employable, synergetically critical additions to human knowledge and their technologically realizable insights.

986.020 Elementary School Definitions

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My first mathematics and geometry teachers taught me games that I learned to play well enough to obtain swiftly the answers for which their (only-axiomatically- argued) assumptions called. Webster’s dictionary states tersely the definitions of the games they taught me. Webster’s definitions are carefully formulated by leading academic authorities and represent the up-to-the-minute concensus of what the educational system assumes geometry, mathematics, and science to consist.

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Webster defines geometry as “the mathematics of the properties, measurements, and relationships of points, lines, angles, surfaces, and solids”— none of which we ourselves observe can exist experientially (ergo, science-verifiably), independently of the others; ergo, they cannot be isolatable “properties” or separate characteristics.

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Physics has found no surfaces and no solids: only localized regions of high- frequency, self-interfering, deflecting, and consequently self-knotting energy events. These self-interference patterns occur in pure principle of ultra-high-frequency intervals and on so minuscule a scale as to prohibit intrusion by anything so dimensionally gross and slow as our fingers. We cannot put our fingers between any two of all the numbers occurring serially between the integer 1 and the integer 2,000,000,000,000—two trillion—as aggregated linearly in one inch. This is the approximate number of atomic domains (the x- illion-per-second, electron-orbited atoms’ individual spinout domains) tangentially arrayable in a row within an experience inch.

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Within each of the electron-orbited spheric domains the respective atomic nuclei are centered as remotely distant from their orbiting electrons as is our Sun from its orbiting planets. Within each of these nuclei complex, high-frequency events are occurring in pure principle of interrelationship.

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How do you see through a solid-glass window? Light passes through glass. Light is high-frequency radiation passing unobstructedly at 700 million miles per hour with lots of time and room “to spare” between the set of energy events that constitute the atomic-event constellation known as “glass.” (In lenses the light caroms off atoms to have its course deliberately and angularly altered.)

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Webster’s definition of mathematics is “the science of dealing with quanitites, forms, etc., and their relationships by the use of numbers and symbols.”

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Webster defines science as “systematized knowledge derived from observation and study.”

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In respect to those definitions I was taught, between 1905 and 1913 at the private preparatory school then most highly regarded by Harvard, that “the properties of a point” are nonexistent—that a point is nondimensional or infradimensional, weightless, and timeless. The teacher had opened the day’s lesson by making a white chalk mark on the cleanly washed-off blackboard and saying, “This is a point.” I was next taught that a line is one dimensional and consists of a “straight” row of nondimensional points—and I am informed that today, in 1978, all schoolchildren around the world are as yet being so taught. Since such a line lacks three-dimensionality, it too is nonexistent to the second power or to “the square root of nonexistence.” We were told by our mathematics teacher that the plane is a raft of tangentially parallel rows of nonexistent lines—ergo, either a third power or a “cube root of nonexistence”—while the supposedly “real” cube of three dimensions is a rectilinear stack of those nonexistent planes and therefore must be either a fourth power or a fourth root of nonexistence. Since the cube lacked weight, temperature, or duration in time, and since its empty 12-edged frame of nonexistent lines would not hold its shape, it was preposterously nondemonstrable—ergo, a treacherous device for students and useful only in playing the game of deliberate self-deception. Since it was arbitrarily compounded of avowedly nonexistent points, the socially accepted three- dimensional reality of the academic system was not “derived from observation and study”— ergo, was to me utterly unscientific.

986.030 Abstraction

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The scientific generalized eternal principle of leverage can be experientially demonstrated, and its rate of lifting-advantage-gain per each additional modular increment of lifting-arm length can be mathematically expressed to cover any and all special case temporal realizations of the leverage principle. Biological species can be likewise generalizingly defined. So in many ways humanity has been able to sort out its experiences and identify various prominent sets and subsets of interrelationship principles. The special- case “oriole on the branch of that tree over there,” the set of all the orioles, the class of all birds, the class of all somethings, the class of all anythings—any one of which anythings is known as X … that life’s experiences lead to the common discovery of readily recognized, differentiated, and remembered generalizable sets of constantly manifest residual interrelationship principles—swiftly persuaded mathematical thinkers to adopt the symbolism of algebra, whose known and unknown components and their relationships could be identified by conveniently chosen empty-set symbols. The intellectuals call this abstraction.

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Abstraction led to the discovery of a generalized family of plus-and-minus interrelationship phenomena, and these generalized interrelationships came to be expressed as ratios and equations whose intermultiplicative, divisible, additive, or subtractive results could—or might—be experimentally (objectively) or experientially (subjectively) verified in substantive special case interquantation relationships.

986.040 Greek Geometry

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It was a very different matter, however, when in supposed scientific integrity mathematicians undertook to abstract the geometry of structural phenomena. They began their geometrical science by employing only three independent systems: one supposedly “straight”-edged ruler, one scribing tool, and one pair of adjustable-angle dividers.

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Realistically unaware that they were on a spherical planet, the Greek geometers were first preoccupied with only plane geometry. These Greek plane geometers failed to recognize and identify the equally important individual integrity of the system upon whose invisibly structured surface they were scribing. The Euclidean mathematicians had a geocentric fixation and were oblivious to any concept of our planet as an includable item in their tool inventory. They were also either ignorant of—or deliberately overlooked—the systematically associative minimal complex of inter-self-stabilizing forces (vectors) operative in structuring any system (let alone our planet) and of the corresponding cosmic forces (vectors) acting locally upon a structural system. These forces must be locally coped with to insure the local system’s structural integrity, which experientially demonstrable force-interaction requirements are accomplishable only by scientific intertriangulations of the force vectors. Their assumption that a square or a cube could hold its own structural shape proves their oblivousness to the force (vector) interpatternings of all structurally stable systems in Universe. To them, structures were made only of stone walls—and stone held its own shape.

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The Ionian Greeks seem to have been self-deceived into accepting as an absolute continuum the surface of what also seemed to them to be absolutely solid items of their experience—whether as randomly fractured, eroded, or ground-apart solids or as humanly carved or molded symmetrical shapes. The Ionian Greeks did not challenge the self-evident axiomatic solid integrity of their superficial-continuum, surface-face-area assumptions by such thoughts as those of the somewhat later, brilliantly intuitive, scientific speculation of Democritus, which held that matter might consist of a vast number of invisible minimum somethings—to which he gave the name “atoms.” All of the Euclidean geometry was based upon axioms rather than upon experimentally redemonstrable principles of physical behavior.

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Webster’s dictionary defines axiom (etymologically from the Greek “to think worthy”) as (1) a maxim widely accepted on its intrinsic merit, and (2) a proposition regarded as self-evident truth. The dictionary defines maxim as (1) a general truth, fundamental principle, or rule of conduct, and (2) a saying of a proverbial nature. Maxim and maximum possibly integratingly evolved as “the most important axiom.” Max + axiom = maxim. The assumption of commonly honored, customarily accredited axioms as the fundamental “building-blocks” of Greek geometry circumvented the ever- experimentally-redemonstrable qualifying requirement of all serious scientific considerations.

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The Ionian Greeks assumed as fundamental geometric components their line- surrounded areas. These areas’ surfaces could be rough, smooth, or polished—just as the smooth surface of the water of the sea could be roughened without losing its identity to them as “the surface.” Looking upon plane geometry as the progenitor of subsequently-to- be-developed solid geometry, it seemed never to have occurred to the Euclideans that the surface on which they scribed had shape integrity only as a consequence of its being a component of a complex polyhedral system, the system itself consisting of myriads of subvisible structural systems, whose a priori structural integrity complex held constant the shape of the geometrical figures they scribed upon—the polyhedral system, for instance, the system planet Earth upon whose ground they scratched their figures, or the stone block, or the piece of bark on which they drew. Even Democritus’s brilliant speculative thought of a minimum thing smaller than our subdimensional but point-to-able speck was speculative exploration a priori to any experimentally induced thinking of complex dynamic interactions of a plurality of forces that constituted structuring in its most primitive sense. Democritus did not think of the atom as a kinetic complex of structural shaping interactions of energy events operating at ultra-high-frequency in pure principle.

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Cubical forms of wood and stone with approximately flat faces and corner angles seemed to the Euclidean-led Ionians to correspond satisfactorily with what was apparently a flat plane world to which trees and humanly erected solid wooden posts and stone columns were obviously perpendicular—ergo, logically parallel to one another. From these only-axiomatically-based conclusions the Ionians developed their arbitrarily shaped, nonstructural, geometrical abstractions and their therefrom-assumed generalizations.

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The Greeks’ generalized geometry commenced with the planar relationships and developed therefrom a “solid” geometry by in effect standing their planes on edge on each of the four sides of a square base and capping this vertical assembly with a square plane. This structure was then subdivided by three interperpendicularly coordinate lines—X, Y, and Z—each with its corresponding sets of modularly interspaced and interparalleled planes. Each of these three sets of interparallel and interperpendicular planes was further subdivisible into modularly interspaced and interparallel lines. Their sets of interparallel and interperpendicular planar and linear modulations also inherently produced areal squares and volumetric cubes as the fundamental, seemingly simplest possible area-and-volume standards of uniform mensuration whose dimensioning increments were based exclusively on the uniform linear module of the coordinate system—whose comprehensive interrelationship values remained constant—ergo, were seemingly generalizable mathematically quite independently of any special case experiential selection of special case lengths to be identified with the linear modules.

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The Euclidean Greeks assumed not only that the millions of points and instant planes existed independently of one another, but that the complex was always the product of endlessly multipliable simplexes—to be furnished by an infinite resource of additional components. The persistence of the Greeks’ original misconceptioning of geometry has also so distorted the conditioning of the human brain-reflexing as to render it a complete 20th-century surprise that we have a finite Universe: a finite but nonunitarily- and-nonsimultaneously accomplished, eternally regenerative Scenario Universe. In respect to such a scenario Universe multiplication is always accomplished only by progressively complex, but always rational, subdivisioning of the initially simplest structural system of Universe: the sizeless, timeless, generalized tetrahedron. Universe, being finite, with energy being neither created nor lost but only being nonsimultaneously intertransformed, cannot itself be multiplied. Multiplication is cosmically accommodated only by further subdivisioning.

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If the Greeks had tried to do so, they would soon have discovered that they could not join tetrahedra face-to-face to fill allspace; whereas they could join cubes face- to-face to fill allspace. Like all humans they were innately intent upon finding the “Building-Block” of Universe. The cube seemed to the Greeks, the Mesopotamians, and the Egyptians to be just what they needed to account their experiences volumetrically. But if they had tried to do so, they would have found that unit-dimensioned tetrahedra could be joined corner-to-corner only within the most compact omnidirectional confine permitted by the corner-to-corner rule, which would have disclosed the constant interspace form of the octahedron, which complements the tetrahedron to fill allspace; had they done so, the Ionians would have anticipated the physicists’ 1922 discovery of “fundamental complementarity” as well as the 1956 Nobel-winning physics discovery that the complementarity does not consist of the mirror image of that which it complements. But the Greeks did not do so, and they tied up humanity’s accounting with the cube which now, two thousand years later, has humanity in a lethal bind of 99 percent scientific illiteracy.

986.050 Unfamiliarity with Tetrahedra

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The distorted conditioning of human reflexing and reasoning persisted in overwhelming the academic point of view—and still does so up to this moment in history. This is nowhere more apparent than in the official reaction to the data and photographs taken on planet Mars by the planet Earth’s scientists from their multistage-rocket- despatched Mariner 9 and Viking orbiters:

But even at the present limits of resolution, some surprising formations have been seen, the most inexplicable of which are the three-sided pyramids found on the plateau of Elysium. Scientists have tried to find a natural geological process that would account for the formation of these pyramids, some of which are two miles across at the base, but as yet their origin is far from being explained. Such tantalizing mysteries may not be fully solved until astronauts are able to make direct observations on the Martian surface.¹

(Footnote 1: David L. Chandler, “Life on Mars,” Atlantic, June 1977.)

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Fig. 986.052 Robot Camera Photograph of Tetrahedra on Mars

Fig. 986.052 Robot Camera Photograph of Tetrahedra on Mars: On their correct but awkward description of these gigantic polyhedra as “three-sided pyramids” the NASA scientists revealed their unfamiliarity with tetrahedra

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In 1977 the NASA scientists scrutinized the robot-photographed pictures of the close-in Martian scene and reported the—to them—surprise presence on Mars of two (two-mile-base-edged) three-sided pyramids the size of Mount Fuji. The NASA scientists were unfamiliar with the tetrahedron. They remarked that these forms, with whose simplest, primitive character they were unacquainted, must have been produced by wind- blown sand erosion, whereas we have discovered that tetrahedra are always and only a priori to nature’s processes of alteration of her simplest and most primitive polyhedral systems.

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Also suggestive of the same blindness to nature’s reality suffered by the academic world and the scientists who lead it, was van’t Hoff’s late 19th-century identification of the primitive significance of the tetrahedron in the structuring of organic chemistry. (See Sec. 931.60.) His hypothesis was at first scoffed at by scientists. Fortunately, through the use of optical instruments he was able to present visual proof of the tetrahedral configuration of carbon bonds-which experimentally reproduced evidence won him the first Nobel prize awarded a chemist. The Greeks of three millennia ago and today’s “educated” society are prone to assume that nature is primitively disorderly and that symmetrical shapes are accomplished only by human contriving.

986.060 Characteristics of Tetrahedra

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Fig. 986.061 Truncation of Tetrahedra

Fig. 986.061 Truncation of Tetrahedra: Only vertexes and edges may be truncated. (Compare Figs. 987.241 and 1041.11.)

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The tetrahedron is at once both the simplest system and the simplest structural system in Universe (see Secs. 402 and 620). All systems have a minimum set of topological characteristics of vertexes, faces, and edges (see Secs. 1007.22 and 1041.10). Alteration of the minimum structural system, the tetrahedron, or any of its structural- system companions in the primitive hierarchy (Sec. 982.61), may be accomplished by either external or internal contact with other systems—which other systems may cleave, smash, break, or erode the simplest primitive systems. Other such polyhedral systems may be transformingly developed by wind-driven sandstorms or wave-driven pebble beach actions. Those other contacting systems can alter the simplest primitive systems in only two topological-system ways:
1. by truncating a vertex or a plurality of vertexes, and
2. by truncating an edge or a plurality of edges.
Faces cannot be truncated. (See Fig. 986.061.)

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Fig. 986.062 Truncated Tetrahedron within Five-frequency Tetra Grid

Fig. 986.062 Truncated Tetrahedron within Five-frequency Tetra Grid: Truncating the vertexes of the tetrahedron results in a polyhedron with four triangular faces and four hexagonal faces. (Compare Figs. 1041.11 and 1074.13.)

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As we have learned regarding the “Platonic solids” carvable from cheese (Sec. 623.10), slicing a polyhedron parallel to one of its faces only replaces the original face with a new face parallel to the replaced face. Whereas truncating a vertex or an edge eliminates those vertexes and edges and replaces them with faces—which become additional faces effecting a different topological abundance inventory of the numbers of vertexes and edges as well. For every edge eliminated by truncation we gain two new edges and one new face. For every corner vertex eliminated by truncation our truncated polyhedron gains three new vertexes, three new edges, and one new face.

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The cheese tetrahedron (Sec. 623.13) is the only one of the primitive hierarchy of symmetrical polyhedral systems that, when sliced parallel to only one of its four faces, maintains its symmetrical integrity. It also maintains both its primitive topological and structural component inventories when asymmetrically sliced off parallel to only one of its four disparately oriented faces. When the tetrahedron has one of its vertexes truncated or one of its edges truncated, however, then it loses its overall system symmetry as well as both its topological and structural identification as the structurally and topologically simplest of cosmic systems.

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We may now make a generalized statement that the simplest system in Universe, the tetrahedron, can be design-altered and lose its symmetry only by truncation of one or more of its corners or edges. If all the tetrahedron’s four vertexes and six edges were to be similarly truncated (as in Fig. 1041.11) there would result a symmetrical polyhedron consisting of the original four faces with an addition of 10 more, producing a 14-faceted symmetrical polyhedron known as the tetrakaidecahedron, or Kelvin’s “solid,” which (as shown in Sec. 950.12 and Table 954.10) is an allspace filler—as are also the cube, the rhombic dodecahedron, and the tetrahedral Mites, Sytes, and Couplers. All that further external alteration can do is produce more vertex and edge truncations which make the individual system consist of a greater number of smaller-dimension topological aspects of the system. With enough truncations—or knocking off of corners or edges—the system tends to become less angular and smoother (smoother in that its facets are multiplying in number and becoming progressively smaller and thus approaching subvisible identification). Further erosion can only “polish off” more of the only-microscopically- visible edges and vertexes. A polished beach pebble, like a shiny glass marble or like a high-frequency geodesic polyhedral “spheric” structure, is just an enormously high- frequency topological inventory-event system.

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Joints, Windows, and Struts: As we have partially noted elsewhere (Secs. 536 and 604), Euler’s three primitive topological characteristics—texes, faces, and *lines—*are structurally identifiable as joints, windows, and push-pull struts, respectively. When you cannot see through the windows (faces), it is because the window consists of vast numbers of subvisible windows, each subvisible-magnitude window being strut- mullion-framed by a complex of substructural systems, each with its own primitive topological and structural components.

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Further clarifying those structural and topological primitive componentation characteristics, we identify the structural congruences of two or more joined-together- systems’ components as two congruent single vertexes (or joints) producing one single, univalent, universal-joint intersystem bonding. (See Secs. 704, 931.20, and Fig. 640.41B.) Between two congruent pairs of interconnected vertexes (or joints) there apparently runs only one apparent (because congruent) line, or interrelationship, or push-pull strut, or hinge.

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Returning to our early-Greek geometry initiative and to the as-yet-persistent academic misconditioning by the Greeks’ oversights and misinterpretations of their visual experiences, we recall how another non-Ionian Greek, Pythagoras, demonstrated and “proved” that the number of square areas of the unit-module-edged squares and the number of cubical module volumes of the unit-module-edged cubes correspond exactly with arithmetic’s second-powerings and third-powerings. The Greeks, and all mathematicians and all scientists, have ever since misassumed these square and cube results to be the only possible products of such successive intermultiplying of geometry’s unit-edge-length modular components. One of my early mathematical discoveries was the fact that all triangles—regular, isosceles, or scalene—may be modularly subdivided to express second-powering. Any triangle whose three edges are each evenly divided into the same number of intervals, and whose edge-interval marks are cross-connected with lines that are inherently parallel to the triangle’s respective three outer edges—any triangle so treated will be subdivided by little triangles all exactly similar to the big triangle thus subdivided, and the number of small similar triangles subdividing the large master triangle will always be the second power of the number of edge modules of the big triangle. In other words, we can say “triangling” instead of “squaring,” and since all squares are subdivisible into two triangles, and since each of those triangles can demonstrate areal second-powering, and since nature is always most economical, and since nature requires structural integrity of her forms of reference, she must be using “triangling” instead of “squaring” when any integer is multiplied by itself. (See Sec. 990.)

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This seemed to be doubly confirmed when I discovered that any nonequiedged quadrangle, with each of its four edges uniformly subdivided into the same number of intervals and with those interval marks interconnected, produced a pattern of dissimilar quadrangles. (See Fig. 990.01.) In the same manner I soon discovered experimentally that all tetrahedra, octahedra, cubes, and rhombic dodecahedra—regular or skew—could be unitarily subdivided into tetrahedra with the cube consisting of three tetra, the octahedron of four tetra, and the rhombic dodecahedron of six similar tetra; and that when any of these regular or skew polyhedras’ similar or dissimilar edges and faces were uniformly subdivided and interconnected, their volumes would always be uniformly subdivided into regular or skew tetrahedra, and that N³ could and should be written and spoken of as N${}^{\text{tetrahedroned}}$ and not as N${}^{\text{cubed}}$.

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Nature would use the tetrahedron as the module of subdivision because nature has proven to the physicists and the other physical scientists that she always chooses the most economic realization. Cubes require three times as much Universe as do tetrahedra to demonstrate volumetric content of systems because cubic identification with third-powering used up three times as much volume as is available in Universe. As a result of cubic mensuration science has had to invent such devices as “probability” and “imaginary numbers.” Thus “squaring” and “cubing,” instead of nature’s “triangling” and “tetrahedroning,” account for science’s using mathematical tools that have no physical- model demonstrability—ergo, are inherently “unscientific.”

986.070 Buildings on Earth’s Surface

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In the practical fortress and temple building of the earliest known Mesopotamians, Egyptians, and Greeks their cubes and omnirectilinear blocks seemed readily to fill allspace as they were assembled into fortress or temple walls with plumb bobs, water-and-bubble levels, straightedges, and right-triangle tools. No other form they knew—other than the cube—seemed to fill allspace as demonstrated in practical masonry; wherefore they assumed this to be scientifically demonstrated proof of the generalizability of their mathematically abstracted plane- and solid-geometry system and its XYZ coordination.

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Because of the relatively diminutive size of humans in respect to the size of our planet, world-around society as yet spontaneously cerebrates only in terms of our immediate world’s seeming to demonstrate itself to be a flat plane base, all of the perpendiculars of which—such as trees and humans and human-built local structures- appear to be rising from the Earth parallel to one another—ergo, their ends point in only two possible directions, “up” or “down.” … It’s “a wide, wide world,” and “the four corners of the Earth.”

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It was easy and probably unavoidable for humanity to make the self- deceptive blunders of assuming that a cube held its shape naturally, and not because the stone-cutters or wood-cutters had chosen quite arbitrarily to make it in this relatively simple form. Human’s thought readily accepted—and as yet does—the contradictory abstract state “solid.” The human eye gave no hint of the energetic structuring of the atomic microcosm nor of the omnidynamic, celestial-interpositioning transformations of both macro- and micro-Universe.

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Prior to steel-framed or steel-reinforced-concrete construction methods, humans’ buildings that were constructed only of masonry could not be safely built to a height of over 20 stories—approximately 200 feet high. Such a masonry building was Chicago’s turn-of-the-20th-century world-record Monadnock Building, whose base covered a small but whole city block. It is not until we reach a height of 100 stories—approximately 1000 feet high—that two exactly vertical square columns, each with base edges of 250 feet, built with exactly vertical walls, and touching one another only along one of each of their base edges, will show a one-inch space between them. The rate their vertical walls part from one another is only 1/1000th of an inch for each foot of height.

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Masons’ and carpenters’ linear measuring devices are usually graduated only to 1/16th of an inch, and never finer than 1/32nd of an inch. Thus differentials of a thousandth of an inch are undetectable and are altogether inadvertently overlooked; ergo, they get inadvertently filled-in, or cross-joined, never to have been known to exist even on the part of the most skilled and conscientious of building craftsmen, whose human eyes cannot see intervals of less than 1/100th of an inch.

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Fig. 986.076 Diagram of Verrazano Bridge

Fig. 986.076 Diagram of Verrazano Bridge: The two towers are not parallel to each other.

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If two exactly-vertical-walled city skyscrapers are built side by side, not until they are two and one-half miles high (the height of Mount Fuji) will there be a space of one foot between the tops of their two adjacent walls. (See Fig. 986.076.) Of course, the farther apart the centers of their adjacent bases, the more rapidly will the tops of such high towers veer away from one another:
The twin towers of New York’s Verrazano Bridge are 693 feet high … soaring as high as a 70-story skyscraper … set almost a mile from each other, the two towers, though seemingly parallel, are an inch and five-eighths farther apart at their summits than at their bases because of the Earth’s curvature.²

(Footnote 2: The Engineer (New York: Time-Life Books, 1967.) If the towers are 12,000 miles apart-that is, halfway around the world from one another-their tops will be built in exactly opposite directions ergo, at a rate of two feet farther apart for each foot of their respective heights.)

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It is easy to understand how humans happened to think it “illogical” to have to consider that all the perpendiculars to a sphere are radii of that sphere—ergo, never parallel to one another. Our humans-in-Universe scale is inherently self-deceptive—ergo, difficult to cope with rigorously.

986.080 Naive Perception of Childhood

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The inventory of experimentally demonstrated discoveries of science which had accrued by the time of my childhood gave me reason to question many of the “abstractions” of geometry as I was being instructed in that subject. Axioms were based on what only seemed “self-evident,” such as the stone block or the “cubical” wooden play blocks of my nursery. To society they “obviously held their shape.” I do not think that I was precocious or in any way a unique genius. I had one brother; he was three years younger than I. His eyesight was excellent; mine was atrocious. I did not get my first eyeglasses until my younger brother was running around and talking volubly. He could see things clearly; I could not. Our older sister could also see things clearly. I literally had to feel my way along—tactilely—in order to recognize the “things” of my encountered environment-ergo, my deductions were slow in materializing. My father called my younger brother “stickly-prickly” and he called me “slow-and-solid”-terms he adopted from “The Jaguar and the Armadillo” in Kipling’s Just So Stories.

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I was born cross-eyed on 12 July 1895. Not until I was four-and-a-half years old was it discovered that I was also abnormally farsighted. My vision was thereafter fully corrected with lenses. Until four-and-a-half I could see only large patterns—houses, trees, outlines of people—with blurred coloring. While I saw two dark areas on human faces, I did not see a human eye or a teardrop or a human hair until I was four. Despite my newly gained ability—in 1899—to apprehend details with glasses, my childhood’s spontaneous dependence upon only big-pattern clues has persisted. All that I have to do today to reexperience what I saw when I was a child is to take off my glasses, which, with some added magnification for age, have exactly the same lens corrections as those of my first five-year-old pair of spectacles. This helps me to recall vividly my earliest sensations, impressions, and tactical assumptions.

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I was sent to kindergarten before I received my first eyeglasses. The teacher, Miss Parker, had a large supply of wooden toothpicks and semidried peas into which you could easily stick the sharp ends of the toothpicks. The peas served as joints between the toothpicks. She told our kindergarten class to make structures. Because all of the other children had good eyesight, their vision and imagination had been interconditioned to make the children think immediately of copying the rectilinearly framed structures of the houses they saw built or building along the road. To the other children, horizontally or perpendicularly parallel rectilinear forms were structure. So they used their toothpicks and peas to make cubic and other rectilinear models. The semidried peas were strong enough to hold the angles between the stuck-in toothpicks and therefore to make the rectilinear forms hold their shapes—despite the fact that a rectangle has no inherent self-structuring capability.

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In my poor-sighted, feeling-my-way-along manner I found that the triangle—I did not know its name-was the only polygon—I did not know that word either-that would hold its shape strongly and rigidly. So I naturally made structural systems having interiors and exteriors that consisted entirely of triangles. Feeling my way along I made a continuous assembly of octahedra and tetrahedra, a structured complex to which I was much later to give the contracted name “octet truss.” (See Sec. 410.06). The teacher was startled and called the other teachers to look at my strange contriving. I did not see Miss Parker again after leaving kindergarten, but three-quarters of a century later, just before she died, she sent word to me by one of her granddaughters that she as yet remembered this event quite vividly.

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Three-quarters of a century later, in 1977, the National Aeronautics and Space Administration (NASA), which eight years earlier had put the first humans on the Moon and returned them safely to our planet Earth, put out bids for a major space-island platform, a controlled-environment structure. NASA’s structural specifications called for an “octet truss” —my invented and patented structural name had become common language, although sometimes engineers refer to it as “space framing.” NASA’s scientific search for the structure that had to provide the most structural advantages with the least pounds of material—ergo, least energy and seconds of invested time-in order to be compatible and light enough to be economically rocket-lifted and self-erected in space—had resolved itself into selection of my 1899 octet truss. (See Sec. 422.)

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It was probable also that my only-insectlike, always-slow, cross-referencing strategy of touching, tasting, smelling, listening, and structurally testing by twisting and pounding and so forth—to which I spontaneously resorted—made me think a great deal about the fact that- when I broke a piece of glass or a stone or a wooden cube apart, it did not separate naturally into little cubes but usually into sharp pointed shapes. In the earliest of my memories I was always suspicious of the integrity of cubes, which only humans seemed to be introducing into the world. There were no cubical roses, eggs, trees, clouds, fruits, nuts, stones, or anything else. Cubes to me were unnatural: I observed humans deliberately sawing ice into large rectilinear cakes, but window glass always broke itself into predominantly triangular pieces; and snowflakes formed themselves naturally into a myriad of differently detailed, six-triangled, hexagonal patterns.

986.087

I was reacting normally in combining those spontaneous feelings of my childhood with the newly discovered knowledge of the time: that light has speed (it is not instantaneous, and comes in smallest packages called photons); that there is something invisible called electricity (consisting of “invisible behaviors” called electrons, which do real work); and that communication can be wireless, which Marconi had discovered the year I was born—and it is evident that I was reacting normally and was logically unable to accept the customarily honored axioms that were no longer “self-evident.”

986.088

My contemporaries and I were taught that in order to design a complete and exact sphere and have no materials left over, we must employ the constant known as pi (pi), which I was also taught was a “transcendentally irrational number,” meaning it could never be resolved. I was also informed that a singly existent bubble was a sphere; and I asked, To how many places does nature carry out pi when she makes each successive bubble in the white-cresting surf of each successive wave before nature finds out that pi can never be resolved? … And at what moment in the making of each separate bubble in Universe does nature decide to terminate her eternally frustrated calculating and instead turn out a fake sphere? I answered myself that I don’t think nature is using pi or any of the other irrational fraction constants of physics. Chemistry demonstrates that nature always associates or disassociates in whole rational increments… Those broken window shards not only tended to be triangular in shape, but also tended to sprinkle some very fine polyhedral pieces. There were wide ranges of sizes of pieces, but there were no pieces that could not “make up their minds” or resolve which share of the original whole was theirs. Quite the contrary, they exploded simultaneously and unequivocally apart.

986.089

At first vaguely, then ever more excitedly, precisely, and inclusively, I began to think and dream about the optimum grand strategy to be employed in discovering nature’s own obviously elegant and exquisitely exact mathematical coordinate system for conducting the energetic transactions of eternally regenerative Universe. How does nature formulate and mass-produce all the botanical and zoological phenomena and all the crystals with such elegant ease and expedition?

986.090 The Search for Nature’s Coordinate System

986.091

Several things were certain: nature is capable of both omnidirectional disorderly, dispersive, and destructive expansion and omnidirectional collective, selective sorting and constructive contraction; and rays of candlelight are not parallel to one another. I decided to initiate my search for nature’s coordinate system by assuming that the coordinate system must be convergently and divergently interaccommodative. That the seasons of my New England childhood brought forth spectacular transformations in nature’s total interpatterning; that the transformations were not simultaneous nor everywhere the same; that there were shaded and Sun-shined-upon area variables; and that they were all embraced by a comprehensive coordination—altogether made me dream of comprehending the comprehensively accommodating coordinate system that had no separate departments of chemistry, physics, biology, mathematics, art, history, or languages. I said nature has only one department and only one language.

986.092

These thoughts kept stimulating my explorations for the totally accommodative coordinate system. Einstein’s conclusion-that the definitive, maximum possible speed of light rendered astronomical phenomena an aggregate of nonsimultaneous and only partially overlapping, differently enduring energy events—greatly affected the increasing inventory of my tentative formulations of the interaccommodative requirements of the cosmic coordination system which I sought. I was driven by both consciously and subconsciously sustained intuition and excitement. This was very private, however. I talked to no one about it. It was all very remote from that which seemed to characterize popular interest.

986.093

The youthful accruals of these long-sustained private observations, cogitations, and speculations were enormously helpful when I decided at the age of 32, in my crisis year of 1927, to abandon the game of competitive survival (a game I had been taught to believe in as thought-out, managed, and evolved entirely by others) and instead to rely completely upon my own thinking and experience-suggested inclinations … to find out how Universe is organized and what it is doing unbeknownst to humans. Why are humans here in Universe? What should we be doing to fulfill our designed functioning in Universe? Surely all those stars and galaxies were not designed only to be romantic scenery for human moods. What am I designed to be able to comprehend about Universe? What are we humans designed to be able to do for one another and for our Universe?

986.094

Expanding Universe: My determination to commit myself completely to the search for nature’s raison d’ˆtre and for its comprehensive coordinate system’s mathematics was greatly reinforced by the major discovery of the astronomer E. P. Hubble in 1924. He discovered an expanding macrocosmic system with all the myriads of galaxies and their respective stellar components at all times maintaining the same interangular orientations and relative interdistancings from one another while sum-totally and omnisymmetrically expanding and moving individually away from one another, and doing so at astronomical speeds. This discovery of Hubble’s became known as Expanding Universe.

986.095

The only way humans can expand their houses is by constructing lopsided additions to their rectilinearly calculated contriving. People found that they could “blow up” rubber-balloon spheres to increase their radii, but they couldn’t blow up their buildings except by dynamite. They called their wooden “2 × 4,” and “2 × 6,” and “2 × 8- inch” cross-section, wooden-timber nail-ups “balloon framing,” but why they selected that name was difficult to explain.

986.096

Fig. 986.096 4-D Symbol

Fig. 986.096 4-D Symbol: Adopted by the author in 1928 to characterize his fourth-dimensional mathematical explorings.

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My insights regarding nature’s coordinate system were greatly enhanced by two of Milton Academy’s greatest teachers: Homer LeSourd in physics and William Lusk Webster Field (“Biology Bill”) in biology. During the summer vacation of 1906, at 11 years of age I designed and built my first small but exciting experimental dwelling on our family’s small mid-Penobscot Bay island. Living all my youthful summers on that island, with its essential boat-building, boat-modifying, boat-upkeep, and boat-sailing, followed by five years as a line officer in the regular U.S. Navy with some of my own smaller-craft commands, some deck-officering on large craft of the new era’s advanced technology ships, together with service involving airplanes, submarines, celestial navigation, ballistics, radio, and radiotelephone; then resignation from the Navy followed by five more private- enterprise years developing a new building system, inventing and installing its production tools, managing the production of the materials, and erecting therewith 240 residences and small commercial buildings—altogether finally transformed my sustained activity into full preoccupation with my early-boyhood determination some day to comprehend and codify nature’s omniintertransformative, omnidirectional, cosmic coordination system and its holistic, only-experientially-proven mathematics. In 1928, inspired and fortified by Hubble’s Expanding Universe discovery, I gave the name and its symbol 4-D to my mathematical preoccupations and their progressively discovered system codifying. In 1936 I renamed my discipline “Energetic Vectorial Geometry.” In 1938 I again renamed it “Energetic-synergetic Geometry,” and in 1970 for verbal economy contracted that title to “Synergetics.” (See Fig. 986.096.)

986.100 Sequence of Considerations

986.101

At the outset of my lifelong search for nature’s omnirational coordination system of mathematical interaccounting and intertransformability I proceeded through a sequence of considerations which may be enumerated as follows:

986.110 Consideration 1: Energetic Vectors

986.111

I first determined to employ only vectors for lines. I realized that operationally all lines are always trajectories of energy events, either as the energy invested by humans in the work of carving or depositing linearly—which we call “drawing” a line—or as the inanimately articulated energy of force lines.

986.112

Vectors always represent energy forces of given magnitude operating at given angles upon given entities at given loci, and vectors may always be demonstrated by lines representing given mass moving at given velocity in unique angular direction in respect to a given axis of observation. Vectors do not occur singly: They occur only as the total family of forces interacting in any given physical circumstance.

986.113

Vectors always have unique length, that length being the product of the mass and the velocity as expressed in a given modular system of measurement. Vectors do not have inherent extendibility to infinity—as did the lines of my boyhood’s schoolteachers. Vectors are inherently terminal. Vectors bring into geometry all the qualities of energetic reality lacking in Euclidean geometry: time and energy-energy differentially divergent as radiation (velocity) and energy integratively convergent as matter (mass). Velocity and mass could be resolved into numerically described time and temperature components.

986.120 Consideration 2: Avogadro’s Constant Energy Accounting

986.121

Avogadro discovered that under identical conditions of pressure and heat all elements in their gaseous state always consist of the same number of molecules per given volume. Since the chemical elements are fundamentally different in electron-proton componentation, this concept seemed to me to be the “Grand Central Station” of nature’s numerical coordinate system’s geometric volume-that numerically exact volumes contain constant, exact numbers of fundamental energy entities. This was the numerical and geometrical constancy for which I was looking. I determined to generalize Avogadro’s experimentally proven hypothesis that “under identical conditions of heat and pressure all gases disclose the same number of molecules per given volume.” (See Secs. 410.03-04.)

986.122

Here were Physical Universe’s natural number quantations being constantly related to given volumes. Volumes are geometrical entities. Geometrically defined and calculated volumes are polyhedral systems. Polyhedra are defined by edge lines, each of which must be a vector.

986.123

Within any given volumetrically contained gaseous state the energy kinetics of molecules are everywhere the same. The outward pressure of air against the enclosing tube wall and casing of any one automobile tire is everywhere the same. Pressure and heat differentials involve isolated conditions—isolated by containers; ergo, special cases. To me this meant that we could further generalize Avogadro by saying that “under identical, uncontained, freely self-interarranging conditions of energy all chemical elements will disclose the same number of fundamental somethings per given volume.” This constant- volume-population-and-omniequilibrious-energy relationship would require physically demonstrable, substantive, geometrical combining of a given number of unique energetic- event entities per unit volume with constant-angularly-defined positional orientation integrities. This meant that the vectorially structured shapes of the volumes accommodating given numbers of most primitive energy events must be experientially demonstrable.

986.130 Consideration 3: Angular Constancy

986.131

I said that since vectors are physically modelable structural components, they produce conceptual structural models of energy events, and since my hypothetical generalization of Avogadro’s law requires that “all the conditions of energy be everywhere the same,” what does this condition look like as structured in vectorial geometry? Obviously all the vectors must be the same length and all of them must interact at the same angles. I said: It will make no difference what length is employed so long as they are all the same length. Linear size is special case. Special case occurs only in time. Angles are cosmically constant independently of time-size considerations.

986.140 Consideration 4: Isotropic Vector Model

986.141

I said, Can you make a vector model of this generalization of Avogadro? And I found that I had already done so in that kindergarten event in 1899 when I was almost inoperative visually and was exploring tactilely for a structural form that would hold its shape. This I could clearly feel was the triangle—with which I could make systems having insides and outsides. This was when I first made the octet truss out of toothpicks and semidried peas, which interstructuring pattern scientists decades later called the “isotropic vector matrix,” meaning that the vectorial lengths and interanglings are everywhere the same. (See Sec. 410.06.)

986.142

This matrix was vectorially modelable since its lines, being vectors, did not lead to infinity. This isotropic vector matrix consists of six-edged tetrahedra plus 12-edged octahedra—multiples of six. Here is an uncontained omniequilibrious condition that not only could be, but spontaneously would be, reverted to anywhen and anywhere as a six- dimensional frame of transformative-evolution reference, and its vector lengths could be discretely tuned by uniform modular subdivisioning to accommodate any desired special case wavelength time-size, most economically interrelated, transmission or reception of physically describable information. (Compare Secs. 639.02 and 1075.10.)

986.143

Since the vectors are all identical in length, their intersection vertexes become the nuclear centers of unit-radius spheres in closest-packed aggregation—which closest packing is manifest by atoms in their crystal growth. All the foregoing brought the adoption of my vectorial geometry’s everywhere-the-same (isotropic) vector matrix as the unified field capable of accommodating all of Physical Universe’s intertransformative requirements.

986.150 Consideration 5: Closest Packing of Spheres

986.151

I had thus identified the isotropic vector matrix with the uniform linear distances between the centers of unit radius spheres, which aggregates became known later—in 1922—as “closest-packed” unit-radius spheres (Sec. 410.07 ), a condition within which we always have the same optimum number of the same “somethings”—spheres or maybe atoms—per given volume, and an optimally most stable and efficient aggregating arrangement known for past centuries by stackers of unit-radius coconuts or cannonballs and used by nature for all time in the closest packing of unit-radius atoms in crystals.

986.160 Consideration 6: Diametric Unity

986.161

Fig. 986.161 Diametric Unity

Fig. 986.161 Diametric Unity: The vectors of the isotropic vector matrix interconnect the spheric centers of any two tangentially adjacent spheres. The radii of the two spheres meet at the kissing point and are each one-half of the system vector. Unity is plural and at minimum two.

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The installation of the closest-packed unit-radius spheres into their geometrical congruence with the isotropic vector matrix showed that each of the vectors always reaches between the spheric centers of any two tangentially adjacent spheres. This meant that the radius of each of the kissing spheres consists of one-half of the interconnecting vectors. Wherefore, the radius of our closest-packed spheres being half of the system vector, it became obvious that if we wished to consider the radius of the unit sphere as unity, we must assume that the value of the vector inherently interconnecting two unit spheres is two. Unity is plural and at minimum two. Diameter means dia- meter — unit of system measurement is two.

986.162

Fig. 986.161 shows one vector D whose primitive value is two. Vectors are energy relationships. The phenomenon relationship exists at minimum between two entities, and the word unity means union, which is inherently at minimum two. “Unity is plural and at minimum two” also at the outset became a prime concept of synergetics vectorial geometry. (See Sec. 540.10.)

986.163

1 R + 1 R = 2 R

2 R = Diameter

Diameter is the relative-conceptual-size determinant of a system. A diameter is the prime characteristic of the symmetrical system. The separate single system = unity. Diameter describes unity. Unity = 2. (See Secs. 905.10 and 1013.10.)

986.164

One by itself is nonexistent. Existence begins with awareness. Awareness begins with observable otherness. (See Secs. 264 and 981.)

986.165

Understanding means comprehending the interrelationship of the observer and the observed. Definitive understanding of interrelationships is expressed by ratios.

986.166

At the outset of my explorations I made the working assumption that unity is two, as combined with the experimentally demonstrable fact that every system and every systemic special case sphere is at once both a concave and a convex sphere—ergo, always inherently two spheres. Reflective concave surfaces convergently concentrate all impinging radiation, and reflective convex surfaces divergently diffuse all impinging radiation. Though concave and convex are inherently congruent as they are always-and- only coexisting, they are also diametrically opposed physical behavior phenomena—ergo, absolutely different because the one diffuses the energies of Universe, producing macrocosmic dispersion, and the other concentrates the energies of Universe, producing microcosmic convergence. Concave and convex are explicitly two opposites cosituate (congruent) geometrically as one. This led me to the working assumption at the outset of my—thus far—60-year exploration for nature’s own coordinate system, that unity is inherently plural and at minimum is to be dealt with as the value two, which twoness might well coexist with other numbers of inherent properties of primary-existence systems.

986.170 Consideration 7: Vector Equilibrium

986.171

I then identified this closest-packed-spheres isotropic vector matrix as a generalized field condition of the everywhere-and-everywhen most economically interaccommodating of any plurality of nuclearly convergent-divergent, importively organizing, and exportingly info-dispensing energy events—while also providing for any number of individually discrete, overlappingly co-occurrent, frequency differentiated info- interexchangings—ergo, to be always accommodative of any number of co-occurrent, individual-pattern-integrity evolutionary scenarios.

986.172

Thus the eternally regenerative Universe, embracing the minimum complex of intercomplementary transformations necessary to effect total regeneration, becomes comprehensively accommodated by the only generalizably definable Scenario Universe as the condition of the vector equilibrium, an everywhere-everywhen condition at which nature refuses to pause, but through which most economically accommodating field of operational reference she pulsates her complex myriads of overlapping, concurrent, local intertransformings and aberrative structurings. I then invented the symbol to identify vector equilibrium.

986.180 Consideration 8: Concentric Polyhedral Hierarchy

986.181

Thereafter I set about sorting out the relative numbers and volumes of the most primitive hierarchy of symmetrically structured polyhedral-event “somethings”—all of which are always concentrically congruent and each and all of which are to be discovered as vertexially defined and structurally coexistent within the pre-time-size, pre- frequency-modulated isotropic vector matrix. (See Sec. and Fig. 982.61.)

986.190 Consideration 9: Synergetics

986.191

This book Synergetics (volumes 1 and 2) embraces the record of the lifetime search, research, sorting-outs, and structural-intertransforming experiments based upon the foregoing eight considerations, all of which I had adopted by 1927. This 1927 inventory has been progressively amplified by subsequent experience-induced considerations.

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986.200 Narrative Exposition of Spherical Accommodation

986.201 Consideration 10: The Spheric Experience: Energetic-reality Accounting vs Abstract-cubic Accounting

986.202

In Synergetics 1, Secs. 962 through 966, I developed the first-, second-, and third-power values of my numerical factors for converting the XYZ coordinate system’s edge lengths, square areas, and cubical volumes to my 1927-discovered synergetic system’s unit VE vectorial edge lengths, triangular areas, and tetrahedral volumes.³ (See Table 963.10.)

(Footnote 3: My chart of these conversion factors, which I at first called the Dymaxion constants, was privately published in 1950 at North Carolina State University, and again in 1959 in The Dymaxion World of Buckminster Fuller, written with Robert W. Marks.)

986.203

The synergetics coordinate system-in contradistinction to the XYZ coordinate system-is linearly referenced to the unit-vector-length edges of the regular tetrahedron, each of whose six unit vector edges occur in the isotropic vector matrix as the diagonals of the cube’s six faces. We also recall that the eight corners of the cube are defined and structured omnitriangularly by the symmetrically interarrayed and concentric pairs of positive and negative tetrahedra (Figs. 110A and 110B).

986.204

Since the cube-face diagonal is the edge of the six-vector-edged, four- planes-of-symmetry tetrahedron, and since synergetics finds the unit-vector-edged tetrahedron to be the simplest structural system in Universe, the tetrahedron’s vector edge logically becomes the most economically primitive simplex module of relative length in synergetics’ coordinate system of exploratory reference. Thus the tetrahedron’s unit vector edge of unity 2 is manifest as nature’s coordinate primitive-length module for assessing:
second-power triangular area, 2² = 4
as well as for assessing that vector’s
third-power tetrahedral volume, 2³ = 8,
These areas and volumes become the logical unit of areal and volumetric reference in accounting the relative geometrical area and volume values of the entire hierarchy of primitive, concentrically congruent, symmetrical polyhedra as these naturally occur around any vertex of the isotropic vector matrix, and that matrix’s experimentally demonstrable, maximum-limit set of seven axes of polyhedral symmetries, which seven symmetries (Sec. 1040) accommodate and characterize the energetic special case formulations of all great- circle gridding.

986.205

The synergetics hierarchy of topological characteristics as presented in Table 223.64 of Synergetics 1 (which was contracted for with Macmillan in 1961 and published by them in 1975), discloses the rational values of the comprehensive coordinate system of nature, which my 60-year exploration discovered. In 1944 I published a paper disclosing this rational system. At that time I was counseled by some of my scientist friends, who were aware that I was continuing to make additional refinements and discoveries, that premature publication of a treatise of disclosure might result in the omission of one or more items of critical information which might be later discovered and which might make the difference between scientific acceptance or rejection of the disclosures. Reminded by those scientist-artist friends that we have only one opportunity in a lifetime out of many lifetimes to publish a prime-science-reorienting discovery, I postponed publishing a comprehensive treatise until in 1970, at the age of 75, I felt it could no longer be delayed.

986.206

The eleventh-hour publishing of Synergetics 1 coincided with my busiest years of serving other obligations over a period calling for a vast number of tactical decisions regarding the methodology of producing what proved to be a 780-page book. Typical of the problems to be swiftly resolved are those shortly to be herewith recounted. The accounting also discloses the always surprisingly productive events that ensue upon mistake-making that are not only discovered and acknowledged, but are reexplored in search of the significance of the mistakes’ having occurred.

986.207

Because the XYZ-coordinate, three-dimensional system values are arrived at by successive multiplying of the dimensions, volume in that system is an inherently three- dimensional phenomenon. But in synergetics the primitive values start holistically with timeless-sizeless tetrahedral volume unity in respect to which the cube’s primitive value is 3, the octahedron’s relative timeless-sizeless value is 4, the rhombic triacontrahedron’s is 5, and the rhombic dodecahedron’s is 6. In synergetics, when time-size special-case realizations enter into the consideration, then the (only-interrelated-to-one-another) primitive volumes of the synergetic hierarchy are multiplied by frequency of the edge modulation to the third power. Since innate primitive volume is a base-times-altitude three-dimensional phenomenon, and since all the synergetics hierarchy’s time-size realization volumes are inherently six-dimensional, I was confronted with an exploratory tactical quandary.

986.208

The problem was to arrive at the numerical volume value for the sphere in the synergetics hierarchy, and the dilemma was whether I should apply my synergetics’ volumetric constant to the first power or to the third power of the XYZ-coordinate system’s volumetric values as arrived at by the conventional XYZ-coordinate system’s method of calculating the volume of a sphere of radius vector = 1. This operation is recorded in Sec. 982.55 of Synergetics 1, where I misconceptualized the operation, and (without reviewing how I had calculated the constant for converting XYZ to synergetics) redundantly took the number 1.192324, which I assumed (again in mistaken carelessness) to be the third-power value of the synergetics-conversion constant, and I applied it to the volumetric value of a sphere of unit vector diameter as already arrived at by conventional XYZ-referenced mathematics, the conventional XYZ-coordinate volumetric value for the volume of a sphere of radius 1 being 4.188, which multiplied by 1.192324 gave the product 4.99—a value so close to 5 that I thought it might possibly have been occasioned by the unresolvability of tail-end trigonometric interpolations, wherefore I tentatively accepted 4.99 as probably being exactly 5, which, if correct, was an excitingly significant number as it would have neatly fitted the sphere into the hierarchy of primitive polyhedra (Sec. 982.61). My hindsight wisdom tells me that my subconscious demon latched tightly onto this 5 and fended off all subconsciously challenging intuitions.

986.209

But what I had mistakenly assumed to be the third-power synergetics constant was in fact the ninth power of that constant, as will be seen in the following list of the synergetics constant raised to varying powers:

$$\begin{array}{c}\text{Table 986.209}\\ \text{Synergetics Power Constants}\\ \\ \begin{array}{rl}{S}_1& =1.019824451\\ S_2& =1.040041912\\ S_3& =1.060660172\\ S_4& =1.081687178\\ S_5& =1.103131033\\ S_6& =1.125\\ S_9& =1.193242693\\ S_{12}& =1.265625\end{array}\end{array}$$

986.210

Fig. 986.210 Diagonal of Cube as Unity in Synergetic Geometry

Fig. 986.210 Diagonal of Cube as Unity in Synergetic Geometry: In synergetic geometry mensural unity commences with the tetra edge as prime vector. Unity is taken not from the cube edge but from the edge of one of the two tetra that structure it. (Compare Fig. 463.01.) Proportionality exactly known to us is not required in nature’s structuring. Parts have no existence independent of the polyhedra they constitute.

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In our always-experimental-evidenced science of geometry we need only show ratio of proportion of parts, for parts of primitive polyhedra have no independent existence. Ergo, no experimental proof is required for (square) roots and (square) roots. Though those numbers are irrational, their irrationality could not frustrate the falling apart of the polyhedral parts, because the parts are nonexistent except as parts of wholes, and exact proportionality is not required in the structuring.

986.211

Whatever the workings of my subconscious may have been, the facts remain that I had erroneously concluded that the 5 was the tetravolume of the sphere whose diameter was our unit vector whose value was 2. In due course I received a letter from a mathematician, Ramsey Campbell, whose conventional calculations seemed to show that I was wrong. But I was not convinced that his conventional results were not also erroneous, inasmuch as they had been “cubically” arrived at rather than tetrahedrally referenced.

986.212

At this point a young associate of mine, Robert Grip—who was convinced that I was misconvinced—and who knew that I would alter my position only as confronted by physically demonstrable evidence, made a gallon-sized, water-holding tetrahedron and a sphere whose diameter was identical with the prime vector length of the tetrahedron’s edge. The water content-the volume of the sphere was indeed 4.43 units—0.57 less than 5.

986.213

The cubically-arrived-at spherical volume (A) of a sphere of diameter equal to the unit edge of the XYZ coordinate system’s cube is 4.188. To convert that spherical volume value (A) to that of sphere (B) whose diameter is equal to the diagonal of the face of the XYZ system’s cube, we multiply the volume of sphere (A) by the synergetics hierarchy’s volumetric constant, which is obtained by taking synergetics’ unit VE vector linear constant 1.0198 and raising it to its third-power—or volumetric—dimension, which is 1.0198 × 1.0198 × 1.0198, which equals 1.0606. Multiplying the XYZ system’s cube- edge-diametered (A) sphere’s volume of 4.1888 by the synergetics’ volumetric constant of 1.0606 gives us 4.4429, which is the sought-for volume of the sphere (B). I thanked Mr. Campbell and acknowledged my error.

986.214

I then said to my mathematical associates, Robert Grip and Chris Kitrick, that there is no single item that more effectively advances research than the unblocking of our thought processes—through experiential evidence—of a previously held erroneous assumption. Wherefore my intuition told me that my error may have been stubbornly clung to because there might be something very important to be discovered in this region of investigation. There is possibly some enlightening significance in the fact that I had intuitively applied (and again forsaking the first correction, had doubly reapplied) my third-power synergetics’ conversion factor to an already-three-dimensional cubic-volume quantation, which on the occasion of these retreatments had erroneously seemed to me to be as yet three powers short of the minimum primitive realizable somethingness.

986.215

Why did I think as I did? Why was I puzzled? I was not confused about arithmetical operations per se. We conventionally arrive at the area of a square by multiplying the square’s edge length by itself, and we arrive at the volume of a cube by multiplying its edge length times itself twice—that is, we identify the square’s area by the second power of its edge length, and we identify a cube’s volume by the third power of its edge length. All that seems simple and clear … until we discover that the cube does not exist and cannot exist until it has at least three other observable attributes: weight, duration, and temperature. Given the quantitative inputs for those coordinate factors, the cube as yet fails to “exist,” because as calculated it is now “solid,” and physics has discovered and proven that no such solid phenomenon exists; wherefore the cubical domain has to be substantively populated by atoms which have a variety of interspacing and interpositioning behavioral patterns.

986.216

Also, in order to exist the cube must have both tension and compression forces so arranged and quantated as to produce a self-stabilizing, independent behavior in the presence of the cosmic complex of coexisting force events. For it to exist there also must be introduced coordinate factors that account for the fact that this special case cube is keeping locatable company with the planet Earth with which it is traveling around the Sun at approximately 60,000 miles per hour.

986.217

As the Earth and the Sun whirl circumferentially in company with the other hundred billion stars of the galactic system, and as all the while the galactic system keeps company with all the now-known billion such galaxies whose uniformly angled retreat from one another at an astronomical speed altogether constitutes what is called the Expanding Universe … if we wrote out the formula for integrating all those quantities and for realistically diagramming its geometry and its dimensions, we would have to admit that the dimensions of the cube did not as yet produce existence. There would as yet be required the set of coordinate factors stating when and where the cube was born, how old it was at the moment of its dimensioning, and what its exact remaining longevity would be—and with all that, we have not disclosed its smell, its resonance factor, its electromagnetic-wave propagation length and frequencies. My quandary was one of adequately identifying and calculating the magnitude of relevant dimensions for the “considered set” (Section 509).

986.218

My quandary also included, “Which exactly are the attributes that are being disclosed by the successive powerings?” With all the foregoing considerations I resolved upon the following set as that which I would employ in publishing Synergetics.

986.219

Since our dimensional control is the prime vector, and since a vector’s relative size represents mass times velocity, and since mass has a priori both volume and weight, it inherently introduces one more dimension to velocity’s a priori two-dimensional product of time and distance. Ergo, vectors are in themselves primitive, pre-time-size, potentially energizable, three-dimensional phenomena. Any special case time-size phenomena must also be multiplied by frequency of subdivision of the primitive system taken volumetrically to the third power. We seem thus to have arrived at nine dimensions—i.e., ninth powering—and we have altogether identified geometrical realization as being at least nine-dimensional.

986.220

This is how I came to adopt my ninth-power factor for conversion from XYZ coordination to synergetics coordination. Employing the XYZ coordinated volume of 4.188790205, I multiplied it by the appropriate factor (see table 986.209, where we find that S⁹ = 1.193242693), which produced the inherently imperfect (only chord-describable rather than arc-describable) sphere of 4.998243305. This I knew was not a primitive three- dimensional or six-dimensional volume, and I assumed it to be the value of potential energy embraceable by a sphere of vector radius = 1. Ergo, both my conscious and subconscious searchings and accountings were operating faultlessly, but I was confusing the end product, identifying it as volume instead of as potential energy.

986.221

I was astonished by my error but deeply excited by the prospect of reviewing the exponentially powered values. Looking over the remaining valid trail blazings, I ruminated that the proximity to 5 that provoked the 4.998243305 figure might have other significance—for instance, as a real ninth-dimensional phenomenon. There was some question about that constant 1.193242693 being a sixth-dimension figure: N³ · N³ = N⁶, which operation I had—in my forgetfulness and carelessness—inadvertently performed. Or the figure I had arrived at could be taken as nine-dimensional if you assume primitive demonstrability of minimum something always to have a combined a priori volumetric- and-energetic existence value, which is indeed what synergetics vectorial structuring does recognize to be naturally and demonstrably true. (See Sec. 100.20.)

986.222

Synergetics demonstrates that the hierarchy of vectorially defined, primitive, triangularly self-stabilized, structural-system polyhedra is initially sixth-dimensional, being both a vectorially six-way coordinate system (mass × velocity) as well as being tetrahedrally—ergo, four-dimensionally-coordinate⁴ —ergo, N⁶ · N⁴ = N¹⁰ somethings; and that they grow expansively in time-size—ergo, in volume at the rate of F³ —ergo, in time-size D¹⁰ D³ = D¹³, a 13-dimensional special-case-somethingness of reality.

(Footnote 4: It was a mathematical requirement of XYZ rectilinear coordination that in order to demonstrate four-dimensionality, a fourth perpendicular to a fourth planar facet of the symmetric system must be found—which fourth symmetrical plane of the system is not parallel to one of the already-established three planes of symmetry of the system. The tetrahedron, as synergetics’ minimum structural system, has four symmetrically interarrayed planes of symmetry—ergo, has four unique perpendiculars—ergo, has four dimensions.)

986.223

We have learned in synergetics by physical experiment that in agglomerating unit-radius, closest-packed spheres around a nuclear sphere of the same unit radius, successively concentric symmetrical layers of the nuclear surroundment occur in a pattern in which the number of spheres in the outer shell is always the second power of the frequency of modular-system subdivision of the vector-defined edges of the system, and that when the primitive interhierarchy’s relative volumetric values are multiplied by frequency to the third power—and an additional factor of six—it always gives the symmetrical system’s total cumulative volume growth, not only of all its progressively concentric, closest-packed, unit-radius spheres’ combined shells, but also including the volume of the unit-radius, closest-packed sphere shells’ interstitial spaces, as altogether embraced by the exterior planes of the primitive polyhedra of reference. (See Sec. 971 and, in the drawings section, Fig. 970.20, “Dymax Nuclear Growth” (10 June 1948), and “Light Quanta Particle Growth” (7 May 1948); also drawings published in 1944 appearing as end papers to Synergetics 2.)

986.230 System Spinnability

986.231

Synergetics assumes an a priori to time-size, conceptually primitive, relative volumetric value of all the hierarchy of primitive polyhedra; and it also assumes that when we introduce frequency, we are also introducing time and size (see Secs. 782.50 and 1054.70), and we are therefore also introducing all the degrees of freedom inherent in time-size realizations of energetic-system behavior—as for instance the phenomenon of inherent system spinnability.

986.232

With the introduction of the phenomenon of system spinnability around any one or several or all of the hierarchy of concentric symmetric systems’ seven axes of symmetry (Sec. 1040), we observe experientially that such inherent system spinnability produces a superficially spherical appearance, whose time-size realizations might be thought of as being only the dynamic development in time-size aspects of the primitive static polyhedral states. We recall the scientific nondemonstrability of the Greek sphere as defined by them (Secs. 981.19 and 1022.11). We also recall having discovered that the higher the frequency of the unit-radius-vertexed, symmetrical polyhedra of our primitive cosmic hierarchy, the more spherical do such geodesic-structured polyhedra appear (compare Sec. 986.064). I realized that under these recalled circumstances it could be safely assumed that a sphere does not exist in the primitive hierarchy of pre-time-size polyhedral conceptioning, whose timeless-sizeless—ergo, eternal—perfection alone permitted consideration of the vector equilibrium’s isotropic vector matrix as the four- dimensional frame of reference of any time-size intertransforming aberrations of realizable physical experience. Such perfection can be only eternal and timeless.

986.233

Timeless but conceptually primitive polyhedra of differently-lengthed-and- radiused external vertexes can be dynamically spinnable only in time, thereby to produce circular profiles some of whose longer radii dominantly describe the superficial, illusory continuity whose spherical appearance seems to be radially greater than half the length of the prime vector. (See Fig. 986.314.)

986.234

Thus the only-superficially-defined spherical appearance is either the consequence of the multiplicity of revolving vertexes of the polyhedron occurring at a distance outwardly of the unit vector radius of the prime polyhedral hierarchy, or it could be inherent in the centrifugal deformation of the polyhedral structure. Wherefore I realized that my having unwittingly and redundantly applied the synergetics constant of the sixth power—rather than only of the third power—and my having applied that sixth-power factor to the theretofore nonexistent static sphere of the Greeks’ energy-and-time deprived three-dimensionality, was instinctively sound. Thus the erroneous result I had obtained must not discourage my intuitive urge to pursue the question further. I had inadvertently produced the slightly-greater-than-vector-radiused, highfrequency “spheric” polyhedron.

986.235

It seemed ever more evident that it could be that there is no true sphere in Universe. This seemed to be confirmed by the discovery that the sum of the angles around all the vertexes of any system will always be 720 degrees—one tetra—less than the number of the system’s external vertexes times 360 degrees (Sec. 224). It could be that the concept conjured up by the mouthed-word sphere itself is scientifically invalid; ergo, it could be that the word sphere is not only obsolete but to be shunned because it is meaningless and possibly disastrously misleading to human thought.

986.240 The Sphere Experimentally Defined

986.241

The best physically demonstrable definition of the “spheric” experience is: an aggregate of energy events approximately equidistant, multidirectionally outwardly from approximately the same central event of an only approximately simultaneous set of external events-the more the quantity of external points measuringly identified and the more nearly simultaneous the radius-measuring events, the more satisfactorily “spherical.” With each of all the outward unit-radius events most economically and most fully triangularly interchorded with their most immediate neighbors—chords being shorter than their corresponding arcs—we find that the “spheric” experience inherently describes only high-frequency, omnitriangularly faceted polyhedra. By geometrical definition these are geodesic structures whose volumes will always be something less than a theoretically perfect omni-arc-embraced sphere of the same radius as an omni-chord-embraced geodesic sphere’s uniformly radiused outer vertexes.

986.242

As is demonstrated in Sec. 224, the sum of the angles around all the vertexes of any system will always be 720 degrees less than the number of vertexes multiplied by 360 degrees. By the mathematicians’ definition a perfect arc-embraced sphere would have to have 360 degrees around every point on its surface, for the mathematicians assume that for an infinitesimal moment a sphere’s surface is congruent with the tangent plane. Trigonometry errs in that it assumes 360 degrees around every spherical surface point.

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986.300 Minimum-Maximum System Limits

986.301 Consideration 11: Maximum-limit Case

986.302

The explorer gains assurance by discovering the relevant minimum-maximum limit cases—the min-max limits of the variables—of the system under consideration.

986.303

For instance, we have learned through experimental evidence, the cosmic hierarchy of primitive polyhedra has a limit set of seven axes of great-circle symmetries and spinnabilities. They are the 3, 4, 6 (VE), 12, 10, 15, 6 (icosa) great-circle-spinnable systems. (See Table 986.304 and Sec. 1040.) Within that inherently limited hierarchy of seven symmetries, the triacontahedron, with its 15 different great circles’ self- hemispherings and 120 triangular interconfigurings, produces the maximum-limit number of identical polyhedral surface self-facetings of all great-circle systems in Universe (Sec. 400).

Table 986.304: Limit Set of Seven Axes of Spinnability

Generalized Set of All Symmetrical Systems

Spinnable System	Great Circles
#1	3
#2	4
#3	6 (VE)
#4	12
#5	10
#6	15
#7	6 (icosa)

(Compare Secs. 1041.01 and 1042.05.)

986.310 Strategic Use of Min-max Cosmic System Limits

986.311

The maximum limit set of identical facets into which any system can be divided consists of 120 similar spherical right triangles ACB whose three corners are 60 degrees at A, 90 degrees at C, and 36 degrees at B. Sixty of these right spherical triangles are positive (active), and 60 are negative (passive). (See Sec. 901.)

986.312

These 120 right spherical surface triangles are described by three different central angles of 37.37736814 degrees for arc AB, 31.71747441 degrees for arc BC, and 20.90515745 degrees for arc AC—which three central-angle arcs total exactly 90 degrees. These 120 spherical right triangles are self-patterned into producing 30 identical spherical diamond groups bounded by the same central angles and having corresponding flat-faceted diamond groups consisting of four of the 120 angularly identical (60 positive, 60 negative) triangles. Their three surface corners are 90 degrees at C, 31.71747441 degrees at B, and 58.2825256 degrees at A. (See Fig. 986.502.)

986.313

These diamonds, like all diamonds, are rhombic forms. The 30-symmetrical- diamond system is called the rhombic triacontahedron: its 30 mid-diamond faces (right- angle cross points) are approximately tangent to the unit-vector-radius sphere when the volume of the rhombic triacontahedron is exactly tetravolume-5. (See Fig. 986.314.)

986.314

Fig. 986.314

Fig. 986.314 Polyhedral Profiles of Selected Polyhedra of Tetravolume-5 and Approximately Tetravolume-5: A graphic display of the radial proximity to one another of exact and neighboring tetravolume-5 polyhedra, showing central angles and ratios to prime vector.

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I therefore asked Robert Grip and Chris Kitrick to prepare a graphic comparison of the various radii and their respective polyhedral profiles of all the symmetric polyhedra of tetravolume 5 (or close to 5) existing within the primitive cosmic hierarchy (Sec. 982.62) —i.e. other than those of tetravolumes 1, 2, 3, 4, and 6—which carefully drafted drawing of the tetravolume-5 polyhedra (and those polyhedra “approximately” tetravolume S) my colleagues did prepare (see Fig. 986.314). These exactly tetravolume-5 polyhedra are, for example—
a. the icosahedron with outer edges of unit vector length;
b. the icosahedron of outer vertex radius of unit vector length;
c. the regular dodecahedron of unit vector edge; and
d. the regular dodecahedron of unit vector radius
—all of which show that they have only a slightly greater radius length than that of the prime vector.

986.315

The chart of the polyhedral profiles (Fig. 986.314) shows the triacontahedron of tetravolume 5 having its mid-diamond-face point C at a distance outward radially from the volumetric center that approximately equals the relative length of the prime vector. I say “approximately” because the trigonometrically calculated value is .999483332 instead of 1, a 0.0005166676 radial difference, which—though possibly caused in some very meager degree by the lack of absolute resolvability of trigonometric calculations themselves—is on careful mathematical review so close to correct as to be unalterable by any known conventional trigonometric error allowance. It is also so correct as to hold historical significance, as we shall soon discover. Such a discrepancy is so meager in relation, for instance, to planet Earth’s spheric diameter of approximately 8,000 miles that the spherical surface aberration would be approximately the same as that existing between sea level and the height of Mount Fuji, which is only half the altitude of Mount Everest. And even Mount Everest is invisible on the Earth’s profile when the Earth is photographed from outer space. The mathematical detection of such meager relative proportioning differences has time and again proven to be of inestimable value to science in first detecting and then discovering cosmically profound phenomena. In such a context my “spherical energy content” of 4.99, instead of exactly 5, became a thought-provoking difference to be importantly remembered.

986.316

By careful study of the Grip-Kitrick drawings of tetravolume-5 polyhedra it is discovered that the graphically displayed zones of radial proximity to one another of all the tetravolume-5 symmetric polyhedra (Fig. 986.314) describe such meager radial differences at their respective systems’ outermost points as to suggest that their circumferential zone enclosed between the most extremely varied and the most inwardly radiused of all their axially spun vertexes of the exact tetravolume-5 polyhedra may altogether be assumed to constitute the zone of limit cases of radiantly swept-out and pulsating tetravolume-5 kinetic systems.

986.317

Recognizing that polyhedra are closed systems and that there are only seven cases of symmetrical subdivisioning of systems by the most economical great-circle spinnings (and most economically by the chords of the great-circle arcs), we discover and prove structurally that the maximum-limit abundance of a unit-symmetrical-polyhedral- system’s identical facetings is the rhombic triacontahedron, each of whose 30 symmetrical diamond planar faces may be symmetrically subdivided into four identical right triangles (30 × 4=120), and we find that the triacontahedron’s 120-spherical-right-triangled frame of system reference is the maximum-limit case of identical faceting of any and all symmetrical polyhedral systems in Universe. This maximum-limit-system structuring proof is accomplished by the physically permitted, great-circle-spun, hemispherical self-halvings, as permitted by any and all of the seven cosmic limit cases of symmetric systems’ being spun-defined around all the respective system’s geometrically definitive (ergo, inherent) axes of symmetrical spinnability. It is thus that we learn experimentally how all the symmetric systems of Universe self-fractionate their initial system unities into the maximum number of omniangularly identical surface triangles outwardly defining their respective internal-structure tetrahedra whose angles-central or surface-are always independent of a system’s time-size considerations. And because they are independent of time-size considerations, such minimum-maximum limit-case ranges embrace all the symmetrical polyhedral systems’ generalized-primitive-conceptuality phenomena.

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986.400 T Quanta Module

986.401 Consideration 12: Dynamic Spinning of Rhombic Triacontahedron

986.402

I then speculated that the only-by-spinning-produced, only superficially apparent “sphericity” could be roundly aspected by spinning the rhombic triacontahedron of tetravolume 5. This rational volumetric value of exactly 5 tetravolumes placed the rhombic triacontahedron neatly into membership in the primitive hierarchy family of symmetric polyhedra, filling the only remaining vacancy in the holistic rational-number hierarchy of primitive polyhedral volumes from 1 through 6, as presented in Table 1053.51A.

986.403

In the isotropic vector matrix system, where R = radius and PV = prime vector, PV = 1 = R—ergo, PVR = prime vector radius, which is always the unity of VE. In the 30-diamond-faceted triacontahedron of tetravolume 5 and the 12-diamond-faceted dodecahedron of tetravolume 6, the radius distances from their respective symmetric polyhedra’s volumetric centers O to their respective mid-diamond faces C (i.e., their short- and-long-diamond-axes’ crossing points) are in the rhombic triacontahedral case almost exactly PVR—i.e., 0.9994833324 PVR—and in the rhombic dodecahedral case exactly PVR, 1.0000 (alpha) PVR.

986.404

In the case of the rhombic dodecahedron the mid-diamond-face point C is exactly PVR distance from the polyhedral system’s volumetric (nucleic) center, while in the case of the rhombic triacontahedron the point C is at approximately PVR distance from the system’s volumetric (nucleic) center. The distance outward to C from the nucleic center of the rhombic dodecahedron is that same PVR length as the prime unit vector of the isotropic vector matrix. This aspect of the rhombic triacontahedron is shown at Fig. 986.314.

986.405

Fig. 986.405

Fig. 986.405 Respective Subdivision of Rhombic Dodecahedron (A) and Rhombic Triacontahedron (B) into Diamond-faced Pentahedra: O is at the respective volumetric centers of the two polyhedra, with the short axes A-A and the long axes B-B (diagrams on the right). The central surface angles of the two pentahedra differ as shown.

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The symmetric polyhedral centers of both the rhombic dodecahedron and the rhombic triacontahedron may be identified as 0, and both of their respective external diamond faces’ short axes may be identified as A-A and their respective long axes as B-B. Both the rhombic dodecahedron’s and the triacontahedron’s external diamond faces ABAB and their respective volumetric centers O describe semiasymmetric pentahedra conventionally labeled as OABAB. The diamond surface faces ABA of both OABAB pentahedra are external to their respective rhombic-hedra symmetrical systems, while their triangular sides OAB (four each) are internal to their respective rhombichedra systems. The angles describing the short A-A axis and the long B-B axis, as well as the surface and central angles of the rhombic dodecahedron’s OABAB pentahedron, all differ from those of the triacontahedron’s OABAB pentahedron.

986.410 T Quanta Module

986.411

Fig. 986.411A T and E Quanta Modules Lengths

Fig. 986.411A T and E Quanta Modules: Edge Lengths: This plane net for the T Quanta Module and the E Quanta Module shows their edge lengths as ratioed to the octa edge. Octa edge = tetra edge = unity.

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Fig. 986.411B T and E Quanta Module Angles

Fig. 986.411B T and E Quanta Module Angles: This plane net shows the angles and the foldability of the T Quanta Module and the E Quanta Module.

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Fig. 986.411C

Fig. 986.411C T and E Quanta Modules in Context of Rhombic Triacontahedron

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The respective 12 and 30 pentahedra OABAB of the rhombic dodecahedron and the triacontahedron may be symmetrically subdivided into four right-angled tetrahedra ABCO, the point C being surrounded by three right angles ABC, BCO, and ACO. Right- angle ACB is on the surface of the rhombic-hedra system and forms the face of the tetrahedron ABCO, while right angles BCO and ACO are internal to the rhombic-hedra system and from two of the three internal sides of the tetrahedron ABCO. The rhombic dodecahedron consists of 48 identical tetrahedral modules designated ABCO${}^{\text{d}}$. The triacontahedron consists of 120 (60 positive and 60 negative) identical tetrahedral modules designated ABCO${}^{\text{t}}$, for which tetrahedron ABCO${}^{\text{t}}$ we also introduce the name T Quanta Module.

986.412

The primitive tetrahedron of volume 1 is subdivisible into 24 A Quanta Modules. The triacontahedron of exactly tetravolume 5, has the maximum-limit case of identical tetrahedral subdivisibility—i.e., 120 subtetra. Thus we may divide the 120 subtetra population of the symmetric triacontahedron by the number 24, which is the identical subtetra population of the primitive omnisymmetrical tetrahedron: 120/24=5. Ergo, volume of the A Quanta Module = volume of the T Quanta Module.

986.413

Fig. 986.413

Fig. 986.413 Regular Tetrahedron Composed of 24 Quanta Modules: Compare Fig. 923.10.

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The rhombic dodecahedron has a tetravolume of 6, wherefore each of its 48 identical, internal, asymmetric, component tetrahedra ABCO${}^{\text{d}}$ has a regular tetravolume of 6/48 = 1/8 The regular tetrahedron consists of 24 quanta modules (be they A, B, C, D,⁵ * or T Quanta Modules; therefore ABCO${}^{\text{d}}$, having l/8-tetravolume, also equals three quanta modules. (See Fig. 986.413.)

(Footnote 5: C Quanta Modules and D Quanta Modules are added to the A and B Quanta Modules to compose the regular tetrahedron as shown in drawing B of Fig. 923.10.)

986.414

The vertical central-altitude line of the regular, primitive, symmetrical tetrahedron may be uniformly subdivided into four vertical sections, each of which we may speak of as quarter-prime-tetra altitude units-each of which altitude division points represent the convergence of the upper apexes of the A, B, C, D, A’, B’, C’, D’, A”, B”, C”, D” … equivolume modules (as illustrated in Fig. 923.10B where—prior to the discovery of the E “Einstein” Module—additional modules were designated E through H, and will henceforth be designated as successive ABCD, A’B’C’D’, A”B”C”D” … groups). The vertical continuance of these unit-altitude differentials produces an infinite series of equivolume modules, which we identify in vertical series continuance by groups of four repetitive ABCD groups, as noted parenthetically above. Their combined group-of- four, externally protracted, altitude increase is always equal to the total internal altitude of the prime tetrahedron.

986.415

The rhombic triacontahedron has a tetravolume of 5, wherefore each of its 120 identical, internal, asymmetric, component tetrahedra ABCO${}^{\text{t}}$, the T Quanta Module, has a tetravolume of 5/120 = 1/24 tetravolume—ergo, the volume of the T Quanta Module is identical to that of the A and B Quanta Modules. The rhombic dodecahedron’s 48 ABCO${}^{\text{d}}$ asymmetric tetrahedra equal three of the rhombic triacontahedron’s 120 ABCO${}^{\text{t}}$ , T Quanta Module asymmetric tetrahedra. The rhombic triacontahedron’s ABCO${}^{\text{t}}$ T Quanta Module tetrahedra are each 1/24 of the volume of the primitive “regular” tetrahedron—ergo, of identical volume to the A Quanta Module. The A Mod, like the T Mod, is structurally modeled with one of its four corners omnisurrounded by three right angles.

986.416

1 A Module = 1 B Module = 1 C Module = 1 D Module = 1 T Module = any one of the unit quanta modules of which all the hierarchy of concentric, symmetrical polyhedra of the VE family are rationally comprised. (See Sec. 910).

986.417

I find that it is important in exploratory effectiveness to remember—as we find an increasingly larger family of equivolume but angularly differently conformed quanta modules—that our initial exploration strategy was predicated upon our generalization of Avogadro’s special-case (gaseous) discovery of identical numbers of molecules per unit volume for all the different chemical-element gases when individually considered or physically isolated, but only under identical conditions of pressure and heat. The fact that we have found a set of unit-volume, all-tetrahedral modules—the minimum-limit structural systems—from which may be aggregated the whole hierarchy of omnisymmetric, primitive, concentric polyhedra totally occupying the spherically spun and interspheric accommodation limits of closest-packable nuclear domains, means that we have not only incorporated all the min-max limit-case conditions, but we have found within them one unique volumetric unit common to all their primitive conformational uniqueness, and that the volumetric module was developed by vectorial—i.e., energetic—polyhedral-system definitions.

986.418

None of the tetrahedral quanta modules are by themselves allspace-filling, but they are all groupable in units of three (two A’s and one B—which is called the Mite) to fill allspace progressively and to combine these units of three in nine different ways—all of which account for the structurings of all but one of the hierarchy of primitive, omniconcentric, omnisymmetrical polyhedra. There is one exception, the rhombic triacontahedron of tetravolume 5—i.e., of 120 quanta modules of the T class, which T Quanta Modules as we have learned are of equivolume to the A and B Modules.

986.419

Fig. 986.419

Fig. 986.419 T Quanta Modules within Rhombic Triacontahedron: The 120 T Quanta Modules can be grouped two different ways within the rhombic triacontahedron to produce two different sets of 60 tetrahedra each: 60 BAAO and 60 BBAO.

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The 120 T Quanta Modules of the rhombic triacontahedron can be grouped in two different ways to produce two different sets of 60 tetrahedra each: the 60 BAAO tetrahedra and the 60 BBAO tetrahedra. But rhombic triacontahedra are not allspace-filling polyhedra. (See Fig. 986.419.)

986.420 Min-max Limit Hierarchy of Pre-time-size Allspace-fillers

986.421

Fig. 986.421

Fig. 986.421 A and B Quanta Modules. The top drawings present plane nets for the modules with edge lengths of the A Modules ratioed to the tetra edge and edge lengths of the B Modules ratioed to the octa edge. The middle drawings illustrate the angles and foldability. The bottom drawings show the folded assembly and their relation to each other. Tetra edge=octa edge. (Compare Figs. 913.01 and 916.01.)

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Of all the allspace-filling module components, the simplest are the three- quanta-module Mites, consisting of two A Quanta Modules (one A positive and one A negative) and of one B Quanta Module (which may be either positive or negative). Thus a Mite can be positive or negative, depending on the sign of its B Quanta Module. The Mites are not only themselves tetrahedra (the minimum-sided polyhedra), but they are also the simplest minimum-limit case of allspace-filling polyhedra of Universe, since they consist of two energy-conserving A Quanta Modules and one equivolume energy- dispersing B Quanta Module. The energy conservation of the A Quanta Module is provided geometrically by its tetrahedral form: four different right-triangled facets being all foldable from one unique flat-out whole triangle (Fig. 913.01), which triangle’s boundary edges have reflective properties that bounce around internally to those triangles to produce similar smaller triangles: Ergo, the A Quanta Module acts as a local energy holder. The B Quanta Module is not foldable out of one whole triangle, and energies bouncing around within it tend to escape. The B Quanta Module acts as a local energy dispenser. (See Fig. 986.421.)

986.422

Fig. 986.422 MITE

MITE (See color plate 17.)

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Mite: The simplest allspace-filler is the Mite (see Secs. 953 and 986.418). The positive Mite consists of 1 A + mod, 1 A - mod, and B + mod; the negative Mite consists of 1 A + mod, 1A - mod, and B-mod. Sum-total number of modules…3.

986.423

Around the four corners of the tetrahedral Mites are three right triangles. Two of them are similar right triangles with differently angled acute corners, and the third right triangle around that omni-right-angled corner is an isosceles.

986.424

The tetrahedral Mites may be inter-edge-bonded to fill allspace, but only because the spaces between them are inadvertent capturings of Mite-shaped vacancies. Positive Mite inter-edge assemblies produce negative Mite vacancies, and vice versa. The minimum-limit case always provides inadvertent entry into the Negative Universe. Sum- total number of modules is…1½

986.425

Mites can also fill allspace by inter-face-bonding one positive and one negative Mite to produce the Syte. This trivalent inter-face-bonding requires twice as many Mites as are needed for bivalent inter-edge-bonding. Total number of modules is…3

986.426

Syte: The next simplest allspace-filler is the Syte. (See Sec. 953.40.) Each Syte consists of one of only three alternate ways of face-bonding two Mites to form an allspace-filling polyhedron, consisting of 2 A + mods, 2 A - mods, 1 B + mod, and 1 B - mod. Sum-total number of modules…6

986.427

Fig. 986.427 Bite, Rite, Lite

Fig. 986.427 Bite, Rite, Lite

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Two of the three alternate ways of combining two Mites produce tetrahedral Sytes of one kind:
BITE (See color plate 17), RITE (See color plate 19)
while the third alternate method of combining will produce a hexahedral Syte.
LITE (See color plate 18)

986.428

Kite: The next simplest allspace-filler is the Kite. Kites are pentahedra or half-octahedra or half-Couplers, each consisting of one of the only two alternate ways of combining two Sytes to produce two differently shaped pentahedra, the Kate and the Kat, each of 4 A + mods, 4 A - mods, 2 B + mods, and 2 B-mods. Sum-total number of modules…12

986.429

Fig. 986.429 Kate, Kat

Fig. 986.429 Kate, Kat.

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Two Sytes combine to produce two Kites as KATE (See color plate 20) KAT (See color plate 21)

986.430

Fig. 986.430 OCTET

OCTET (See color plate 22)

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Octet: The next simplest allspace-filler is the Octet, a hexahedron consisting of three Sytes—ergo, 6 A + mods, 6 A - mods, 3 B + mods, and 3 B-mods. Sum-total number of modules…18

986.431

Fig. 986.431 COUPLER

COUPLER (See color plate 23)

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Coupler: The next simplest allspace-filler is the Coupler, the asymmetric octahedron. (See Secs. 954.20-.70.) The Coupler consists of two Kites—ergo, 8 A + mods, 8 A - mods, 4 B + mods, and 4 B - mods. Sum-total number of modules…24

986.432

Fig. 986.432 CUBE

CUBE (See color plate 24)

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Cube: The next simplest allspace-filler is the Cube, consisting of four Octets—ergo, 24 A + mods, 24 A - mods, 12 B + mods, and 12 B - mods. Sum-total number of modules…72

986.433

Fig. 986.433 RHOMBIC DODECAHEDRON

RHOMBIC DODECAHEDRON (See color plate 25)

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Rhombic Dodecahedron: The next and last of the hierarchy of primitive allspace-fillers is the rhombic dodecahedron. The rhombic dodecahedron is the domain of a sphere (see Sec. 981.13). The rhombic dodecahedron consists of 12 Kites—ergo, 48 A + mods, 48 A - mods, 24 B + mods, and 24 B - mods. Sum-total number of modules…144

986.434

This is the limit set of simplest allspace-fillers associable within one nuclear domain of closest-packed spheres and their respective interstitial spaces. There are other allspace-fillers that occur in time-size multiplications of nuclear domains, as for instance the tetrakaidecahedron. (Compare Sec. 950.12.)

986.440 Table: Set of Simple Allspace-fillers

This completes one spheric domain (i.e., sphere plus interstitial space) of one unit-radius sphere in closest packing, each sphere being centered at every other vertex of the isotropic vector matrix.

Name	Face Triangles	Type Hedra	A Quanta Modules	B Quanta Modules	Sum- Total Modules
MITE	4	tetrahedron	2	1	3
SYTE
BITE	4	tetrahedron	4	2	6
RITE	4	tetrahedron	4	2	6
LITE	6	hexahedron	4	2	6
KITE
KATE	5	pentahedron	8	4	12
KAT	5	pentahedron	8	4	12
OCTET	6	hexahedron	12	6	18
COUPLER	8	octahedron	16	8	24
CUBE	6	hexahedron	48	24	72
RHOMBIC DODECAHEDRON	12	dodecahedron	96	48	144

(For the minimum time-size special case realizations of the two-frequency systems. multiply each of the above Quanta Module numbers by eight.)

986.450 Energy Aspects of Spherical Modular Arrays

986.451

The rhombic dodecahedron has an allspace-filling function as the domain of any one sphere in an aggregate of unit-radius, closest-packed spheres; its 12 mid-diamond- face points C are the points of intertangency of all unit-radius, closest-packed sphere aggregates; wherefore that point C is the midpoint of every vector of the isotropic vector matrix, whose every vertex is the center of one of the unit-radius, closest-packed spheres.

986.452

These 12 inter-closest-packed-sphere-tangency points—the C points—are the 12 exclusive contacts of the “Grand Central Station” through which must pass all the great-circle railway tracks of most economically interdistanced travel of energy around any one nuclear center, and therefrom—through the C points—to other spheres in Universe. These C points of the rhombic dodecahedron’s mid-diamond faces are also the energetic centers-of-volume of the Couplers, within which there are 56 possible unique interarrangements of the A and B Quanta Modules.

986.453

We next discover that two ABABO pentahedra of any two tangentially adjacent, closest-packed rhombic dodecahedra will produce an asymmetric octahedron OABABO’ with O and O’ being the volumetric centers (nuclear centers) of any two tangentially adjacent, closest-packed, unit-radius spheres. We call this nucleus-to-nucleus, asymmetric octahedron the Coupler, and we found that the volume of the Coupler is exactly equal to the volume of one regular tetrahedron—i.e., 24 A Quanta Modules. We also note that the Coupler always consists of eight asymmetric and identical tetrahedral Mites, the minimum simplex allspace-filling of Universe, which Mites are also identifiable with the quarks (Sec. 1052.360).

986.454

We then discover that the Mite, with its two energy-conserving A Quanta Modules and its one energy-dispersing B Quanta Module (for a total combined volume of three quanta modules), serves as the cosmic minimum allspace-filler, corresponding elegantly (in all ways) with the minimum-limit case behaviors of the nuclear physics’ quarks. The quarks are the smallest discovered “particles”; they always occur in groups of three, two of which hold their energy and one of which disperses energy. This quite clearly identifies the quarks with the quanta module of which all the synergetics hierarchy of nuclear concentric symmetric polyhedra are co-occurrent.

986.455

In both the rhombic triacontahedron of tetravolume 5 and the rhombic dodecahedron of tetravolume 6 the distance from system center O at AO is always greater than CO, and BO is always greater than AO.

986.456

With this information we could reasonably hypothesize that the triacontahedron of tetravolume 5 is that static polyhedral progenitor of the only- dynamically-realizable sphere of tetravolume 5, the radius of which (see Fig. 986.314) is only 0.04 of unity greater in length than is the prime vector radius OC, which governs the dimensioning of the triacontahedron’s 30 midface cases of 12 right-angled corner junctions around mid-diamond-vertex C, which provides the 12 right angles around C-the four right-angled corners of the T Quanta Module’s ABC faces of their 120 radially arrayed tetrahedra, each of which T Quanta Module has a volume identical to that of the A and B Quanta Modules.

986.457

We also note that the radius OC is the same unitary prime vector with which the isotropic vector matrix is constructed, and it is also the VE unit-vector-radius distance outwardly from O, which O is always the common system center of all the members of the entire cosmic hierarchy of omniconcentric, symmetric, primitive polyhedra. In the case of the rhombic triacontahedron the 20 OA lines’ distances outwardly from O are greater than OC, and the 12 OB lines’ distances are even greater in length outwardly from O than OA. Wherefore I realized that, when dynamically spun, the greatcircle chord lines AB and CB are centrifugally transformed into arcs and thus sprung apart at B, which is the outermost vertex—ergo, most swiftly and forcefully outwardly impelled. This centrifugal spinning introduces the spherical excess of 6 degrees at the spherical system vertex B. (See Fig. 986.405) Such yielding increases the spheric appearance of the spun triacontahedron, as seen in contradistinction to the diamond-faceted, static, planar-bound, polyhedral state aspect.

986.458

The corners of the spherical triacontahedron’s 120 spherical arc-cornered triangles are 36 degrees, 60 degrees and 90 degrees, having been sprung apart from their planar-phase, chorded corners of 31.71747441 degrees, 58.28252559 degrees, and 90 degrees, respectively. Both the triacontahedron’s chorded and arced triangles are in notable proximity to the well-known 30-, 60-, and 90-degree-cornered draftsman’s flat, planar triangle. I realized that it could be that the three sets of three differently-distanced- outwardly vertexes might average their outward-distance appearances at a radius of only four percent greater distance from O-thus producing a moving-picture-illusioned “dynamic” sphere of tetravolume 5, having very mildly greater radius than its static, timeless, equilibrious, rhombic triacontahedron state of tetravolume 5 with unit-vector- radius integrity terminaled at vertex C.

986.459

In the case of the spherical triacontahedron the total spherical excess of exactly 6 degrees, which is one-sixtieth of unity = 360 degrees, is all lodged in one corner. In the planar case 1.71747441 degrees have been added to 30 degrees at corner B and subtracted from 60 degrees at corner A. In both the spherical and planar triangles—as well as in the draftsman’s triangle—the 90-degree corners remain unchanged.

986.460

The 120 T Quanta Modules radiantly arrayed around the center of volume of the rhombic triacontahedron manifest the most spherical appearance of all the hierarchy of symmetric polyhedra as defined by any one of the seven axially rotated, great circle system polyhedra of the seven primitive types of great-circle symmetries.

986.461

What is the significance of the spherical excess of exactly 6 degrees? In the transformation from the spherical rhombic triacontahedron to the planar triacontahedron each of the 120 triangles releases 6 degrees. 6 × 120 = 720. 720 degrees = the sum of the structural angles of one tetrahedron = 1 quantum of energy. The difference between a high-frequency polyhedron and its spherical counterpart is always 720 degrees, which is one unit of quantum—ergo, it is evidenced that spinning a polyhedron into its spherical state captures one quantum of energy—and releases it when subsiding into its pre-time- size primitive polyhedral state.

986.470 Geodesic Modular Subdivisioning

986.471

Fig. 986.471

Fig. 986.471 Modular Subdivisioning of Icosahedron as Maximum Limit Case: The 120 outer surface right spherical triangles of the icosahedron’s 6, 10, and 15 great circles generate a total of 242 external vertexes, 480 external triangles, and 480 internal face-congruent tetrahedra, constituting the maximum limit of regular spherical system surface omnitriangular self-subdivisioning into centrally collected tetrahedral components.

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A series of considerations leads to the definition of the most spherical- appearing limit of triangular subdivisioning:
1. recalling that the experimentally demonstrable “most spherically-appearing” structure is always in primitive reality a polyhedron;
2. recalling that the higher the modular frequency of a system the more spheric it appears, though it is always polyhedral and approaching not a “true sphere” limit but an unlimited multiplication of its polyhedral facetings;
3. recalling that the 120 outer surface triangles of the icosahedron’s 15 great circles constitute the cosmic maximum limit of system-surface omni-triangular- self-subdivisioning into centrally collected tetrahedron components; and
4. recalling that the icosahedron’s 10- and 6-great-circle equators of spin further subdivide the 15 great circles’ outer 120 LCD triangles into four different right triangles, ADC, CDE, CFE, and EFB (see Fig. 901.03),

then it becomes evident that the icosahedron’s three sets of symmetrical greatcircle spinnabilities—i.e., 6 + 10 + 15 (which totals 31 great circle self-halvings)—generate a total of 242 unit-radius, external vertexes, 480 external triangles, and 720 internal triangles (which may be considered as two congruent internal triangles, each being one of the internal triangular faces of the 480 tetrahedra whose 480 external triangular faces are showing-in which case there are 1440 internal triangles). The 480 tetrahedra consist of 120 OCAD, 120 OCDE, 120 OCEF, and 120 OFEB tetrahedra. (See Fig. 986.471.) The 480 internal face-congruent tetrahedra therefore constitute the “most spheric-appearing” of all the hemispheric equators’ self-spun, surface-subdividing entirely into triangles of all the great circles of all the primitive hierarchy of symmetric polyhedra.

986.472

In case one thinks that the four symmetrical sets of the great circles of the spherical VE (which total 25 great circles in all) might omnisubdivide the system surface exclusively into a greater number of triangles, we note that some of the subdivision areas of the 25 great circles are not triangles (see quadrant BCEF in Fig. 453.01 —third printing of Synergetics 1—of which quadrangles there are a total of 48 in the system); and note that the total number of triangles in the 25-great-circle system is 288—ergo, far less than the 31 great circles’ 480 spherical right triangles; ergo, we become satisfied that the icosahedron’s set of 480 is indeed the cosmic maximum-limit case of system-self-spun subdivisioning of its self into tetrahedra, which 480 consist of four sets of 120 similar tetrahedra each.

986.473

It then became evident (as structurally demonstrated in reality by my mathematically close-toleranced geodesic domes) that the spherical trigonometry calculations’ multifrequenced modular subdividing of only one of the icosahedron’s 120 spherical right triangles would suffice to provide all the basic trigonometric data for any one and all of the unit-radius vertex locations and their uniform interspacings and interangulations for any and all frequencies of modular subdividings of the most symmetrical and most economically chorded systems’ structuring of Universe, the only variable of which is the special case, time-sized radius of the special-case system being considered.

986.474

This surmise regarding nature’s most-economical, least-effort design strategy has been further verified by nature’s own use of the same geodesics mathematics as that which I discovered and employed in my domes. Nature has been using these mathematical principles for eternity. Humans were unaware of that fact. I discovered these design strategies only as heretofore related, as an inadvertent by-product of my deliberately undertaking to find nature’s coordination system. That nature was manifesting icosahedral and VE coordinate patterning was only discovered by other scientists after I had found and demonstrated geodesic structuring, which employed the synergetics’ coordinate-system strategies. This discovery by others that my discovery of geodesic mathematics was also the coordinate system being manifest by nature occurred after I had built hundreds of geodesic structures around the world and their pictures were widely published. Scientists studying X-ray diffraction patterns of protein shells of viruses in 1959 found that those shells disclosed the same patterns as those of my widely publicized geodesic domes. When Dr. Aaron Klug of the University of London—who was the one who made this discovery—communicated with me, I was able to send him the mathematical formulae for describing them. Klug explained to me that my geodesic structures are being used by nature in providing the “spherical” enclosures of her own most critical design-controlling programming devices for realizing all the unique biochemical structurings of all biology—which device is the DNA helix.

986.475

The structuring of biochemistry is epitomized in the structuring of the protein shells of all the viruses. They are indeed all icosahedral geodesic structures. They embracingly guard all the DNA-RNA codified programming of all the angle-and-frequency designing of all the biological, life-accommodating, life-articulating structures. We find nature employing synergetics geometry, and in particular the high-frequency geodesic “spheres,” in many marine organisms such as the radiolaria and diatoms, and in structuring such vital organs as the male testes, the human brain, and the eyeball. All of these are among many manifests of nature’s employment on her most critically strategic occasions of the most cosmically economical, structurally effective and efficient enclosures, which we find are always mathematically based on multifrequency and three-way-triangular gridding of the “spherical”—because high-frequenced—icosahedron, octahedron, or tetrahedron.

986.476

Comparing the icosahedron, octahedron, and tetrahedron-the icosahedron gives the most volume per unit weight of material investment in its structuring; the high- frequency tetrahedron gives the greatest strength per unit weight of material invested; and the octahedron affords a happy—but not as stable-mix of the two extremes, for the octahedron consists of the prime number 2, 2² = 4; whereas the tetrahedron is the odd prime number 1 and the icosahedron is the odd prime number 5. Gear trains of even number reciprocate, whereas gear trains of an odd number of gears always lock; ergo, the tetrahedral and icosahedral geodesic systems lock-fasten all their structural systems, and the octahedron’s compromise, middle-position structuring tends to yield transformingly toward either the tetra or the icosa locked-limit capabilities—either of which tendencies is pulsatively propagative.

986.480 Consideration 13: Correspondence of Surface Angles and Central Angles

986.481

It was next to be noted that spherical trigonometry shows that nature’s smallest common denominator of system-surface subdivisioning by any one type of the seven great-circle-symmetry systems is optimally accomplished by the previously described 120 spherical-surface triangles formed by the 15 great circles, whose central angles are approximately

$$\begin{array}{r}20.9^{\circ }\\ 37.4^{\circ }\\ 31.7^{\circ }\\ 90.0^{\circ }\end{array}$$

whereas their surface angles are 36 degrees at A, 60 degrees at B, and 90 degrees at C.

986.482

We recall that the further self-subdividing of the 120 triangles, as already defined by the 15 great circles and as subdividingly accomplished by the icosahedron’s additional 6- and 10-great-circle spinnabilities, partitions the 120 LCD triangles into 480 right triangles of four types: ADC, CDE, CFE, and EFB-with 60 positive and 60 negative pairs of each. (See Figs. 901.03 and 986.314.) We also recall that the 6- and 10-great- circle-spun hemispherical gridding further subdivided the 120 right triangles—ACB—formed by the 15 great circles, which produced a total of 12 types of surface angles, four of them of 90 degrees, and three whose most acute angles subdivided the 90-degree angle at C into three surface angles: ACD—31.7 degrees; DCE—37.4 degrees; and ECB—20.9 degrees, which three surface angles, we remember, correspond exactly to the three central angles COB, BOA, and COA, respectively, of the triacontahedron’s tetrahedral T Quanta Module ABCO${}^{\text{t}}$.

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986.500 E Quanta Module

986.501 Consideration 14: Great-circle Foldable Discs

986.502

Fig. 986.502

Fig. 986.502 Thirty Great-circle Discs Foldable into Rhombic Triacontahedron System: Each of the four degree quadrants, when folded as indicated at A and B, form separate T Quanta Module tetrahedra. Orientations are indicated by letter on the great-circle assembly at D.

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With all the foregoing events, data, and speculative hypotheses in mind, I said I think it would be worthwhile to take 30 cardboard great circles, to divide them into four 90-degree quadrants, then to divide each of the quadrants into three angles—COA, 20.9 degrees; AOB, 37.4 degrees; and BOC, 31.7 degrees—and then to score the cardboard discs with fold lines in such a manner that the four lines CO will be negatively outfolded, while the lines AO and BO will be positively infolded, so that when they are altogether folded they will form four similar-arc-edged tetrahedra ABCO with all of their four CO radii edges centrally congruent. And when 30 of these folded great-circle sets of four T Quanta Module tetrahedra are each triple-bonded together, they will altogether constitute a sphere. This spherical assemblage involves pairings of the three intercongruent interface triangles AOC, COB, and BOA; that is, each folded great-circle set of four tetra has each of its four internal triangular faces congruent with their adjacent neighbor’s corresponding AOC, COB, and BOC interior triangular faces. (See Fig. 986.502.)

986.503

I proceeded to make 30 of these 360-degree-folding assemblies and used bobby pins to lock the four CO edges together at the C centers of the diamond-shaped outer faces. Then I used bobby pins again to lock the 30 assemblies together at the 20 convergent A vertexes and the 12 convergent B sphere-surface vertexes. Altogether they made a bigger sphere than the calculated radius, because of the accumulated thickness of the foldings of the construction paper’s double-walled (trivalent) interfacing of the 30 internal tetrahedral components. (See Fig. 986.502D.)

986.504

Fig. 986.504

Fig. 986.504 Profile of Quadrants of Sphere and Rhombic Triacontahedron: Central angles and ratios of radii are indicated at A. Orientation of modules in spherical assembly is indicated by letters at B.

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Instead of the just previously described 30 assemblies of four identical spherically central tetrahedra, each with all of their 62 vertexes in the unit-radius spheres, I next decided to make separately the 120 correspondingly convergent (non-arc-edged but chorded) tetrahedra of the tetravolume-5 rhombic triacontahedron, with its 30 flat ABAB diamond faces, the center C of which outer diamond faces is criss-crossed at right angles at C by the short axis A-A of the diamond and by its long axis B-B, all of which diamond bounding and criss-crossing is accomplished by the same 15 greatcircle planes that also described the 30 diamonds’ outer boundaries. As noted, the criss-crossed centers of the diamond faces occur at C, and all the C points are at the prime-vector-radius distance outwardly from the volumetric center O of the rhombic triacontahedron, while OA is 1.07 of vector unity and OB is 1.17 of vector unity outward, respectively, from the rhombic triacontahedron’s symmetrical system’s center of volume O. (See Figs. 986.504A and 986.504B.)

986.505

Fig. 986.505

Fig. 986.505 Six Intertangent Great-circle Discs in 12-inch Module Grid: The four 90 degree quadrants are folded at the central angles indicated for the T Quanta Module.

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To make my 120 OABC tetrahedra I happened to be using the same construction paperboard I had used before in making the 30 arc-edged great-circle components. The construction paperboard happened to come in sheets 24 by 36 inches, i.e., two feet by three feet. In making the previously described spherical triacontahedron out of these 24-by-36-inch sheets, I had decided to get the most out of my material by using a 12-inch-diameter circle, so that I could lay out six of them tangentially within the six 12-inch-square modules of the paperboard to produce the 30 foldable great circles. This allowed me to cut out six intertangent great circles from each 24-by-36-inch construction paper sheet. Thirty great circles required only five sheets, each sheet producing six circles. To make the 12 separate T Quanta Module tetrahedra, I again spontaneously divided each of the same-size sheets into six squares with each of the six circles tangent to four edges of each square (Fig. 986.505).

986.506

In starting to make the 120 separate tetrahedra (60 positive, 60 negative—known as T Quanta Modules) with which to assemble the triacontahedron- which is a chord-edged polyhedron vs the previous “spherical” form produced by the folded 15-great-circle patterning—I drew the same 12-inch-edge squares and, tangentially within the latter, drew the same six 12-inch-diameter circles on the five 24-by-36-inch sheets, dividing each circle into four quadrants and each quadrant into three subsections of 20.9 degrees, 37.4 degrees, and 31.7 degrees, as in the T Quanta Modules.

986.507

I planned that each of the quadrants would subsequently be cut from the others to be folded into one each of the 120 T Quanta Module tetrahedra of the triacontahedron. This time, however, I reminded myself not only to produce the rhombic triacontahedron with the same central angles as in the previous spheric experiment’s model, but also to provide this time for surfacing their clusters of four tetrahedra ABCO around their surface point C at the mid-crossing point of their 30 flat diamond faces. Flat diamond faces meant that where the sets of four tetra came together at C, there would not only have to be four 90-degree angles on the flat surface, but there would be eight internal right angles at each of the internal flange angles. This meant that around each vertex C corner of each of the four T Quanta Modules OABC coming together at the diamond face center C there would have to be three 90-degree angles.

986.508

Fig. 986.508 Six Intertangent Great-circle Discs

Fig. 986.508 Six Intertangent Great-circle Discs: Twelve-inch module grids divided into 24 quadrant blanks at A Profile of rhombic triacontahedron superimposed on quadrant at B.

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Looking at my “one-circle-per-each-of-six-squares” drawing, I saw that each sheet was divided into 24 quadrant blanks, as in Fig. 986.508A. Next I marked the centers of each of the six circles as point O, O being the volumetric center of the triacontahedral system. Then I realized that, as trigonometrically calculated, the flat, diamond-centered, right-angled, centrally criss-crossed point C of the triacontahedron’s outer faces had to be at our primitive unit-vector-length distance outwardly from the system center O, whereas in the previous arc-edged 30-great-circle-folded model the outer vertex C had been at full- spherical-system-radius distance outwardly from O. In the spherical 15-great-circle-model, therefore, the triacontahedron’s mid-flat-diamond-face C would be at 0.07 lesser radial distance outwardly from O than would the diamond corner vertexes A and vertex A itself at a lesser radial distance outwardly from O than diamond corner vertex B. (See Fig. 986.504A.)

986.509

Thinking about the C corner of the described tetrahedron consisting entirely of 90-degree angles as noted above, I realized that the line C to A must produce a 90- degree-angle as projected upon the line OC”, which latter ran vertically outward from O to C”, with O being the volumetric center of the symmetrical system (in this case the rhombic triacontahedron) and with C” positioned on the perimeter exactly where vertex C had occurred on each of the previous arc-described models of the great circles as I had laid them out for my previous 15 great-circle spherical models. I saw that angle ACO must be 90 degrees. I also knew by spherical trigonometry that the angle AOC would have to be 20.9 degrees, so I projected line OA outwardly from O at 20.9 degrees from the vertical square edge OC.

986.510

At the time of calculating the initial layout I made two mistaken assumptions: first, that the 0.9995 figure was critically approximate to 1 and could be read as 1; and second (despite Chris Kitrick’s skepticism born of his confidence in the reliability of his calculations), that the 0.0005 difference must be due to the residual incommensurability error of the inherent irrationality of the mathematicians’ method of calculating trigonometric functions. (See the Scheherazade Numbers discussed at Sec. 1230.) At any rate I could not lay out with drafting tools a difference of 0.0005 of six inches, which is 0.0030 of an inch. No draftsman can prick off a distance even ten times that size. (I continue to belabor these mistaken assumptions and the subsequent acknowledgments of the errors because it is always upon the occasion of my enlightened admission of error that I make my greatest discoveries, and I am thus eager to convey this truth to those seeking the truth by following closely each step of this development, which leads to one of the most exciting of known discoveries.)

986.511

In order to produce the biggest model possible out of the same 24-by-36- inch construction paper blanks, I saw that vertex A of this new T Quanta Module model would have to lie on the same 12-inch circle, projecting horizontally from A perpendicularly (i.e., at right angles), upon OX at C. I found that the point of 90-degree impingement of AC on OX occurred slightly inward (0.041, as we learned later by/trigonometry), vertically inward, from X. The symbol X now occurs on my layout at the point where the previous spherical model’s central diamond vertex C had been positioned—-on the great-circle perimeter. Trigonometric calculation showed this distance between C and X to be 0.041 of the length of our unit vector radius. Because (1) the distance CO is established by the right-angled projection of A upon OX; and because (2) the length CO is also the prime vector of synergetics’ isotropic vector matrix itself, we found by trigonometric calculation that when the distance from O to C is 0.9995 of the prime vector’s length, that the tetravolume of the rhombic triacontahedron is exactly 5.

986.512

When the distance from O to C is 0.9995, then the tetravolume of the rhombic triacontahedron is exactly 5. OC in our model layout is now exactly the same as the vector radius of the isotropic vector matrix of our “generalized energy field.” OC rises vertically (as the right-hand edge of our cut-out model of our eventually-to-be-folded T Quanta Module’s model designing layout) from the eventual triacontahedron’s center O to what will be the mid-diamond face point C. Because by spherical trigonometry we know that the central angles of our model must read successively from the right-hand edge of the layout at 20.9 degrees, 37.4 degrees, and 31.7 degrees and that they add up to 90 degrees, therefore line OC’ runs horizontally leftward, outward from O to make angle COC’ 90 degrees. This is because all the angles around the mid-diamond criss-cross point C are (both externally and internally) 90 degrees. We also know that horizontal OC’ is the same prime vector length as vertical OC. We also know that in subsequent folding into the T Quanta Module tetrahedron, it is a mathematical requirement that vertical OC be congruent with horizontal OC’ in order to be able to have these edges fold together to be closed in the interior tetrahedral form of the T Quanta Module. We also know that in order to produce the required three 90-degree angles (one surface and two interior) around congruent C and C’ of the finished T Quanta Module, the line C’B of our layout must rise at 90 degrees vertically from C’ at the leftward end of the horizontal unit vector radius OC’. (See Fig. 986.508C.)

986.513

This layout now demonstrates three 90-degree comers with lines OC vertical and OC’ horizontal and of the same exact length, which means that the rectangle COC’C” must be a square with unit-vector-radius edge length OC. The vertical line C’C” rises from C’ of horizontal OC’ until it encounters line OB, which—to conform with the triacontahedron’s interior angles as already trigonometrically established—must by angular construction layout run outwardly from O at an angle of 31.7 degrees above the horizontal from OC’ until it engages vertical C’C” at B. Because by deliberate construction requirement the angle between vertical OC and OA has been laid out as 20.9 degrees, the angle AOB must be 37.4 degrees-being the remainder after deducting both 20.9 degrees and 31.7 degrees from the 90-degree angle Lying between vertical OC and horizontal OC’. All of this construction layout with OC’ horizontally equaling OC vertically, and with the thus-far-constructed layout’s corner angles each being 90 degrees, makes it evident that the extensions of lines CA and C’B will intersect at 90 degrees at point C”, thus completing the square OC `C”C of edge length OC, which length is exactly 0.999483332 of the prime vector of the isotropic vector matrix’s primitive cosmic- hierarchy system.

986.514

Since ACO, COC’, and OC’B are all 90-degree angles, and since vertical CO = horizontal C’O in length, the area COC’C” must be a square. This means that two edges of each of three of the four triangular faces of the T Quanta Module tetrahedron, and six of its nine prefolded edges (it has only six edges after folding), are congruent with an exactly square paperboard blank. The three triangles OCA, OAB, and OBC’ will be folded inwardly along AO and BO to bring the two CO and CO’ edges together to produce the three systemically interior faces of the T Quanta Module.

986.515

Fig. 986.515

Fig. 986.515 T Quanta Module Foldable from Square: One of the triangular corners may be hinged and reoriented to close the open end of the folded tetrahedron.

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This construction method leaves a fourth right-triangular corner piece AC”B, which the dividers indicated-and subsequent trigonometry confirmed—to be the triangle exactly fitting the outer ABC-triangular-shaped open end of the folded-together T Quanta Module OABC. O” marks the fourth corner of the square blank, and trigonometry showed that C”A = C’B and C”B = AC, while AB of triangle OBA by construction is congruent with AB of triangle AC”B of the original layout. So it is proven that the vector- edged square COC’C” exactly equals the surface of the T Quanta Module tetrahedron CABO. (See Fig. 986.515.)

986.516

The triangle AC”B is hinged to the T Quanta Module along the mutual edge AB, which is the hypotenuse of the small AC”B right triangle. But as constructed the small right triangle AC”B cannot be hinged (folded) to close the T Quanta Module tetrahedron’s open-end triangular area ABC—despite the fact that the hinged-on triangle AC”B and the open triangle ABC are dimensionally identical. AC”B is exactly the right shape and size and area and can be used to exactly close the outer face of the T Quanta Module tetrahedron, if—but only if—it is cut off along line BA and is then turned over so that its faces are reversed and its B corner is now where its A corner had been. This is to say that if the square COC’C” is made of a cardboard sheet with a red top side and a gray underside, when we complete the tetrahedron folding as previously described, cut off the small corner triangle AC”B along line BA, reverse its face and its acute ends, and then address it to the small triangular ABC open end of the tetrahedron CABO, it will fit exactly into place, but with the completed tetrahedron having three gray faces around vertex O and one red outer face CAB. (See Fig. 986.508C.)

986.517

Following this closure procedure, when the AC”B triangles of each of the squares are cut off from COC’C” along line AB, and right triangle AC”B is reversed in face and its right-angle corner C” is made congruent with the right-angle corner C of the T Quanta Module’s open-end triangle, then the B corner of the small triangle goes into congruence with the A corner of the open-end triangle, and the A corner of the small triangle goes into congruence with the B corner of the open-end triangle—with the 90- degree corner C becoming congruent with the small triangle’s right-angle corner C”. When all 120 of these T Quanta Module tetrahedra are closed and assembled to produce the triacontahedron, we will have all of the 360 gray faces inside and all of the 120 red faces outside, altogether producing an externally red and an internally gray rhombic triacontahedron.

986.518

In developing the paper-folding pattern with which to construct any one of these 120 identical T Quanta Module tetrahedra, we inadvertently discovered it to be foldable out of an exact square of construction paper, the edge of which square is almost (0.9995 of the prime vector 1) identical in length to that of the prime vector radius of synergetics’ closest-packed unit-radius spheres, and of the isotropic vector matrix, and therefore of the radii and chords of the vector equilibrium—which synergetics’ vector (as with all vectors) is the product of mass and velocity. While the unit-vector length of our everywhere-the-same energy condition conceptually idealizes cosmic equilibrium, as prime vector (Sec. 540.10) it also inherently represents everywhere-the-same maximum cosmic velocity unfettered in vacuo—ergo, its linear velocity (symbolized in physics as lower-case c) is that of all radiation—whether beamed or piped or linearly focused—the velocity of whose unbeamed, omnidirectionally outward, surface growth rate always amounts to the second-powering of the linear speed. Ergo, omniradiance’s wave surface growth rate is c².

986.519

Since the edge length of the exactly 5.0000 (alpha) volumed T Quanta Module surface square is 0.9995 of the prime vector 1.0000 (alpha), the surface-field energy of the T Quanta Module of minimum energy containment is 0.9995 V² , where 1.0000 (alpha) V is the prime vector of our isotropic vector matrix. The difference—0.0005—is minimal but not insignificant; for instance, the mass of the electron happens also to be 0.0005 of the mass of the proton.

986.520 Einstein’s Equation

986.521

Remembering that in any given dimensional system of reference the vector’s length represents a given mass multiplied by a given velocity, we have in the present instance the physical evidence that the surface area of the T Quanta Module tetrahedron exactly equals the area of the edge length—0.9995—“squared.” In this case of the T Quanta Module the edge length of 0.9995 of the foldable square (the visibly undetectable) is 0.0005 less than the length of the prime vector of 1.000.

986.522

The generalized isotropic vector matrix’s prime vector to the second power—“squared”— becomes physically visible in the folded-square T tetra modules. (Try making one of them yourself.) This visible “squaring” of the surface area of the exactly one-energy-quantum module tetrahedron corresponds geometrically to what is symbolically called for in Einstein’s equation, which language physics uses as a nonengineering-language symbolism (as with conventional mathematics), and which does not preintermultiply mass and velocity to produce a vector of given length and angular direction-ergo, does not employ the integrated vectorial component VE—ergo, must express V² in separate components as M (mass) times the velocity of energy unfettered in vacuo to the second power, c². However, we can say Mc² = V², the engineering expression V² being more economical. When T = the T Quanta Module, and when the T Quanta Module = one energy quantum module, we can say: $\text{one module}=0.99952$

986}.523

In the Einstein equation the velocity—lower-case c—of all radiation taken to the second power is omnidirectional-ergo, its quasispheric surface-growth rate is at the second power of its radial-linear-arithmetic growth rate—ergo, c². (Compare Secs. 1052.21 and 1052.30.) Thus Einstein’s equation reads E = Mc², where E is the basic one quantum or one photon energy component of Universe.

986.524

With all the foregoing holding true and being physically demonstrable, we find the vector minus 0.0005 of its full length producing an exactly square area that folds into a tetrahedron of exactly one quantum module, but, we must remember, with a unit- integral-square-surface area whose edge length is 0.0005 less than the true V² vector, i.e., less than Mc². But don’t get discouraged; as with the French Vive la Différence, we find that difference of 0.0005 to be of the greatest possible significance … as we shall immediately learn.

986.540 Volume-surface Ratios of E Quanta Module and Other Modules

986.541

Now, reviewing and consolidating our physically exploratory gains, we note that in addition to the 0.9995 V²-edged “square”-surfaced T Quanta Module tetrahedron of exactly the same volume as the A, B, C, or D Quanta Modules, we also have the E Quanta Module—or the “Einstein Module” —whose square edge is exactly vector V = 1.0000 (alpha), but whose volume is 1.001551606 when the A Quanta Module’s volume is exactly 1.0000 (alpha), which volume we have also learned is uncontainable by chemical structuring, bonding, and the mass-attraction law.

986.542

When the prime-unit vector constitutes the radial distance outward from the triacontahedron’s volumetric center O to the mid-points C of each of its mid-diamond faces, the volume of the rhombic triacontahedron is then slightly greater than tetravolume 5, being actually tetravolume 5.007758031. Each of the rhombic triacontahedron’s 120 internally structured tetrahedra is called an E Quanta Module, the “E” for Einstein, being the transformation threshold between energy convergently self-interfering as matter = M, and energy divergently dispersed as radiation = c². Let us consider two rhombic triacontahedra: (1) one of radius 0.9995 V of exact tetravolume 5; and (2) one of radius 1.0000 (alpha) of tetravolume 5.007758031. The exact prime-vector radius 1.0000 (alpha) rhombic triacontahedron volume is 0.007758031 (1/129th) greater than the tetravolume 5—i.e., tetravolume 5.007758031. This means that each E Quanta Module is 1.001551606 when the A Quanta Module is 1.0000.

986.543

The 0.000517 radius difference between the 0.999483-radiused rhombic triacontahedron of exactly tetravolume 5 and its exquisitely minute greater radius-1.0000 (alpha) prime vector, is the exquisite difference between a local-in-Universe energy-containing module and that same energy being released to become energy radiant. Each of the 120 right-angle-cornered T Quanta Modules embraced by the tetravolume-5 rhombic triacontahedron is volumetrically identical to the A and B Quanta Modules, of which the A Modules hold their energy and the B Modules release their energy (Sec. 920). Each quanta module volume is 0.04166—i.e., 1/24 of one regular primitive tetrahedron, the latter we recall being the minimum symmetric structural system of Universe. To avoid decimal fractions that are not conceptually simple, we multiply all the primitive hierarchy of symmetric, concentric, polyhedral volumes by 24—after which we can discuss and consider energetic-synergetic geometry in always-whole-rational-integer terms.

986.544

We have not forgotten that radius I is only half of the prime-unit vector of the isotropic vector matrix, which equals unity 2 (Sec. 986.160). Nor have we forgotten that every square is two triangles (Sec. 420.08); nor that the second-powering of integers is most economically readable as “triangling”; nor that nature always employs the most economical alternatives—but we know that it is momentarily too distracting to bring in these adjustments of the Einstein formula at this point.

986.545

To discover the significance of the “difference” all we have to do is make another square with edge length of exactly 1.000 (alpha) (a difference completely invisible at our one-foot-to-the-edge modeling scale), and now our tetrahedron folded out of the model is an exact geometrical model of Einstein’s E = Mc², which, expressed in vectorial engineering terms, reads E = V² ; however, its volume is now 0.000060953 greater than that of one exact energy quanta module. We call this tetrahedron model folded from one square whose four edge lengths are each exactly one vector long the E Module, naming it for Einstein. It is an exact vector model of his equation.

986.546

The volumetric difference between the T Module and the E Module is the difference between energy-as-matter and energy-as-radiation. The linear growth of 0.0005 transforms the basic energy-conserving quanta module (the physicists’ particle) from matter into one minimum-limit “photon” of radiant energy as light or any other radiation (the physicists’ wave).

986.547

Einstein’s equation was conceived and calculated by him to identify the energy characteristics derived from physical experiment, which defined the minimum radiation unit—the photon—E = Mc². The relative linear difference of 0.000518 multiplied by the atoms’ electrons’ nucleus-orbiting diameter of one angstrom (a unit on only l/40-millionth of an inch) is the difference between it is matter or it is radiation… Vastly enlarged, it is the same kind of difference existing between a soap bubble existing and no longer existing—“bursting,” we call it—because it reached the critical limit of spontaneously coexistent, cohesive energy as-atoms-arrayed-in-liquid molecules and of atoms rearranged in dispersive behavior as gases. This is the generalized critical threshold between it is and it isn’t… It is the same volume-to-tensional-surface-enclosing-capability condition displayed by the soap bubble, with its volume increasing at a velocity of the third power while its surface increases only as velocity to the second power. Its tension- embracement of molecules and their atoms gets thinned out to a one-molecule layer, after which the atoms, behaving according to Newton’s mass-interattraction law, become circumferentially parted, with their interattractiveness decreasing acceleratingly at a second-power rate of the progressive arithmetical distance apart attained—an increase that suddenly attains critical demass point, and there is no longer a bubble. The same principle obtains in respect to the T Quanta Module → E Quanta Module—i.e., matter transforming into radiation.

986.548

The difference between the edge length of the square from which we fold the E Quanta Module and the edge length of the square from which we fold the T Quanta Module is exquisitely minute: it is the difference between the inside surface and the outside surface of the material employed to fabricate the model. In a 20-inch-square model employing aluminum foil

1/200th of an inch thick, the E Module would be congruent with the outside surface and the T Module would be congruent with the inside surface, and the ratio of the edge lengths of the two squares is as 1 is to 0.0005, or 0.0005 of prime vector radius of our spherical transformation. This minuscule modelable difference is the difference between it is and it isn’t—which is to say that the dimensional difference between matter and radiation is probably the most minute of all nature’s dimensioning: it is the difference between inside-out and outside-out of positive and negative Universe.

986.549

Because we have obtained an intimate glimpse of matter becoming radiation, or vice versa, as caused by a minimum-structural-system tetrahedron’s edge-length growth of only 129 quadrillionths of an inch, and because we have been paying faithful attention to the most minute fractions of difference, we have been introduced to a whole new frontier of synergetics exploration. We have discovered the conceptual means by which the 99 percent of humanity who do not understand science may become much more intimate with nature’s energetic behaviors, transformations, capabilities, and structural and de-structural strategies.

986.550

Table: Relative Surface Areas Embracing the Hierarchy of Energetic Quanta Modules: Volumes are unit. All Module Volumes are 1, except the radiant E Module, whose Surface Area is experimentally evidenced Unity:

$$\begin{array}{c}\text{ENERGY PACKAGE / SURFACE AREA}\\ \mathrm{V}=\text{Vector (linear)}\\ \mathrm{V}=\text{Mass}\times \text{velocity}=\text{Energy Package}\\ {\mathrm{V}}^2=\text{Energy package’s surface}\end{array}$$

1 Unit vector of isotropic vector matrix

Vector × Vector = Surface (Energy as local energy system-containment capability)
= Outer array of energy packages.⁶

Mass = F = Relative frequency of primitive-system-subdivision energy-event occupation.

Module	SURFACE	AREA   VOLUME
A Quanta Module	0.9957819158	1 HOLD
T Quanta Module	0.9989669317	1 ENERGY
”Einstein” E Module	1.0000000000	1.00155
B Quanta Module	1.207106781
C Quanta Module	1.530556591
D Quanta Module	1.896581995
A’ Module	2.280238966
B’ Module	2.672519302	1 RELEASE
C’ Module	3.069597104	1 ENERGY
D’ Module	3.469603759
A” Module	3.871525253	1
B” Module	4.27476567	1
C” Module	4.678952488	1
D” Module	5.083841106	1

(For a discussion of C and D Modules see Sec. 986.413.)

(Footnote 6: The VE surface displays the number of closest-packed spheres of the outer layer. That surface = f²; ergo, the number of energy-package spheres in outer layer shell = surface, there being no continuum or solids.)

986.560 Surprise Nestability of Minimod T into Maximod T

986.561

Fig. 986.561 T and E Modules - Minimod Nestabilities

Fig. 986.561 T and E Modules: Minimod Nestabilities: Ratios of Angles and Edges: The top face remains open: the triangular lid will not close, but may be broken off and folded into smaller successive minimod tetra without limit.

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The 6 + 10 + 15 = 31 great circles of icosahedral symmetries (Fig. 901.03) produce the spherical-surface right triangle AC”B; CAB is subdivisible into four spherical right triangles CDA, CDE, DFE, and EFB. Since there are 120 CAB triangles, there are 480 subdivision-right-surface triangles. Among these subdivision-right triangles there are two back-to-back 90-degree surface angles at D—CDA and CDE—and two back-to-back degree surface angles at F—CFE and EFB. The surface chord DE of the central angle DOE is identical in magnitude to the surface chord EB of the central angle EOB, both being 13.28 degrees of circular azimuth. Surface chord FB of central angle FOB and surface chord AD of central angle AOD are identical in magnitude, both being 10.8 degrees azimuth. In the same manner we find that surface chord EF of central angle EOF constitutes the mutual edge of the two surface right triangles CFE and BFE, the central- angle magnitude of EOF being 7.77 degrees azimuth. Likewise, the central angles COA and COF of the surface chords CA and CF are of the same magnitude, 20.9 degrees. All the above data suggest a surprising possibility: that the small corner triangle AC”B itself can be folded on its three internal chord lines CD, CE, and EF, while joining its two edges AC and CF, which are of equal magnitude, having central angles of 20.9 degrees. This folding and joining of F to A and of B to D cancels out the congruent-letter identities F and D to produce the tetrahedron ABEC. (See Fig. 986.561.)

986.562

We find to our surprise that this little flange-foldable tetrahedron is an identically angled miniature of the T Quanta Module OABCt and that it can fit elegantly into the identically angled space terminating at O within the inner reaches of vacant OABC, with the miniature tetrahedron’s corner C becoming congruent with the system’s center O. The volume of the Minimod T is approximately 1/18 that of the Maximod T Quanta Module or of the A or B Modules.

986.570 Range of Modular Orientations

986.571

Now we return to Consideration 13 of this discussion and its discovery of the surface-to-central-angle interexchanging wave succession manifest in the cosmic hierarchy of ever-more-complex, primary structured polyhedra—an interchanging of inside-out characteristics that inherently produces positive-negative world conditions; ergo, it propagates—inside-to-outside-to-in—pulsed frequencies. With this kind of self- propagative regenerative function in view, we now consider exploring some of the implications of the fact that the triangle C’AB is foldable into the E Quanta Module and is also nestable into the T Quanta Module, which produces many possibilities:

1. The triangle AC’B will disconnect and reverse its faces and complete the enclosure of the T Quanta Module tetrahedron.
2. The 120 T Quanta Modules, by additional tension-induced twist, take the AC”B triangles AB ends end-for-end to produce the additional radius outwardly from O to convert the T Quanta Modules into “Einstein” E Quanta Modules, thus radiantly exporting all 120 modules as photons of light or other radiation.
3. The triangle AC”B might disconnect altogether, fold itself into the miniature T Quanta Module, and plunge inwardly to fill its angularly matching central tetrahedral vacancy.
4. The outer triangle may just stay mishinged and flapping, to leave the tetrahedron’s outer end open.
5. The outer triangle might come loose, fold itself into a miniature T Quanta Module, and leave the system.
6. The 120 miniature T Quanta Modules might fly away independently__as, for instance, cosmic rays, i.e., as minimum modular fractions of primitive systems.
7. All 120 of these escaping miniature T Quanta Modules could reassemble themselves into a miniature 1/120 triacontahedron, each of whose miniature T Module’s outer faces could fold into mini-mini T Modules and plunge inwardly in ever-more-concentrating demonstration of implosion, ad infinitum.

There are 229,920 other possibilities that any one of any other number of the 120 individual T Module tetrahedra could behave in any of the foregoing seven alternate ways in that vast variety of combinations and frequencies. At this borderline of ultrahigh frequency of intertransformability between matter and electromagnetic radiation we gain comprehension of how stars and fleas may be designed and be born.

986.580 Consideration 15: Surface Constancy and Mass Discrepancy

986.581

Those AC”B triangles appear in the upper left-hand corner of either the T Module’s or the E Module’s square areas COC’C”, one of which has the edge length 0.994 V and the other the edge length of 1.0000 (alpha) V. Regardless of what those AC”B triangles may or may not do, their AC”B areas, together with the areas of the triangles ACO, ABO, and BCO, exactly constitute the total surface area of either the T Module or the E Module.

Surface of T Module =.994 V²

Surface of E Module = 1.00000 (alpha) V²

986.582

The outer triangle AC”B of the T Quanta Module is an inherent energy conserver because of its foldability into one (minimum-something) tetrahedron. When it folds itself into a miniature T Module with the other 119 T Modules as a surface-closed rhombic triacontahedron, the latter will be a powerful energy conserver—perhaps reminiscent of the giant-to-dwarf-Star behavior. The miniature T Module behavior is also similar to behaviors of the electron’s self-conservation. This self-conserving and self- contracting property of the T Quanta Modules, whose volume energy (ergo, energy quantum) is identical to that of the A and B Modules, provides speculative consideration as to why and how electron mass happens to be only 1/1836 the mass of the proton.

986.583

Certain it is that the T Quanta Module → E Quanta Module threshold transformation makes it clear how energy goes from matter to radiation, and it may be that our little corner triangle AC”B is telling us how radiation retransforms into matter.

986.584

The volume of the T Quanta Module is identical with the volumes of the A and B Quanta Modules, which latter we have been able to identify with the quarks because of their clustering in the cosmically minimum, allspace-filling three-module Mites as A +, A -, and B, with both A’s holding their energy charges and B discharging its energy in exact correspondence with the quark grouping and energy-holding-and-releasing properties, with the A Modules’ energy-holding capabilities being based on their foldability from only one triangle, within which triangle the reflection patterning guarantees the energy conserving. (See Secs. 921 and 986.414)

986.585

As we study the hierarchy of the surface areas of constant volume 1 and their respective shapes, we start with the least-surface A Quanta Module which is folded out of one whole triangle, and we find that no other triangle is enclosed by one triangle except at the top of the hierarchy, where in the upper left-hand corner we find our Minimod T or Minimod E tetrahedron foldable out of our little triangle AC”B, whose fold-line patterning is similar to that of the triangle from which the A Quanta Module is folded. In between the whole foldable triangular blank of the A Quanta Module and the whole foldable triangular blank of the Minimod T or Minimod E, we have a series of only asymmetrical folding blanks-until we come to the beautiful squares of the T and E Quanta Modules, which occur just before we come to the triangles of the minimod tetrahedra, which suggests that we go from radiation to matter with the foldable triangle and from matter to radiation when we get to the squares (which are, of course, two triangles).

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986.600 Surface-Volume Ratios in the Atomic Theater

986.610 Considerations, Recalls, and Discoveries

986.611

Our inventory of considerations, recalls, and discoveries is now burgeoning but remains omniinterrelevant. Wherefore we continue recalling and reconsidering with a high probability factor that we will make further discovery based on our past experience.

986.620 Demass Breakpoint Model of Macrotude-microtude Difference Between Matter and Radiation

986.621

Let me here introduce a physical experiment that will give us a personal feeling of appreciation of the importance to all humanity of all humanity’s being able to see with its own eyes what Einstein’s equation represents—the breakpoint between matter and radiation (critical mass and atomic-energy release)—and above all to give all humanity experienceable, knowable access to all that science has ever discovered regarding Universe, plus much more than science has ever discovered. With all this experienceability of most advanced scientific discovery all humanity will come to appreciate the otherwise utterly incredible exquisiteness of mathematical exactitude with which Universe (which is nature) functions.

986.622

What we employ for such self-instruction at a human-sense-detectable level to appreciate the meager difference between the “T” square’s 0.9995 edge length and the “E” square’s 1.00000 (alpha) edge length is to perform the physical task of producing two squares, which the human eyes can see and fingers can feel are of different sizes. Unaided by a lens, only the most skilled human eyes can see something that is one one-hundredth of an inch (expressed as 0.01 inch). A carpenter works at no finer than 1/32nd of an inch. To make a difference of 0.0005 undeniably visible to any average human we would have to use the popularly adopted 1/16th of an inch, which is that of the common school ruler. This 1/16th of an inch is expressed decimally as 0.0625. To make 0.005 of an inch visible we multiply it by 100, which makes it 0.05. One hundred inches is eight and a half feet—the average room-ceiling height. If we make two squares with 100-inch edges (8 l/2 feet “square”) out of wooden planks and timber, we cannot hold their dimensions to such a close tolerance of error because the humidity and temperature variations will be greater than 0.05 inch. Even if we make the 8 l/2-foot squares of steel and aluminum plate, the expansion and contraction under common weather temperature changes will be greater than 0.05 inch.

986.623

Using machine tools machinists can “dress” their products to tolerances as fine as 0.0001 inch.

986.624

Fiberglass-and-epoxy resin is the substance that has the minimum presently known temperature-and-humidity-caused expansion and contraction rates of all practically producible materials. Wherefore: two square plates two inches thick with edge lengths of 8 l/2 feet could be machine-tool “dressed” and placed vertically face to face in a temperature-controlled slot with one of each of both of their bottom innermost 90-degree corners jammed tightly into a “machined” corner slot, which would then make it possible to “see” with human eyes the difference in square size between the “T” and the “E” squares.

986.625

Even if we “machined” two steel cubes with an edge-length difference of .0005 inch, they would stack one on top of the other with their two vertical surfaces appearing as a polished continuum—the space between them being also subvisible.

986.626

But nature’s energy-as-matter transformed into energy-as-radiation are operations conducted at a size scale far different from our experientially imagined experiments. Nature operates her matter-to-radiation energy exchanging at the atomic level. The nucleus of the atom is where energy-as-matter is self-interferingly knotted together in most primitive polyhedral-patterning event systems. The atomic nucleus diameter is 1/100,000 the diameter of its electron-orbited domain—which domain is spoken of by scientists as “the atom.” One atomic diameter is called the angstrom and is the prime measurement unit of the physicists—macrophysicists or microphysicists, astro or nuclear, as they might well be designated.

986.627

Referring to those two 8 l/2-foot (the size of Barnum’s circus human giant) in height and 2-inches-thick square plates of machine-dressed fiberglass-epoxy resin and their minimum-human-sense-detectability difference of dimension, we find that the angstrom-atomic theater of energy-exchanging performance is only 1/126,500,000,000 the size of the minimum average human sense detectability. This figure, put into human- experience-sensing terms, is the distance that a photon of light expanding radially at 186,000 miles per second will travel-reach between the time humans are born and the time they reach their nineteenth birthday.

986.628

What is important for us to realize here is that synergetics mathematics, beginning with the most primitive hierarchy of min-max geometrical relationships, expresses relationships that exist independently of time-size. So we humans can think intimately about structural principles of any size. These primitive structural principles disclose inherent geometrical nuclei in respect to which all of Universe’s convergent- divergent, gravitational and radiational, contracting into matter and expanding into electromagnetics, and vice versa, together with their terminal angular and frequency knotting and unknotting events comprehensively and comprehendingly occur. And since the sum-total of both macro- and micro-physical science evidences 100-percent conservation of the energy of eternally regenerative Scenario Universe, each smallest differential fraction is of infinite importance to the integrity of Universe.

986.629

And since Physical Universe demonstrates the principle of least effort, i.e., maximum efficiency to be infallibly operative, Universe does the most important tasks in the most exquisite manner; ergo, it is in the most exquisitely minute fractions that she hides her most important secrets. For all the foregoing synergetics’ disclosure of a means of comprehending and operating independently of size provides human mind with not only a cosmic advantage but with all the responsibility such a cosmic decision to invest such an advantage in us implies. With these thoughts we address ourselves now to considering not only the critical cosmic surface-volume relationships but also their unique behavior differentials.

986.630 Interkinetic Limits

986.631

In a structural system’s interbalancing of compression and tension forces the tensed components will always embrace the compression components—as does gravity always comprehensively embrace all radiation—ergo, tension is always outermost of all systems, macrocosmic or microcosmic.

986.632

Take any bendable substance and bend it. As you do so, the outer part of the bend stretches and the inner part compresses. Tension always has the greater radius-ergo, leverage advantage—ergo, gravity is always comprehensive of radiation (Compare Sec. 1051.50)

986.633

In experiential structural reality the so-called sphere is always and only an ultra-high-frequency geodesic polyhedron; ergo, it is always chord-circumferenced and chord-convergent-vertexed rather than arc-circumferenced and arc-vertexed; ergo, it is always and only quasispherical, which quasispherical structural form is experimentally demonstrable as enclosing the most volume with the least surface of any and all symmetrical, equiangular, structural systems. Because of the foregoing we find it desirable to rename the spheric experience, using from now on the word spheric in lieu of the nonexistent, experimentally nondemonstrable “sphere.”

986.634

As an asymmetrical or polarized structural system, the hemispheric-ended cylinder has the same surface-to-volume ratio as that of a sphere with an identical diameter—the latter cylinders as well as their hemispherical terminals consist structurally only of high-frequency, triangularly chorded structures. The spheric and the hemispheric- terminalled cylinders alike contain the most volume with the least surface of all symmetrical polyhedra. At the other extreme of the surface-to-volume ratio, the equiangular tetrahedron encloses the least volume with the most surface of any and all omnisymmetrical structural systems. The more asymmetrical the tetrahedron, the more surface is required to envelop a given volume. It may be assumed, therefore, that with a given quantity of the same energy invested as molecularly structured, system-containing capability, it is less tensionally stressful to enclose a regular equiangular tetrahedron than it is to enclose any asymmetrical tetrahedron.

986.635

In respect to total surface areas of asymmetrical tetrahedra of unit (i.e., identical volume) enclosure, it is experimentally demonstrable that the greater the difference between the most acute angle and the most obtuse of its 12 surface angles, the greater the surface-to-volume ratio will be, and therefore the greater the tensional stressing of its outermost cohering components—ergo, the greater the challenge to the containment of its structural-system integrity. (See Sec. 923 and Fig. 923.10.) According to Newton’s law the mass interattraction of two separate bodies deteriorates exponentially as the distance apart decreases arithmetically; ergo, the relative interproximity of the atoms within any molecule, and the relative interproximity of the molecules as structurally interarrayed within any and all volume-containment systems—and the resultant structural- integrity coherences of those systems—trend acceleratingly toward their theoretical atom- and-molecule-interattractive-proximity limits. These chemical-structure-integrity limits are visibly demonstrated to the human eyes by the bursting of bubbles or of children’s overfilled balloons or of any other internally overpressured fluid-pneumatic, molecular- membraned containers when the membrane impinging and ricocheting interkinetic acceleration of an increasingly introduced population of contained gas molecules separates the molecules of the container membrane beyond their critical-proximity limits. These critical-atomic-and-molecular-proximity limits are mathematically and gravitationally similar to the proximity limits governing the velocity and distance outward from planet Earth’s surface at which a rocket-launched vehicle can maintain its orbit and not fall back into the Earth.

986.700 Spheric Nature of Electromagnetic Waves

986.701 Consideration 16: and Realization of Synergetic Significance

986.702

Since we have learned that nature’s second-powering is triangling and not squaring (Sec. 990), and since each square is always two similar triangles, we must express Einstein’s equation, where E is the product of M and c², as:
$\mathrm{E}=2{\mathrm{V}}^2$

986.710 Recapitulation of Geometry-and-energy Recalls

986.711

I must add to the inventory of only-synergetically-interrevealing significant discoveries of this chronicle a recapitulation of additional “recalls ”:

1. The absolute constancy of cheese polyhedra;
2. that the tetrahedron is the quantum of energy;
3. that the nonpolar vertexes of the polar-edge-”tuned” tetrahedron can connect any other two points in Universe;
4. that the unit-volume progression of quanta modules accounts for electromagnetic intertuning;
5. that the tetrahedron in turning itself inside-out accounts for electromagnetic- wave propagation;
6. that polyhedra should be reidentified as polyvertexia, the simplest of which is the tetravertex;
7. that the tetravertex is the simplest spheric system;
8. that the vector equilibrium provides a field for universal energy accommodation; and
9. that the vector equilibrium shell growth rate predicts the proton and neutron population of the elements.

986.720 Absolute Constancy: Cheese Polyhedra

986.721

My first observation of the polyhedral hierarchy was introduced in Sec. 223.64, Table 224.20, and Fig. 400.30. That hierarchy may be considered as cheese polyhedra in which there is an experimental redemonstrability of absolute constancy of areal, volumetric, topological, and symmetry characteristics, which constancy is exclusively unique to triangles and tetrahedra and is maintained despite any and all asymmetrical aberrations of those triangles and tetrahedra, as caused
— by perspective distortion;
— by interproportional variations of relative lengths and angles as manifest in isosceles, scalene, acute, or obtuse system aspects (see quadrangular versus triangular accounting in Figs. 990.01 and 100.301.);
— by truncatings parallel to triangle edges or parallel to tetrahedron faces; or
— by frequency modulations,
in contradistinction to complete loss of symmetry and topological constancy of all polygons other than the triangle and of all polyhedra other than the tetrahedron as caused by any special-case, time-size alterations or changes of the perspective point from which the observations of those systems are taken.

986.722

In connection with this same cheese tetrahedron recall we remember (1) that we could push in on the face A of the tetrahedron at a given rate of radial contraction of the system, while pulling out face B at a matching rate of radial expansion of the system, which “couple” of local alterations of the system left the tetrahedron unaltered in shape or size throughout the transformation (Sec. 623) and just as it was both before and after the “coupled” transformings took place, the only altered consequence of which was that the tetrahedron’s center of volume had migrated; and we remember (2) that we could also push in on the same tetrahedron’s face C and pull out on face D at a coupled rate other than the coupled rate of radial expansion and contraction of the A-B face-coupling’s intercomplementary transformings; by all of which we learn that the tetrahedron can accommodate two disparate rates of change without in any way altering its own size and only altering its center-of-volume positioning in respect to any other system components of the local Universe consideration. (See color plate 26.)

986.723

It must be noted, however, that because of the generalized nonsimultaneity of cosmic events, there exists an inherent lag between the pushing in of face A and the pulling out of face B, which induces an inherent interim wave-depression or a wave- breaking pulsating of the coupling functionings of the tetrahedron’s accommodation of transmission of two disparately frequenced energetic communications.

986.724

Second, I recall—as in Secs. 920.01 and 921.10 and Fig. 923.10 —that the tetrahedron is the quantum of energy.

986.725

Third, I recall that the single-tuned-length axis of the edge-axis-rotatable tetrahedron’s two nonaxis polar vertexes may be deployed to connect up with any two other points in Universe without altering the tetrahedron’s unit volume or its tuned-axis length. (See Sec. 961.30.)

986.726

Fig. 986.726

Constant-unit-volume Progressions of Asymmetric Tetrahedra: In this progression of ever- more-asymmetric tetrahedra, only the sixth edge remains constant. Tetrahedral wavelength and tuning permits any two point in Universe.

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Fourth, I recall that the tetrahedron’s 24 A Modules and the latter’s B, C, D; A, B, C’, D’; A”, B”, C”, D”…(alpha) (see Fig. 986.726, which is a detail and relabeling of Fig. 923.10B) together with the T and E Modules provide transformative significance of being the constant-unit-volume progression of ever-more-asymmetrically-transforming stages of the constant-unit-volume tetrahedra, with the uniform-stage transforming being provided by five of the six edges of each of the constant-volume tetrahedra being covaryingly and ever-progressively-disparately altered—with the sixth edge alone of each and all stages of the transformation remaining unaltered in frequency and wavelength magnitude. The concurrent
— constant-volume-and-wavelength transformings, and
— system rotating around and angular tilting of the constant, unaltered-in-length, sixth edge’s axial altitude in respect to the all-other-in-Universe experiences’ omniinterangular orientations, altogether both permit and accommodate any two other points X and Y in Universe being interconnected not only with one another, but also with the two points A and B that define the unaltered sixth edge AB of the constant-volume and constant-AB-edge-length, omni- Universe-interconnecting tetrahedron ABXY; all of which permits the constant sixth edge AB length to serve as the anywhere and anywhen in Universe to be established transceiver’s wavelength-defining and frequency-selecting and tuning interconnecting any given two points in Universe with any two other points in Universe; ergo, with all other points in Universe, granted only sufficient elapsed time for rotational realization of the frequency of repetition of the wavelength vector’s velocity factor to reach any given loci in Universe with a given volumetric-unit quantum of energy. (This is the significance of Fig. 923.10.)

986.727

Fifth, I recall as recounted in Sec. 961.40 that the more elongated the unit- volume tetrahedron of only one-edge-length-constancy (the sixth edge), the less becomes the unit-volume tetrahedron’s least-altitude aspect as related to its other interdimensional aspects, wherefore there is attained a condition wherein the controlling sixth edge’s wavelength is greater than half the tetrahedron’s least-altitude aspect—at which condition the tetrahedron spontaneously turns itself inside-out, ergo, turns itself out—not out of Universe, but out of tune-in-able range. Prior to this spontaneous tuning-out range we have a vast range of now-partially-tuned-in-and-now-tuned-out, which altogether propagates finitely packaged, tuned-in energy information occurring in packages yet recurring in constant, contained wavelength intervals that introduce what has hitherto been considered to be the paradoxical aspect of electromagnetic phenomena, of which it has been misassumed that as of any one moment we can consider our electromagnetic phenomena as being continuous-wave phenomena or as discontinuous-particle phenomena—both simultaneous. We thus learn that no such paradox exists. (Compare Secs. 541.30, 961.46-48, 973.30, and 1072.32.)

986.728

Sixth, we recall that there are no solids or absolute continuums; ergo, there are no physically demonstrable faces or sides of hedra; ergo, we reidentify the system- conceptioning experiences heretofore spoken of as polyhedra, by the name polyvertexia, the simplest of which is the tetravertex, or “four-fix” system.

986.729

Seventh, we recall that the tetravertex is not only the simplest limit case—i.e., the topologically most economically definable polyvertex system case—but also the simplest spheric-system experience case. (See Secs. 1024.10-25, 1053.40-62, 1054.00, 1054.30, and Fig. 1054.40.)

986.730

The Spheric Experience: We now scientifically redefine the spheric experience as an aggregate of vertex-direction-pointed-to (fixed) sub-tune-in-able microevent centers surrounding a system center at equal-radius distances from the system center. Four such surrounding, vertex-convergence-indicated, microevent fixes are redemonstrably proven to be the minimum number of such a microcenter- surrounding aggregate geometrically adequate to constitute systemic subdivision of Universe into macrocosm and microcosm by convergent envelopment, which inherently excludes the thus-constituted system’s macrocosm and inherently includes the thus- constituted system’s microcosm, in which spheric experiencing the greater the population of equi-radiused-from-system-center microevent fixes, the more spheric the experience, and the earliest and simplest beyond the tetrahedron being the hierarchy of concentric, symmetric, primitive polyhedra.

986.740 Microenergy Transformations of Octet Truss

986.741

These last nine major recalls (Sec. 986.711) are directly related to the matter-to-radiation transitional events that occur as we transit between the T and the E Quanta Modules. First, we note that bubbles are spherics, that bubble envelopes are liquid membranes, and that liquids are bivalent. Bivalent tetrahedral aggregates produce at minimum the octet truss. (See Sec. 986.835 et seq.) The octet truss’s double-bonded vertexes also require two layers of closest-packed, unit-radius spheres, whose two layers of closest-packed spheres produce an octet truss whose interior intermembranes are planar while both the exterior and interior membranes are domical.

986.742

Sufficient interior pressure will stretch out the bivalent two-sphere layer into univalent one-sphere layering, which means transforming from the liquid into the gaseous state, which also means transforming from interattractive proximity to inadequate interattractive proximity—ergo, to self-diffusing, atoms-dispersing gaseous molecules. This is to say that the surface-to-volume relationship as we transform from T Quanta Module to E Quanta Module is a transformative, double-to-single-bond, liquid-to-gas transition. Nothing “bursts.” … Bursting is a neat structural-to-destructural atomic rearrangement, not an undefinable random mess.

986.743

Small-moleculed, gaseous-state, atomic-element, monovalent integrities, wherein the atoms are within mass-interattractive critical-proximity range of one another, interconstitute a cloud that may entrap individual molecules too large for escape through the small-molecule interstices of the cloud. A cloud is a monovalent atomic crowd. Water is a bivalent crowd of atoms. Clouds of gasses, having no external membrane, tend to dissipate their molecule and atom populations expansively, except, for instance, within critical proximity of planet Earth, whose Van Allen belts and ionosphere are overwhelmingly capable of retaining the atmospheric aggregates—whose minienergy events such as electrons otherwise become so cosmically dispersed as to be encountered only as seemingly “random” rays and particles.

986.744

This cosmic dispersion of individual microenergy event components—alpha particles, beta particles, and so on—leads us to what is seemingly the most entropic disorderly state, which is, however, only the interpenetration of the outer ramparts of a plurality of differently tuned or vectored isotropic-vector-matrix VE systems.

986.750 Universal Accommodation of Vector Equilibrium Field: Expanding Universe

986.751

Recalling (a) that we gave the vector equilibrium its name because nature avoids the indeterminate (the condition of equilibrium) by always transforming or pulsating four-dimensionally in 12 different ways through the omnicentral VE state, as in one plane of which VE a pendulum swings through the vertical;
— and recalling (b) that each of the vertexes of the isotropic vector matrix could serve as the nuclear center of a VE;
— and recalling (c) also that the limits of swing, pulse, or transform through aberrations of all the VE nucleus-concentric hierarchy of polyhedra have shown themselves to be of modest aberrational magnitude (see the unzipping angle, etc.);
— and recalling (d) also that post-Hubble astronomical discoveries have found more than a million galaxies, all of which are omniuniformly interpositioned angularly and are omniuniformly interdistanced from one another, while all those distances are seemingly increasing uniformly;
— all of which recalls together relate to, explain, and engender the name Expanding Universe.

986.752

We realize that these last four recalls clearly identify the isotropic vector matrix as being the operative geometrical field, not only when atoms are closest packed with one another but also when they are scattered entropically into the cosmically greatest time-size galaxies consisting of all the thus-far-discovered-to-exist stars, which consist of the thus-far-discovered evidence of existent atoms within each star’s cosmic region—with those atoms interarrayed in a multitude of all-alternately, equi-degrees-of-freedom-and- frequency-permitted, evolutionary patterning displays ranging from interstellar gasses and dusts to planets and stars, from asteroids to planetary turtles…to coral…to fungi… et al… Wherefore the Expanding Universe of uniformly interpositioned galaxies informs us that we are witnessing the isotropic vector matrix and its local vector equilibria demonstrating integrity of accommodation at the uttermost time-size macrolimits thus far generalizable within this local 20-billion-year-episode sequence of eternally regenerative Universe, with each galaxy’s unique multibillions of stars, and each of these stars’ multibillions of atoms all intertransforming locally to demonstrate the adequacy of the isotropic vector matrix and its local vector equilibria to accommodate the totality of all local time aberrations possible within the galaxies’ total system limits, which is to say within each of their vector equilibrium’s intertransformability limits.

986.753

Each of the galaxies is centered within a major VE domain within the greater isotropic vector matrix geometrical field—which major VE’s respective fields are subdividingly multiplied by isotropic matrix field VE centerings to the extent of the cumulative number of tendencies of the highest frequency components of the systems permitted by the total time-size enduring magnitude of the local systems’ individual endurance time limits.

986.754

In the seemingly Expanding Universe the equidistant galaxies are apparently receding from each other at a uniform rate, as accounted for by the pre-time-size VE matrix which holds for the largest scale of the total time. This is what we mean by multiplication only by division within each VE domain and its total degrees of freedom in which the number of frequencies available can accommodate the full history of the cosmogony.

986.755

The higher the frequency, the lower the aberration. With multiplication only by division we can accommodate the randomness and the entropy within an entirely regenerative Universe. The high frequency is simply diminishing our point of view.

986.756

The Expanding Universe is a misnomer. What we have is a progressively diminishing point of view as ever more time permits ever greater frequency of subdivisioning of the totally tunable Universe.

986.757

What we observe sum-totally is not a uniformly Expanding Universe, but a uniformly-contracting-magnitude viewpoint of multiplication only by division of the finite but non-unitarily-conceptual, eternally regenerative Scenario Universe. (See Secs. 987.066 and 1052.62.)

986.758

Because the higher-frequency events have the shortest wavelengths in aberration limits, their field of articulation is more local than the low-frequency, longer- wavelengths aberration limit events—ergo, the galaxies usually have the most intense activities closer into and around the central VE regions: all their entropy tendency is accommodated by the total syntropy of the astrophysical greatest-as-yet-identified duration limit.

986.759

We may now direct our attention to the microcosmic, no-time-size, closest- packed unity (versus the Galactic Universe macro-interdistanced unity). This brings us to the prefrequency, timeless-sizeless VE’s hierarchy and to the latter’s contractability into the geometrical tetrahedron and to that quadrivalent tetrahedron’s ability to turn itself inside-out in pure principle to become the novent tetrahedron—the “Black Hole”—the presently-non-tuned-in phenomena. And now we witness the full regenerative range of generalized accommodatability of the VE’s isotropic matrix and its gamut of “special case” realizations occurring as local Universe episodes ranging from photons to molecules, from red giants to white dwarfs, to the black-hole, self-insideouting, and self-reversing phase of intertransformability of eternally regenerative Universe.

986.760

Next we reexplore and recall our discovery of the initial time-size frequency- multiplication by division only-which produces the frequency F, F², F³ layers of 12, 42, closest-packed spheres around a nuclear sphere… And here we have evidencible proof of the persistent adequacy of the VE’s local field to accommodate the elegantly simple structural regenerating of the prime chemical elements, with the successive shell populations demonstrating physically the exact proton-neutron population accounting of the first minimum-limit case of most symmetrical shell enclosings, which corresponds exactly with the ever-experimentally-redemonstrable structural model assemblies shown in Sec. 986.770.

986.770 Shell Growth Rate Predicts Proton and Neutron Population of the Elements

986.771

Thus far we have discovered the physical modelability of Einstein’s equation and the scientific discovery of the modelability of the transformation from matter to radiation, as well as the modelability of the difference between waves and particles. In our excitement over these discoveries we forget that others may think synergetics to be manifesting only pure coincidence of events in a pure-scientists’ assumed-to-be model-less world of abstract mathematical expressions, a world of meaningless but alluring, simple geometrical relationships. Hoping to cope with such skepticism we introduce here three very realistic models whose complex but orderly accounting refutes any suggestion of their being three successive coincidences, all occurring in the most elegantly elementary field of human exploration-that of the periodic table of unique number behaviors of the proton and neutron populations in successive stages of the complexity of the chemical elements themselves.

986.772

If we look at Fig. 222.01 (Synergetics 1), which shows the three successive layers of closest-packed spheres around the prime nuclear sphere, we find the successive layer counts to be 12, 42, 92 … that is, they are “frequency to the second power times 10 plus 2.” While we have been aware for 40 years that the outermost layer of these concentric layers is 92, and that its first three layers add to

$\begin{array}{r}12\\ 42\\ 92\\ 146\end{array}$

which 146 is the number of neutrons in uranium, and uranium is the 92nd element—as with all elements, it combines its total of inner-layer neutrons with its outer-layer protons. In this instance of uranium we have combined the 149 with 92, which gives us Uranium- 238, from which count we can knock out four neutrons from eight of the triangular faces without disturbing symmetry to give us Uranium-234.

986.773

Recently, however, a scientist who had been studying synergetics and attending my lectures called my attention to the fact that the first closest-packed layer 12 around the nuclear sphere and the second embracing closest-packed layer of 42 follow the same neutron count, combining with the outer layer number of protons—as in the 92 uranium-layer case—to provide a physically conceptual model of magnesium and molybdenum. (See Table 419.21.)

986.774

We can report that a number of scientists or scientific-minded laymen are communicating to us their discovery of other physics-evolved phenomena as being elegantly illustrated by synergetics in a conceptually lucid manner.

986.775

Sum-totally we can say that the curve of such events suggests that in the coming decades science in general will have discovered that synergetics is indeed the omnirational, omniconceptual, multialternatived, omnioptimally-efficient, and always experimentally reevidenceable, comprehensive coordinate system employed by nature.

986.776

With popular conception of synergetics being the omniconceptual coordinate system of nature will come popular comprehension of total cosmic technology, and therefore popular comprehension that a competent design revolution—structurally and mechanically—employing the generalized principles governing cosmic technology can indeed, render all humanity comprehensively—i.e., physically and metaphysically—successful, i.e., becoming like “hydrogen” or “leverage” —regular member functions of an omnisuccessful Universe.

986.800 Behavioral Proclivities of Spheric Experience

986.810 Discard of Abstract Dimensions

986.811

Inspired by the E=Mc² modelability, I did more retrospective reconsideration of what I have been concerned with mathematically throughout my life. This reviewing led me to (1) more discoveries, clarifications, and definitions regarding spheres; (2) the discard of the concept of axioms; and (3) the dismissal of three- dimensional reality as being inherently illusory—and the discard of many of mathematics’ abstract devices as being inherently “roundabout,” “obscurational,” and “inefficient.”

986.812

Reversion to axioms and three-dimensional “reality” usually occurs on the basis of “Let’s be practical…let’s yield to our ill-informed reflex-conditioning…the schoolbooks can’t be wrong…no use in getting out of step with the system…we’ll lose our jobs…we’ll be called nuts.”

986.813

Because they cannot qualify as laws if any exceptions to them are found, the generalizable laws of Universe are inherently eternal-timeless-sizeless. Sizing requires time. Time is a cosmically designed consequence of humanity’s having been endowed with innate slowness of apprehension and comprehension, which lags induce time-lapse-altered concepts. (Compare Sec. 529.09.)

986.814

Time-lapsed apprehension of any and all energy-generated, human-sense- reported, human-brain-image-coordinated, angular-directional realization of any physical experiences, produces (swing-through-zero) momentums of misapprehending, which pulsatingly unbalances the otherwise equilibrious, dimensionless, timeless, zero-error, cosmic intellect perfection thereby only inferentially identified to human apprehending differentiates the conceptioning of all the special case manifests of the generalized laws experienced by each and every human individual.

986.815

Academic thought, overwhelmed by the admitted observational inexactitude of special case human-brain-sense experiences, in developing the particular logic of academic geometry (Euclidean or non-Euclidean), finds the term “identical” to be logically prohibited and adopts the word “similar” to identify like geometrical entities. In synergetics, because of its clearly defined differences between generalized primitive conceptuality and special-case time-size realizations, the word “identical” becomes logically permitted. This is brought about by the difference between the operational procedures of synergetics and the abstract procedures of all branches of conventional geometry, where the word “abstract” deliberately means “nonoperational,” because only axiomatic and non-physically-demonstrable.

986.816

Fig. 986.816

Fig. 986.816 Angles Are Angles Independent of the Length of their Edges. Lines are “size” phenomena and unlimited in length. Angle is only a fraction of one cycle.

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In conventional geometry the linear characteristics and the relative sizes of lines dominate the conceptioning and its nomenclature-as, for instance, using the term “equiangular” triangle because only lengths or sizes of lines vary in time. Lines are unlimited in size and can be infinitely extended, whereas angles are discrete fractions of a discrete whole circle. Angles are angles independently of the lengths of their edges. (See Sec. 515.10.) Lengths are always special time-size cases: angles are eternally generalized… We can say with scientific accuracy: “identical equiangular triangles.” (See Fig. 986.816.)

986.817

In summary, lines are “size” phenomena and are unlimited in length. Size measuring requires “time.” Primitive synergetics deals only in angles, which are inherently whole fractions of whole circular azimuths.

986.818

Angles are angles independent of the length of their edges. Triangles are triangles independent of their size. Time is cyclic. Lacking one cycle there is no time sense. Angle is only a fraction of one cycle.

986.819

Synergetics procedure is always from a given whole to the particular fractional angles of the whole system considered. Synergetics employs multiplication only by division… only by division of finite but non-unitarily-conceptual Scenario Universe, subdivided into initially whole primitive systems that divide whole Universe into all the Universe outside the system, all the Universe inside the system, and the little bit of Universe that provides the relevant set of special case stars of experience that illuminatingly define the vertexes of the considered primitive generalized system of consideration. (See Sec. 509.) Conventional geometry “abstracts” by employment of nonexistent—ergo, nondemonstrable—parts, and it compounds a plurality of those nonexistents to arrive at supposedly real objects.

986.820

Because the proofs in conventional geometry depend on a plurality of divider-stepped-off lengths between scribed, punched, or pricked indefinably sized point- speck holes, and because the lengths of the straightedge-drawn lines are extendible without limit, conventional geometry has to assume that any two entities will never be exactly the same. Primitive synergetics has only one length: that of the prime unit vector of the VE and of the isotropic vector matrix.

986.821

Synergetics identifies all of its primitive hierarchy and their holistic subdivisions only by their timeless-sizeless relative angular fractional subdivisions of six equiangular triangles surrounding a point, which hexagonal array equals 360 degrees, if we assume that the three angles of the equiangular triangle always add up to 180 degrees. Synergetics conducts all of its calculations by spherical trigonometry and deals always with the central and surface angles of the primitive hierarchy of pre-time-size relationships of the symmetrically concentric systems around any nucleus of Universe—and their seven great-circle symmetries of the 25 and 31 great-circle systems (Sec. 1040). The foldability of the four great-circle planes demonstrates the four sets of hexagons omnisurrounding the cosmic nucleus in omni-60-degree angular symmetry. This we call the VE. (See Sec. 840.) Angular identities may be operationally assumed to be identical: There is only one equiangular triangle, all of its angles being 60 degrees. The 60-ness comes from the 60 positive and 60 negative, maximum number of surface triangles or T Quanta Modules per cosmic system into which convergent-divergent nuclear unity may be subdivided. The triangle, as physically demonstrated by the tube necklace polygons (Sec. 608), is the only self-stabilizing structure, and the equiangular triangle is the most stable of all triangular structures. Equiangular triangles may be calculatingly employed on an “identical” basis.

986.830 Unrealizability of Primitive Sphere

986.831

As is shown elsewhere (Sec. 1022.11), synergetics finds that the abstract Greek “sphere” does not exist; nor does the quasisphere—the sense-reported “spheric” experiencings of humans—exist at the primitive stage in company with the initial cosmic hierarchy of timeless-sizeless symmetric polyhedra as defined by the six positive and six negative cosmic degrees of freedom and their potential force vectors for adequately coping with all the conditions essential to maintain the individual integrity of min-max primitive, structural, presubdivision systems of Universe.

986.832

The sphere is only dynamically developed either by profiles of spin or by multiplication of uniformly radiused exterior vertexes of ever-higher frequency of modular subdivisioning of the primitive system’s initial symmetry of exterior topology. Such exclusively time-size events of sufficiently high frequency of modular subdivisioning, or high frequency of revolution, can transform any one of the primitive (eternal, sizeless, timeless) hierarchy of successive = 2½, 1, 2½, 3, 4, 5, 6-tetravolumed concentrically symmetric polyhedra into quasispherical appearances. In respect to each such ever-higher frequency of subdividing or revolving in time, each one of the primitive hierarchy polyhedra’s behavioral appearance becomes more spherical.

986.833

The volume of a static quasisphere of unit vector length (radius = l) is 4.188. Each quasisphere is subexistent because it is not as yet spun and there is as yet no time in which to spin it. Seeking to determine anticipatorily the volumetric value of the as-yet- only-potential sphere’s as-yet-to-be-spun domain (as recounted in Secs. 986.206-214), I converted my synergetics constant 1.0198255 to its ninth power, as already recounted and as intuitively motivated to accommodate the energetic factors involved, which gave me the number 1.192 (see Sec. 982.55), and with this ninth-powered constant multiplied the incipient sphere’s already-third-powered volume of 4.188, which produced the twelfth- powered value 4.99206, which seems to tell us that synergetics’ experimentally evidenceable only-by-high-frequency-spinning polyhedral sphere has an unattainable but ever-more-closely-approached limit tetravolume-5.000 (alpha) with however a physically imperceptible 0.007904 volumetric shortfall of tetravolume-5, the limit 4.99206 being the maximum attainable twelfth-powered dynamism—being a sphericity far more perfect than that of any of the planets or fruits or any other of nature’s myriads of quasispheres, which shortfallers are the rule and not the exceptions. The primitively nonconceptual, only- incipient sphere’s only-potentially-to-be-demonstrated domain, like the square root of minus one, is therefore a useful, approximate-magnitude, estimating tool, but it is not structurally demonstrable. The difference in magnitude is close to that of the T and E Quanta Modules.

986.834

Since structure means an interself-stabilized complex-of-events patterning (Sec. 600.01), the “spheric” phenomenon is conceptually—sensorially—experienceable only as a time-size high-frequency recurrence of events, an only-by-dynamic sweepout domain, whose complex of involved factors is describable only at the twelfth-power stage. Being nonstructural and involving a greater volumetric sweepout domain than that of their unrevolved structural polyhedral domains, all quasispheres are compressible.

986.835

Independently occurring single bubbles are dynamic and only superficially spherical. In closest packing all interior bubbles of the bubble aggregate become individual, 14-faceted, tension-membrane polyhedra, which are structured only by the interaction with their liquid monomer, closed-system membranes of all the trying-to- escape, kinetically accelerated, interior gas molecules—which interaction can also be described as an omniembracing restraint of the trying-to-escape gaseous molecules by the sum-total of interatomic, critical-proximity-interattracted structural cohesion of the tensile strength of the bubble’s double-molecule-layered (double-bonded) membranes, which comprehensive closed-system embracement is similar to the cosmically total, eternally integral, nonperiodic, omnicomprehensive embracement by gravitation of the always-and- only periodically occurring, differentiated, separate, and uniquely frequenced nonsimultaneous attempts to disintegratingly escape Universe enacted by the individually differentiated sum-total entities (photons) of radiation. Gravity is always generalized, comprehensive, and untunable. Radiation is always special case and tunable.

986.836

Bubbles in either their independent spherical shape or their aggregated polyhedral shapes are structural consequences of the omnidirectionally outward pressing (compression) of the kinetic complex of molecules in their gaseous, single-bonded, uncohered state as comprehensively embraced by molecules in their liquid, double-bonded, coherent state. In the gaseous state the molecules operate independently and disassociatively, like radiation quanta—ergo, less effective locally than in their double- bonded, integrated, gravity-like, liquid-state embracement.

986.840 Primitive Hierarchy as Physical and Metaphysical

986.841

A special case is time-size. Generalization is eternal and is independent of time-size “Spheres,” whether as independent bubbles, as highfrequency geodesic polyhedral structures, or as dynamically spun primitive polyhedra, are always and only special case time-size (frequency) physical phenomena. The omnirational primitive- numbered-tetravolume-interrelationships hierarchy of concentric symmetric polyhedra is the only generalized conceptuality that is both physical and metaphysical. This is to say that the prime number and relative abundance characteristics of the topology, angulation, and the relative tetravolume involvements of the primitive hierarchy are generalized, conceptual metaphysics. Physically evidenced phenomena are always special case, but in special cases are manifests of generalized principles, which generalized principles themselves are also always metaphysical.

986.850 Powerings as Systemic-integrity Factors

986.851

Synergetics is everywhere informed by and dependent on experimental evidence which is inherently witnessable—which means conceptual—and synergetics’ primitive structural polyhedra constitute an entire, infra-limit-to-ultra-limit, systemic, conceptual, metaphysical hierarchy whose entire interrelationship values are the generalizations of the integral and the “internal affairs” of all systems in Universe—both nucleated and nonnucleated. Bubbles and subatomic A, B, T, and E Quanta Modules are nonnucleated containment systems. Atoms are nucleated systems.

986.852

The systemically internal interrelationship values of the primitive cosmic hierarchy are all independent of time-size factorings, all of which generalized primitive polyhedra’s structurings are accommodated by and are governed by six positive and six negative degrees of freedom. There are 12 integrity factors that definitively cope with those 12 degrees of freedom to produce integral structural systems—both physical and metaphysical—which integrity factors we will henceforth identify as powerings.

986.853

That is, we are abandoning altogether the further employment of the word dimension, which suggests (a) special case time-size lengths, and (b) that some of the describable characteristics of systems can exist alone and not as part of a minimum system, which is always a part of a priori eternally regenerative Universe. In lieu of the no longer scientifically tenable concept of “dimension” we are adopting words to describe time-size realizations of generalized, timeless, primitive systems as event complexes, as structural selfstabilizations, and structural intertransformings as first, second, third, etc., local powering states and minimum local systemic involvement with conditions of the cosmic totality environment with its planetary, solar, galactic, complex-galactic, and supergalactic systems and their respective macro-micro isotropicities.

986.854

In addition to the 12-powered primitive structurings of the positive and negative primitive tetrahedron, the latter has its primitive hierarchy of six intertransformable, tetravolumed, symmetrical integrities which require six additional powerings to produce the six rational-valued, relative-volumetric domains. In addition to this 18-powered state of the primitive hierarchy we discover the integrally potential six- way intertransformabilities of the primitive hierarchy, any one of which requires an additional powering factor, which brings us thus far to 24 powering states. Realization of the intertransformings requires time-size, special case, physical transformation of the metaphysical, generalized, timeless-sizeless, primitive hierarchy potentials.

986.855

It is demonstrably evidenceable that the physically realized superimposed intertransformability potentials of the primitive hierarchy of systems are realizable only as observed from other systems. The transformability cannot be internally observed. All primitive systems have potential external observability by other systems. “Otherness” systems have their own inherent 24-powered constitutionings which are not additional powerings—just more of the same.

986.856

All systems have external relationships, any one of which constitutes an additional systemic complexity-comprehending-and-defining-and-replicating power factor. The number of additional powering factors involved in systemic self-systems and otherness systems is determined in the same manner as that of the fundamental interrelationships of self- and otherness systems, where the number of system interrelationships is

$\frac{{\mathrm{n}}^2-\mathrm{n}}{2}$

986.857

Not including the
$\frac{{\mathrm{n}}^2-\mathrm{n}}{2}$
additional intersystems-relationship powerings, beyond the 24 systemically integral powers, there are six additional, only- otherness-viewable (and in some cases only multi-otherness viewable and realizable), unique behavior potentials of all primitive hierarchy systems, each of which behaviors can be comprehensively accounted for only by additional powerings. They are:
25th-power = axial rotation of the system
26th-power = orbital travel of the system
27th-power = expansion-contraction of the system
28th-power = torque (axial twist) of the system
29th-power = inside-outing (involuting-evoluting) of the system
30th-power = intersystem precession (axial tilting) of the system
31st-power = external interprecessionings amongst a plurality of systems
32nd-power = self-steering of a system within the galaxy of systems (precessionally accomplished)
33rd-power = universal synergistic totality comprehensive of all intersystem effects and ultimate micro- and macroisotropicity of VE-ness

986.860 Rhombic Dodecahedron 6 Minus Polyhedron 5 Equals Unity

986.861

High-frequency, triangulated unit-radius-vertexed, geodesically interchorded, spherical polyhedral apparencies are also structural developments in time-size. There are therefore two kinds of spherics: the highfrequency-event-stabilized, geodesic, structural polyhedron and the dynamically spun, only superficially “apparent” spheres. The static, structural, multifaceted, polyhedral, geodesic sphere’s vertexes are uniformly radiused only by the generalized vector, whereas the only superficially spun and only apparently profiled spheres have a plurality of vertexial distances outward from their systemic center, some of which distances are greater than unit vector radius while some of the vertexes are at less than unit vector radius distance. (See Fig. 986.861.)

986.862

Among the symmetrical polyhedra having a tetravolume of 5 and also having radii a little more or a little less than that of unit vector radius, are the icosahedron and the enenicontahedron whose mean radii of spherical profiling are less than four percent vector-aberrant. There is, however, one symmetrical primitive polyhedron with two sets of its vertexes at greater than unit radius distance outwardly from their system’s nucleic center; that is the rhombic dodecahedron, having, however, a tetravolume of 6. The rhombic dodecahedron’s tetravolume of 6 may account for the minimum intersystemness in pure principle, being the space between omni-closest-packed unit-radius spheres and the spheres themselves. And then there is one symmetric primitive polyhedron having a volume of exactly tetravolume 5 and an interpattern radius of 0.9995 of one unit vector; this is the T Quanta Module phase rhombic triacontahedron. There is also an additional rhombic triacontahedron of exact vector radius and a tetravolume of 5.007758031, which is just too much encroachment upon the rhombic dodecahedron 6 minus the triacontahedron 5 → 6 - 5 = 1, or one volumetric unit of unassigned cosmic “fail-safe space”: BANG—radiation-entropy and eventual turnaround precessional fallin to syntropic photosynthetic transformation into one of matter’s four states: plasmic, gaseous, liquid, crystalline.

986.863

All the hierarchy of primitive polyhedra were developed by progressive great-circle-spun hemispherical halvings of halvings and trisectings of halvings and quintasectings (see Sec. 100.1041) of halvings of the initial primitive tetrahedron itself. That the rhombic triacontahedron of contact-facet radius of unit vector length had a trigonometrically calculated volume of 4.998 proved in due course not to be a residual error but the “critical difference” between matter and radiation. This gives us delight in the truth whatever it may be, recalling that all the discoveries of this chronicle chapter were consequent only to just such faith in the truth, no matter how initially disturbing to misinformed and misconditioned reflexes it may be.

986.870 Nuclear and Nonnuclear Module Orientations

986.871

The rhombic triacontahedron may be fashioned of 120 trivalently bonded T Quanta Module tetrahedra, or of either 60 bivalently interbonded positive T Modules or of 60 bivalently interbonded negative T Modules. In the rhombic triacontahedron we have only radiantly arrayed basic energy modules, arrayed around a single spheric nuclear- inadequate volumetric domain with their acute “corners” pointed inwardly toward the system’s volumetric center, and their centers of mass arrayed outwardly of the system—ergo, prone to escape from the system.

986.872

In the tetrahedron constructed exclusively of 24 A Modules, and in the octahedron constructed of 48 A and 48 B Modules, the asymmetric tetrahedral modules are in radical groups, with their acute points arrayed outwardly of the system and their centers of mass arrayed inwardly of the system—ergo, prone to maintain their critical mass interattractive integrity. The outer sharp points of the A and B Modules are located at the centers of the four or six corner spheres defining the tetrahedron and octahedron, respectively. The fact that the tetrahedron’s and octahedron’s A and B Modules have their massive centers of volume pointing inwardly of the system all jointly interarrayed in the concentric layers of the VE, whereas in the rhombic triacontahedron (and even more so in the half-Couplers of the rhombic dodecahedron) we have the opposite condition—which facts powerfully suggest that the triacontahedron, like its congruent icosahedron’s nonnuclear closest-possible-packed omniarray, presents the exclusively radiational aspect of a “one” or of a “no” nuclear-sphere-centered and isolated most “spheric” polyhedral system to be uniquely identified with the nonnuclear bubble, the one-molecule-deep, kinetically-escape-prone, gas-molecules-containing bubble.

986.8721

In the case of the rhombic dodecahedra we find that the centers of volume of their half-Couplers’ A and B Modules occur almost congruently with their respective closest-packed, unit-radius sphere’s outward ends and thereby concentrate their energies at several spherical-radius levels in respect to a common nuclear-volume-adequate center—all of which suggests some significant relationship of this condition with the various spherical-radius levels of the electron “shells.”

986.873

The tetrahedron and octahedron present the “gravitational” model of self- and-otherness interattractive systems which inherently provide witnessable evidence of the systems’ combined massive considerations or constellations of their interbindings.

986.874

The highly varied alternate A and B Module groupings permitted within the same primitive rhombic dodecahedron, vector equilibrium, and in the Couplers, permit us to consider a wide spectrum of complexedly reorientable potentials and realizations of intermodular behavioral proclivities Lying in proximity to one another between the extreme radiational or gravitational proclivities, and all the reorientabilities operative within the same superficially observed space (Sec. 954). All these large numbers of potential alternatives of behavioral proclivities may be circumferentially, embracingly arrayed entirely within the same superficially observed isotropic field.

Link to original

986.700 Spheric Nature of Electromagnetic Waves

986.701 Consideration 16: and Realization of Synergetic Significance

986.702

Since we have learned that nature’s second-powering is triangling and not squaring (Sec. 990), and since each square is always two similar triangles, we must express Einstein’s equation, where E is the product of M and c², as:
$\mathrm{E}=2{\mathrm{V}}^2$

986.710 Recapitulation of Geometry-and-energy Recalls

986.711

I must add to the inventory of only-synergetically-interrevealing significant discoveries of this chronicle a recapitulation of additional “recalls ”:

1. The absolute constancy of cheese polyhedra;
2. that the tetrahedron is the quantum of energy;
3. that the nonpolar vertexes of the polar-edge-”tuned” tetrahedron can connect any other two points in Universe;
4. that the unit-volume progression of quanta modules accounts for electromagnetic intertuning;
5. that the tetrahedron in turning itself inside-out accounts for electromagnetic- wave propagation;
6. that polyhedra should be reidentified as polyvertexia, the simplest of which is the tetravertex;
7. that the tetravertex is the simplest spheric system;
8. that the vector equilibrium provides a field for universal energy accommodation; and
9. that the vector equilibrium shell growth rate predicts the proton and neutron population of the elements.

986.720 Absolute Constancy: Cheese Polyhedra

986.721

My first observation of the polyhedral hierarchy was introduced in Sec. 223.64, Table 224.20, and Fig. 400.30. That hierarchy may be considered as cheese polyhedra in which there is an experimental redemonstrability of absolute constancy of areal, volumetric, topological, and symmetry characteristics, which constancy is exclusively unique to triangles and tetrahedra and is maintained despite any and all asymmetrical aberrations of those triangles and tetrahedra, as caused
— by perspective distortion;
— by interproportional variations of relative lengths and angles as manifest in isosceles, scalene, acute, or obtuse system aspects (see quadrangular versus triangular accounting in Figs. 990.01 and 100.301.);
— by truncatings parallel to triangle edges or parallel to tetrahedron faces; or
— by frequency modulations,
in contradistinction to complete loss of symmetry and topological constancy of all polygons other than the triangle and of all polyhedra other than the tetrahedron as caused by any special-case, time-size alterations or changes of the perspective point from which the observations of those systems are taken.

986.722

In connection with this same cheese tetrahedron recall we remember (1) that we could push in on the face A of the tetrahedron at a given rate of radial contraction of the system, while pulling out face B at a matching rate of radial expansion of the system, which “couple” of local alterations of the system left the tetrahedron unaltered in shape or size throughout the transformation (Sec. 623) and just as it was both before and after the “coupled” transformings took place, the only altered consequence of which was that the tetrahedron’s center of volume had migrated; and we remember (2) that we could also push in on the same tetrahedron’s face C and pull out on face D at a coupled rate other than the coupled rate of radial expansion and contraction of the A-B face-coupling’s intercomplementary transformings; by all of which we learn that the tetrahedron can accommodate two disparate rates of change without in any way altering its own size and only altering its center-of-volume positioning in respect to any other system components of the local Universe consideration. (See color plate 26.)

986.723

It must be noted, however, that because of the generalized nonsimultaneity of cosmic events, there exists an inherent lag between the pushing in of face A and the pulling out of face B, which induces an inherent interim wave-depression or a wave- breaking pulsating of the coupling functionings of the tetrahedron’s accommodation of transmission of two disparately frequenced energetic communications.

986.724

Second, I recall—as in Secs. 920.01 and 921.10 and Fig. 923.10 —that the tetrahedron is the quantum of energy.

986.725

Third, I recall that the single-tuned-length axis of the edge-axis-rotatable tetrahedron’s two nonaxis polar vertexes may be deployed to connect up with any two other points in Universe without altering the tetrahedron’s unit volume or its tuned-axis length. (See Sec. 961.30.)

986.726

Circular transclusion detected: Extras/figure-and-table-pages/Fig.-986.726

Fourth, I recall that the tetrahedron’s 24 A Modules and the latter’s B, C, D; A, B, C’, D’; A”, B”, C”, D”…(alpha) (see Fig. 986.726, which is a detail and relabeling of Fig. 923.10B) together with the T and E Modules provide transformative significance of being the constant-unit-volume progression of ever-more-asymmetrically-transforming stages of the constant-unit-volume tetrahedra, with the uniform-stage transforming being provided by five of the six edges of each of the constant-volume tetrahedra being covaryingly and ever-progressively-disparately altered—with the sixth edge alone of each and all stages of the transformation remaining unaltered in frequency and wavelength magnitude. The concurrent
— constant-volume-and-wavelength transformings, and
— system rotating around and angular tilting of the constant, unaltered-in-length, sixth edge’s axial altitude in respect to the all-other-in-Universe experiences’ omniinterangular orientations, altogether both permit and accommodate any two other points X and Y in Universe being interconnected not only with one another, but also with the two points A and B that define the unaltered sixth edge AB of the constant-volume and constant-AB-edge-length, omni- Universe-interconnecting tetrahedron ABXY; all of which permits the constant sixth edge AB length to serve as the anywhere and anywhen in Universe to be established transceiver’s wavelength-defining and frequency-selecting and tuning interconnecting any given two points in Universe with any two other points in Universe; ergo, with all other points in Universe, granted only sufficient elapsed time for rotational realization of the frequency of repetition of the wavelength vector’s velocity factor to reach any given loci in Universe with a given volumetric-unit quantum of energy. (This is the significance of Fig. 923.10.)

986.727

Fifth, I recall as recounted in Sec. 961.40 that the more elongated the unit- volume tetrahedron of only one-edge-length-constancy (the sixth edge), the less becomes the unit-volume tetrahedron’s least-altitude aspect as related to its other interdimensional aspects, wherefore there is attained a condition wherein the controlling sixth edge’s wavelength is greater than half the tetrahedron’s least-altitude aspect—at which condition the tetrahedron spontaneously turns itself inside-out, ergo, turns itself out—not out of Universe, but out of tune-in-able range. Prior to this spontaneous tuning-out range we have a vast range of now-partially-tuned-in-and-now-tuned-out, which altogether propagates finitely packaged, tuned-in energy information occurring in packages yet recurring in constant, contained wavelength intervals that introduce what has hitherto been considered to be the paradoxical aspect of electromagnetic phenomena, of which it has been misassumed that as of any one moment we can consider our electromagnetic phenomena as being continuous-wave phenomena or as discontinuous-particle phenomena—both simultaneous. We thus learn that no such paradox exists. (Compare Secs. 541.30, 961.46-48, 973.30, and 1072.32.)

986.728

Sixth, we recall that there are no solids or absolute continuums; ergo, there are no physically demonstrable faces or sides of hedra; ergo, we reidentify the system- conceptioning experiences heretofore spoken of as polyhedra, by the name polyvertexia, the simplest of which is the tetravertex, or “four-fix” system.

986.729

Seventh, we recall that the tetravertex is not only the simplest limit case—i.e., the topologically most economically definable polyvertex system case—but also the simplest spheric-system experience case. (See Secs. 1024.10-25, 1053.40-62, 1054.00, 1054.30, and Fig. 1054.40.)

986.730

The Spheric Experience: We now scientifically redefine the spheric experience as an aggregate of vertex-direction-pointed-to (fixed) sub-tune-in-able microevent centers surrounding a system center at equal-radius distances from the system center. Four such surrounding, vertex-convergence-indicated, microevent fixes are redemonstrably proven to be the minimum number of such a microcenter- surrounding aggregate geometrically adequate to constitute systemic subdivision of Universe into macrocosm and microcosm by convergent envelopment, which inherently excludes the thus-constituted system’s macrocosm and inherently includes the thus- constituted system’s microcosm, in which spheric experiencing the greater the population of equi-radiused-from-system-center microevent fixes, the more spheric the experience, and the earliest and simplest beyond the tetrahedron being the hierarchy of concentric, symmetric, primitive polyhedra.

986.740 Microenergy Transformations of Octet Truss

986.741

These last nine major recalls (Sec. 986.711) are directly related to the matter-to-radiation transitional events that occur as we transit between the T and the E Quanta Modules. First, we note that bubbles are spherics, that bubble envelopes are liquid membranes, and that liquids are bivalent. Bivalent tetrahedral aggregates produce at minimum the octet truss. (See Sec. 986.835 et seq.) The octet truss’s double-bonded vertexes also require two layers of closest-packed, unit-radius spheres, whose two layers of closest-packed spheres produce an octet truss whose interior intermembranes are planar while both the exterior and interior membranes are domical.

986.742

Sufficient interior pressure will stretch out the bivalent two-sphere layer into univalent one-sphere layering, which means transforming from the liquid into the gaseous state, which also means transforming from interattractive proximity to inadequate interattractive proximity—ergo, to self-diffusing, atoms-dispersing gaseous molecules. This is to say that the surface-to-volume relationship as we transform from T Quanta Module to E Quanta Module is a transformative, double-to-single-bond, liquid-to-gas transition. Nothing “bursts.” … Bursting is a neat structural-to-destructural atomic rearrangement, not an undefinable random mess.

986.743

Small-moleculed, gaseous-state, atomic-element, monovalent integrities, wherein the atoms are within mass-interattractive critical-proximity range of one another, interconstitute a cloud that may entrap individual molecules too large for escape through the small-molecule interstices of the cloud. A cloud is a monovalent atomic crowd. Water is a bivalent crowd of atoms. Clouds of gasses, having no external membrane, tend to dissipate their molecule and atom populations expansively, except, for instance, within critical proximity of planet Earth, whose Van Allen belts and ionosphere are overwhelmingly capable of retaining the atmospheric aggregates—whose minienergy events such as electrons otherwise become so cosmically dispersed as to be encountered only as seemingly “random” rays and particles.

986.744

This cosmic dispersion of individual microenergy event components—alpha particles, beta particles, and so on—leads us to what is seemingly the most entropic disorderly state, which is, however, only the interpenetration of the outer ramparts of a plurality of differently tuned or vectored isotropic-vector-matrix VE systems.

986.750 Universal Accommodation of Vector Equilibrium Field: Expanding Universe

986.751

Recalling (a) that we gave the vector equilibrium its name because nature avoids the indeterminate (the condition of equilibrium) by always transforming or pulsating four-dimensionally in 12 different ways through the omnicentral VE state, as in one plane of which VE a pendulum swings through the vertical;
— and recalling (b) that each of the vertexes of the isotropic vector matrix could serve as the nuclear center of a VE;
— and recalling (c) also that the limits of swing, pulse, or transform through aberrations of all the VE nucleus-concentric hierarchy of polyhedra have shown themselves to be of modest aberrational magnitude (see the unzipping angle, etc.);
— and recalling (d) also that post-Hubble astronomical discoveries have found more than a million galaxies, all of which are omniuniformly interpositioned angularly and are omniuniformly interdistanced from one another, while all those distances are seemingly increasing uniformly;
— all of which recalls together relate to, explain, and engender the name Expanding Universe.

986.752

We realize that these last four recalls clearly identify the isotropic vector matrix as being the operative geometrical field, not only when atoms are closest packed with one another but also when they are scattered entropically into the cosmically greatest time-size galaxies consisting of all the thus-far-discovered-to-exist stars, which consist of the thus-far-discovered evidence of existent atoms within each star’s cosmic region—with those atoms interarrayed in a multitude of all-alternately, equi-degrees-of-freedom-and- frequency-permitted, evolutionary patterning displays ranging from interstellar gasses and dusts to planets and stars, from asteroids to planetary turtles…to coral…to fungi… et al… Wherefore the Expanding Universe of uniformly interpositioned galaxies informs us that we are witnessing the isotropic vector matrix and its local vector equilibria demonstrating integrity of accommodation at the uttermost time-size macrolimits thus far generalizable within this local 20-billion-year-episode sequence of eternally regenerative Universe, with each galaxy’s unique multibillions of stars, and each of these stars’ multibillions of atoms all intertransforming locally to demonstrate the adequacy of the isotropic vector matrix and its local vector equilibria to accommodate the totality of all local time aberrations possible within the galaxies’ total system limits, which is to say within each of their vector equilibrium’s intertransformability limits.

986.753

Each of the galaxies is centered within a major VE domain within the greater isotropic vector matrix geometrical field—which major VE’s respective fields are subdividingly multiplied by isotropic matrix field VE centerings to the extent of the cumulative number of tendencies of the highest frequency components of the systems permitted by the total time-size enduring magnitude of the local systems’ individual endurance time limits.

986.754

In the seemingly Expanding Universe the equidistant galaxies are apparently receding from each other at a uniform rate, as accounted for by the pre-time-size VE matrix which holds for the largest scale of the total time. This is what we mean by multiplication only by division within each VE domain and its total degrees of freedom in which the number of frequencies available can accommodate the full history of the cosmogony.

986.755

The higher the frequency, the lower the aberration. With multiplication only by division we can accommodate the randomness and the entropy within an entirely regenerative Universe. The high frequency is simply diminishing our point of view.

986.756

The Expanding Universe is a misnomer. What we have is a progressively diminishing point of view as ever more time permits ever greater frequency of subdivisioning of the totally tunable Universe.

986.757

What we observe sum-totally is not a uniformly Expanding Universe, but a uniformly-contracting-magnitude viewpoint of multiplication only by division of the finite but non-unitarily-conceptual, eternally regenerative Scenario Universe. (See Secs. 987.066 and 1052.62.)

986.758

Because the higher-frequency events have the shortest wavelengths in aberration limits, their field of articulation is more local than the low-frequency, longer- wavelengths aberration limit events—ergo, the galaxies usually have the most intense activities closer into and around the central VE regions: all their entropy tendency is accommodated by the total syntropy of the astrophysical greatest-as-yet-identified duration limit.

986.759

We may now direct our attention to the microcosmic, no-time-size, closest- packed unity (versus the Galactic Universe macro-interdistanced unity). This brings us to the prefrequency, timeless-sizeless VE’s hierarchy and to the latter’s contractability into the geometrical tetrahedron and to that quadrivalent tetrahedron’s ability to turn itself inside-out in pure principle to become the novent tetrahedron—the “Black Hole”—the presently-non-tuned-in phenomena. And now we witness the full regenerative range of generalized accommodatability of the VE’s isotropic matrix and its gamut of “special case” realizations occurring as local Universe episodes ranging from photons to molecules, from red giants to white dwarfs, to the black-hole, self-insideouting, and self-reversing phase of intertransformability of eternally regenerative Universe.

986.760

Next we reexplore and recall our discovery of the initial time-size frequency- multiplication by division only-which produces the frequency F, F², F³ layers of 12, 42, closest-packed spheres around a nuclear sphere… And here we have evidencible proof of the persistent adequacy of the VE’s local field to accommodate the elegantly simple structural regenerating of the prime chemical elements, with the successive shell populations demonstrating physically the exact proton-neutron population accounting of the first minimum-limit case of most symmetrical shell enclosings, which corresponds exactly with the ever-experimentally-redemonstrable structural model assemblies shown in Sec. 986.770.

986.770 Shell Growth Rate Predicts Proton and Neutron Population of the Elements

986.771

Thus far we have discovered the physical modelability of Einstein’s equation and the scientific discovery of the modelability of the transformation from matter to radiation, as well as the modelability of the difference between waves and particles. In our excitement over these discoveries we forget that others may think synergetics to be manifesting only pure coincidence of events in a pure-scientists’ assumed-to-be model-less world of abstract mathematical expressions, a world of meaningless but alluring, simple geometrical relationships. Hoping to cope with such skepticism we introduce here three very realistic models whose complex but orderly accounting refutes any suggestion of their being three successive coincidences, all occurring in the most elegantly elementary field of human exploration-that of the periodic table of unique number behaviors of the proton and neutron populations in successive stages of the complexity of the chemical elements themselves.

986.772

If we look at Fig. 222.01 (Synergetics 1), which shows the three successive layers of closest-packed spheres around the prime nuclear sphere, we find the successive layer counts to be 12, 42, 92 … that is, they are “frequency to the second power times 10 plus 2.” While we have been aware for 40 years that the outermost layer of these concentric layers is 92, and that its first three layers add to

$\begin{array}{r}12\\ 42\\ 92\\ 146\end{array}$

which 146 is the number of neutrons in uranium, and uranium is the 92nd element—as with all elements, it combines its total of inner-layer neutrons with its outer-layer protons. In this instance of uranium we have combined the 149 with 92, which gives us Uranium- 238, from which count we can knock out four neutrons from eight of the triangular faces without disturbing symmetry to give us Uranium-234.

986.773

Recently, however, a scientist who had been studying synergetics and attending my lectures called my attention to the fact that the first closest-packed layer 12 around the nuclear sphere and the second embracing closest-packed layer of 42 follow the same neutron count, combining with the outer layer number of protons—as in the 92 uranium-layer case—to provide a physically conceptual model of magnesium and molybdenum. (See Table 419.21.)

986.774

We can report that a number of scientists or scientific-minded laymen are communicating to us their discovery of other physics-evolved phenomena as being elegantly illustrated by synergetics in a conceptually lucid manner.

986.775

Sum-totally we can say that the curve of such events suggests that in the coming decades science in general will have discovered that synergetics is indeed the omnirational, omniconceptual, multialternatived, omnioptimally-efficient, and always experimentally reevidenceable, comprehensive coordinate system employed by nature.

986.776

With popular conception of synergetics being the omniconceptual coordinate system of nature will come popular comprehension of total cosmic technology, and therefore popular comprehension that a competent design revolution—structurally and mechanically—employing the generalized principles governing cosmic technology can indeed, render all humanity comprehensively—i.e., physically and metaphysically—successful, i.e., becoming like “hydrogen” or “leverage” —regular member functions of an omnisuccessful Universe.

986.800 Behavioral Proclivities of Spheric Experience

986.810 Discard of Abstract Dimensions

986.811

Inspired by the E=Mc² modelability, I did more retrospective reconsideration of what I have been concerned with mathematically throughout my life. This reviewing led me to (1) more discoveries, clarifications, and definitions regarding spheres; (2) the discard of the concept of axioms; and (3) the dismissal of three- dimensional reality as being inherently illusory—and the discard of many of mathematics’ abstract devices as being inherently “roundabout,” “obscurational,” and “inefficient.”

986.812

Reversion to axioms and three-dimensional “reality” usually occurs on the basis of “Let’s be practical…let’s yield to our ill-informed reflex-conditioning…the schoolbooks can’t be wrong…no use in getting out of step with the system…we’ll lose our jobs…we’ll be called nuts.”

986.813

Because they cannot qualify as laws if any exceptions to them are found, the generalizable laws of Universe are inherently eternal-timeless-sizeless. Sizing requires time. Time is a cosmically designed consequence of humanity’s having been endowed with innate slowness of apprehension and comprehension, which lags induce time-lapse-altered concepts. (Compare Sec. 529.09.)

986.814

Time-lapsed apprehension of any and all energy-generated, human-sense- reported, human-brain-image-coordinated, angular-directional realization of any physical experiences, produces (swing-through-zero) momentums of misapprehending, which pulsatingly unbalances the otherwise equilibrious, dimensionless, timeless, zero-error, cosmic intellect perfection thereby only inferentially identified to human apprehending differentiates the conceptioning of all the special case manifests of the generalized laws experienced by each and every human individual.

986.815

Academic thought, overwhelmed by the admitted observational inexactitude of special case human-brain-sense experiences, in developing the particular logic of academic geometry (Euclidean or non-Euclidean), finds the term “identical” to be logically prohibited and adopts the word “similar” to identify like geometrical entities. In synergetics, because of its clearly defined differences between generalized primitive conceptuality and special-case time-size realizations, the word “identical” becomes logically permitted. This is brought about by the difference between the operational procedures of synergetics and the abstract procedures of all branches of conventional geometry, where the word “abstract” deliberately means “nonoperational,” because only axiomatic and non-physically-demonstrable.

986.816

Circular transclusion detected: Extras/figure-and-table-pages/Fig.-986.816

In conventional geometry the linear characteristics and the relative sizes of lines dominate the conceptioning and its nomenclature-as, for instance, using the term “equiangular” triangle because only lengths or sizes of lines vary in time. Lines are unlimited in size and can be infinitely extended, whereas angles are discrete fractions of a discrete whole circle. Angles are angles independently of the lengths of their edges. (See Sec. 515.10.) Lengths are always special time-size cases: angles are eternally generalized… We can say with scientific accuracy: “identical equiangular triangles.” (See Fig. 986.816.)

986.817

In summary, lines are “size” phenomena and are unlimited in length. Size measuring requires “time.” Primitive synergetics deals only in angles, which are inherently whole fractions of whole circular azimuths.

986.818

Angles are angles independent of the length of their edges. Triangles are triangles independent of their size. Time is cyclic. Lacking one cycle there is no time sense. Angle is only a fraction of one cycle.

986.819

Synergetics procedure is always from a given whole to the particular fractional angles of the whole system considered. Synergetics employs multiplication only by division… only by division of finite but non-unitarily-conceptual Scenario Universe, subdivided into initially whole primitive systems that divide whole Universe into all the Universe outside the system, all the Universe inside the system, and the little bit of Universe that provides the relevant set of special case stars of experience that illuminatingly define the vertexes of the considered primitive generalized system of consideration. (See Sec. 509.) Conventional geometry “abstracts” by employment of nonexistent—ergo, nondemonstrable—parts, and it compounds a plurality of those nonexistents to arrive at supposedly real objects.

986.820

Because the proofs in conventional geometry depend on a plurality of divider-stepped-off lengths between scribed, punched, or pricked indefinably sized point- speck holes, and because the lengths of the straightedge-drawn lines are extendible without limit, conventional geometry has to assume that any two entities will never be exactly the same. Primitive synergetics has only one length: that of the prime unit vector of the VE and of the isotropic vector matrix.

986.821

Synergetics identifies all of its primitive hierarchy and their holistic subdivisions only by their timeless-sizeless relative angular fractional subdivisions of six equiangular triangles surrounding a point, which hexagonal array equals 360 degrees, if we assume that the three angles of the equiangular triangle always add up to 180 degrees. Synergetics conducts all of its calculations by spherical trigonometry and deals always with the central and surface angles of the primitive hierarchy of pre-time-size relationships of the symmetrically concentric systems around any nucleus of Universe—and their seven great-circle symmetries of the 25 and 31 great-circle systems (Sec. 1040). The foldability of the four great-circle planes demonstrates the four sets of hexagons omnisurrounding the cosmic nucleus in omni-60-degree angular symmetry. This we call the VE. (See Sec. 840.) Angular identities may be operationally assumed to be identical: There is only one equiangular triangle, all of its angles being 60 degrees. The 60-ness comes from the 60 positive and 60 negative, maximum number of surface triangles or T Quanta Modules per cosmic system into which convergent-divergent nuclear unity may be subdivided. The triangle, as physically demonstrated by the tube necklace polygons (Sec. 608), is the only self-stabilizing structure, and the equiangular triangle is the most stable of all triangular structures. Equiangular triangles may be calculatingly employed on an “identical” basis.

986.830 Unrealizability of Primitive Sphere

986.831

As is shown elsewhere (Sec. 1022.11), synergetics finds that the abstract Greek “sphere” does not exist; nor does the quasisphere—the sense-reported “spheric” experiencings of humans—exist at the primitive stage in company with the initial cosmic hierarchy of timeless-sizeless symmetric polyhedra as defined by the six positive and six negative cosmic degrees of freedom and their potential force vectors for adequately coping with all the conditions essential to maintain the individual integrity of min-max primitive, structural, presubdivision systems of Universe.

986.832

The sphere is only dynamically developed either by profiles of spin or by multiplication of uniformly radiused exterior vertexes of ever-higher frequency of modular subdivisioning of the primitive system’s initial symmetry of exterior topology. Such exclusively time-size events of sufficiently high frequency of modular subdivisioning, or high frequency of revolution, can transform any one of the primitive (eternal, sizeless, timeless) hierarchy of successive = 2½, 1, 2½, 3, 4, 5, 6-tetravolumed concentrically symmetric polyhedra into quasispherical appearances. In respect to each such ever-higher frequency of subdividing or revolving in time, each one of the primitive hierarchy polyhedra’s behavioral appearance becomes more spherical.

986.833

The volume of a static quasisphere of unit vector length (radius = l) is 4.188. Each quasisphere is subexistent because it is not as yet spun and there is as yet no time in which to spin it. Seeking to determine anticipatorily the volumetric value of the as-yet- only-potential sphere’s as-yet-to-be-spun domain (as recounted in Secs. 986.206-214), I converted my synergetics constant 1.0198255 to its ninth power, as already recounted and as intuitively motivated to accommodate the energetic factors involved, which gave me the number 1.192 (see Sec. 982.55), and with this ninth-powered constant multiplied the incipient sphere’s already-third-powered volume of 4.188, which produced the twelfth- powered value 4.99206, which seems to tell us that synergetics’ experimentally evidenceable only-by-high-frequency-spinning polyhedral sphere has an unattainable but ever-more-closely-approached limit tetravolume-5.000 (alpha) with however a physically imperceptible 0.007904 volumetric shortfall of tetravolume-5, the limit 4.99206 being the maximum attainable twelfth-powered dynamism—being a sphericity far more perfect than that of any of the planets or fruits or any other of nature’s myriads of quasispheres, which shortfallers are the rule and not the exceptions. The primitively nonconceptual, only- incipient sphere’s only-potentially-to-be-demonstrated domain, like the square root of minus one, is therefore a useful, approximate-magnitude, estimating tool, but it is not structurally demonstrable. The difference in magnitude is close to that of the T and E Quanta Modules.

986.834

Since structure means an interself-stabilized complex-of-events patterning (Sec. 600.01), the “spheric” phenomenon is conceptually—sensorially—experienceable only as a time-size high-frequency recurrence of events, an only-by-dynamic sweepout domain, whose complex of involved factors is describable only at the twelfth-power stage. Being nonstructural and involving a greater volumetric sweepout domain than that of their unrevolved structural polyhedral domains, all quasispheres are compressible.

986.835

Independently occurring single bubbles are dynamic and only superficially spherical. In closest packing all interior bubbles of the bubble aggregate become individual, 14-faceted, tension-membrane polyhedra, which are structured only by the interaction with their liquid monomer, closed-system membranes of all the trying-to- escape, kinetically accelerated, interior gas molecules—which interaction can also be described as an omniembracing restraint of the trying-to-escape gaseous molecules by the sum-total of interatomic, critical-proximity-interattracted structural cohesion of the tensile strength of the bubble’s double-molecule-layered (double-bonded) membranes, which comprehensive closed-system embracement is similar to the cosmically total, eternally integral, nonperiodic, omnicomprehensive embracement by gravitation of the always-and- only periodically occurring, differentiated, separate, and uniquely frequenced nonsimultaneous attempts to disintegratingly escape Universe enacted by the individually differentiated sum-total entities (photons) of radiation. Gravity is always generalized, comprehensive, and untunable. Radiation is always special case and tunable.

986.836

Bubbles in either their independent spherical shape or their aggregated polyhedral shapes are structural consequences of the omnidirectionally outward pressing (compression) of the kinetic complex of molecules in their gaseous, single-bonded, uncohered state as comprehensively embraced by molecules in their liquid, double-bonded, coherent state. In the gaseous state the molecules operate independently and disassociatively, like radiation quanta—ergo, less effective locally than in their double- bonded, integrated, gravity-like, liquid-state embracement.

986.840 Primitive Hierarchy as Physical and Metaphysical

986.841

A special case is time-size. Generalization is eternal and is independent of time-size “Spheres,” whether as independent bubbles, as highfrequency geodesic polyhedral structures, or as dynamically spun primitive polyhedra, are always and only special case time-size (frequency) physical phenomena. The omnirational primitive- numbered-tetravolume-interrelationships hierarchy of concentric symmetric polyhedra is the only generalized conceptuality that is both physical and metaphysical. This is to say that the prime number and relative abundance characteristics of the topology, angulation, and the relative tetravolume involvements of the primitive hierarchy are generalized, conceptual metaphysics. Physically evidenced phenomena are always special case, but in special cases are manifests of generalized principles, which generalized principles themselves are also always metaphysical.

986.850 Powerings as Systemic-integrity Factors

986.851

Synergetics is everywhere informed by and dependent on experimental evidence which is inherently witnessable—which means conceptual—and synergetics’ primitive structural polyhedra constitute an entire, infra-limit-to-ultra-limit, systemic, conceptual, metaphysical hierarchy whose entire interrelationship values are the generalizations of the integral and the “internal affairs” of all systems in Universe—both nucleated and nonnucleated. Bubbles and subatomic A, B, T, and E Quanta Modules are nonnucleated containment systems. Atoms are nucleated systems.

986.852

The systemically internal interrelationship values of the primitive cosmic hierarchy are all independent of time-size factorings, all of which generalized primitive polyhedra’s structurings are accommodated by and are governed by six positive and six negative degrees of freedom. There are 12 integrity factors that definitively cope with those 12 degrees of freedom to produce integral structural systems—both physical and metaphysical—which integrity factors we will henceforth identify as powerings.

986.853

That is, we are abandoning altogether the further employment of the word dimension, which suggests (a) special case time-size lengths, and (b) that some of the describable characteristics of systems can exist alone and not as part of a minimum system, which is always a part of a priori eternally regenerative Universe. In lieu of the no longer scientifically tenable concept of “dimension” we are adopting words to describe time-size realizations of generalized, timeless, primitive systems as event complexes, as structural selfstabilizations, and structural intertransformings as first, second, third, etc., local powering states and minimum local systemic involvement with conditions of the cosmic totality environment with its planetary, solar, galactic, complex-galactic, and supergalactic systems and their respective macro-micro isotropicities.

986.854

In addition to the 12-powered primitive structurings of the positive and negative primitive tetrahedron, the latter has its primitive hierarchy of six intertransformable, tetravolumed, symmetrical integrities which require six additional powerings to produce the six rational-valued, relative-volumetric domains. In addition to this 18-powered state of the primitive hierarchy we discover the integrally potential six- way intertransformabilities of the primitive hierarchy, any one of which requires an additional powering factor, which brings us thus far to 24 powering states. Realization of the intertransformings requires time-size, special case, physical transformation of the metaphysical, generalized, timeless-sizeless, primitive hierarchy potentials.

986.855

It is demonstrably evidenceable that the physically realized superimposed intertransformability potentials of the primitive hierarchy of systems are realizable only as observed from other systems. The transformability cannot be internally observed. All primitive systems have potential external observability by other systems. “Otherness” systems have their own inherent 24-powered constitutionings which are not additional powerings—just more of the same.

986.856

All systems have external relationships, any one of which constitutes an additional systemic complexity-comprehending-and-defining-and-replicating power factor. The number of additional powering factors involved in systemic self-systems and otherness systems is determined in the same manner as that of the fundamental interrelationships of self- and otherness systems, where the number of system interrelationships is

$\frac{{\mathrm{n}}^2-\mathrm{n}}{2}$

986.857

Not including the
$\frac{{\mathrm{n}}^2-\mathrm{n}}{2}$
additional intersystems-relationship powerings, beyond the 24 systemically integral powers, there are six additional, only- otherness-viewable (and in some cases only multi-otherness viewable and realizable), unique behavior potentials of all primitive hierarchy systems, each of which behaviors can be comprehensively accounted for only by additional powerings. They are:
25th-power = axial rotation of the system
26th-power = orbital travel of the system
27th-power = expansion-contraction of the system
28th-power = torque (axial twist) of the system
29th-power = inside-outing (involuting-evoluting) of the system
30th-power = intersystem precession (axial tilting) of the system
31st-power = external interprecessionings amongst a plurality of systems
32nd-power = self-steering of a system within the galaxy of systems (precessionally accomplished)
33rd-power = universal synergistic totality comprehensive of all intersystem effects and ultimate micro- and macroisotropicity of VE-ness

986.860 Rhombic Dodecahedron 6 Minus Polyhedron 5 Equals Unity

986.861

High-frequency, triangulated unit-radius-vertexed, geodesically interchorded, spherical polyhedral apparencies are also structural developments in time-size. There are therefore two kinds of spherics: the highfrequency-event-stabilized, geodesic, structural polyhedron and the dynamically spun, only superficially “apparent” spheres. The static, structural, multifaceted, polyhedral, geodesic sphere’s vertexes are uniformly radiused only by the generalized vector, whereas the only superficially spun and only apparently profiled spheres have a plurality of vertexial distances outward from their systemic center, some of which distances are greater than unit vector radius while some of the vertexes are at less than unit vector radius distance. (See Fig. 986.861.)

986.862

Among the symmetrical polyhedra having a tetravolume of 5 and also having radii a little more or a little less than that of unit vector radius, are the icosahedron and the enenicontahedron whose mean radii of spherical profiling are less than four percent vector-aberrant. There is, however, one symmetrical primitive polyhedron with two sets of its vertexes at greater than unit radius distance outwardly from their system’s nucleic center; that is the rhombic dodecahedron, having, however, a tetravolume of 6. The rhombic dodecahedron’s tetravolume of 6 may account for the minimum intersystemness in pure principle, being the space between omni-closest-packed unit-radius spheres and the spheres themselves. And then there is one symmetric primitive polyhedron having a volume of exactly tetravolume 5 and an interpattern radius of 0.9995 of one unit vector; this is the T Quanta Module phase rhombic triacontahedron. There is also an additional rhombic triacontahedron of exact vector radius and a tetravolume of 5.007758031, which is just too much encroachment upon the rhombic dodecahedron 6 minus the triacontahedron 5 → 6 - 5 = 1, or one volumetric unit of unassigned cosmic “fail-safe space”: BANG—radiation-entropy and eventual turnaround precessional fallin to syntropic photosynthetic transformation into one of matter’s four states: plasmic, gaseous, liquid, crystalline.

986.863

All the hierarchy of primitive polyhedra were developed by progressive great-circle-spun hemispherical halvings of halvings and trisectings of halvings and quintasectings (see Sec. 100.1041) of halvings of the initial primitive tetrahedron itself. That the rhombic triacontahedron of contact-facet radius of unit vector length had a trigonometrically calculated volume of 4.998 proved in due course not to be a residual error but the “critical difference” between matter and radiation. This gives us delight in the truth whatever it may be, recalling that all the discoveries of this chronicle chapter were consequent only to just such faith in the truth, no matter how initially disturbing to misinformed and misconditioned reflexes it may be.

986.870 Nuclear and Nonnuclear Module Orientations

986.871

The rhombic triacontahedron may be fashioned of 120 trivalently bonded T Quanta Module tetrahedra, or of either 60 bivalently interbonded positive T Modules or of 60 bivalently interbonded negative T Modules. In the rhombic triacontahedron we have only radiantly arrayed basic energy modules, arrayed around a single spheric nuclear- inadequate volumetric domain with their acute “corners” pointed inwardly toward the system’s volumetric center, and their centers of mass arrayed outwardly of the system—ergo, prone to escape from the system.

986.872

In the tetrahedron constructed exclusively of 24 A Modules, and in the octahedron constructed of 48 A and 48 B Modules, the asymmetric tetrahedral modules are in radical groups, with their acute points arrayed outwardly of the system and their centers of mass arrayed inwardly of the system—ergo, prone to maintain their critical mass interattractive integrity. The outer sharp points of the A and B Modules are located at the centers of the four or six corner spheres defining the tetrahedron and octahedron, respectively. The fact that the tetrahedron’s and octahedron’s A and B Modules have their massive centers of volume pointing inwardly of the system all jointly interarrayed in the concentric layers of the VE, whereas in the rhombic triacontahedron (and even more so in the half-Couplers of the rhombic dodecahedron) we have the opposite condition—which facts powerfully suggest that the triacontahedron, like its congruent icosahedron’s nonnuclear closest-possible-packed omniarray, presents the exclusively radiational aspect of a “one” or of a “no” nuclear-sphere-centered and isolated most “spheric” polyhedral system to be uniquely identified with the nonnuclear bubble, the one-molecule-deep, kinetically-escape-prone, gas-molecules-containing bubble.

986.8721

In the case of the rhombic dodecahedra we find that the centers of volume of their half-Couplers’ A and B Modules occur almost congruently with their respective closest-packed, unit-radius sphere’s outward ends and thereby concentrate their energies at several spherical-radius levels in respect to a common nuclear-volume-adequate center—all of which suggests some significant relationship of this condition with the various spherical-radius levels of the electron “shells.”

986.873

The tetrahedron and octahedron present the “gravitational” model of self- and-otherness interattractive systems which inherently provide witnessable evidence of the systems’ combined massive considerations or constellations of their interbindings.

986.874

The highly varied alternate A and B Module groupings permitted within the same primitive rhombic dodecahedron, vector equilibrium, and in the Couplers, permit us to consider a wide spectrum of complexedly reorientable potentials and realizations of intermodular behavioral proclivities Lying in proximity to one another between the extreme radiational or gravitational proclivities, and all the reorientabilities operative within the same superficially observed space (Sec. 954). All these large numbers of potential alternatives of behavioral proclivities may be circumferentially, embracingly arrayed entirely within the same superficially observed isotropic field.

Link to original

986.800 Behavioral Proclivities of Spheric Experience

986.810 Discard of Abstract Dimensions

986.811

Inspired by the E=Mc² modelability, I did more retrospective reconsideration of what I have been concerned with mathematically throughout my life. This reviewing led me to (1) more discoveries, clarifications, and definitions regarding spheres; (2) the discard of the concept of axioms; and (3) the dismissal of three- dimensional reality as being inherently illusory—and the discard of many of mathematics’ abstract devices as being inherently “roundabout,” “obscurational,” and “inefficient.”

986.812

Reversion to axioms and three-dimensional “reality” usually occurs on the basis of “Let’s be practical…let’s yield to our ill-informed reflex-conditioning…the schoolbooks can’t be wrong…no use in getting out of step with the system…we’ll lose our jobs…we’ll be called nuts.”

986.813

Because they cannot qualify as laws if any exceptions to them are found, the generalizable laws of Universe are inherently eternal-timeless-sizeless. Sizing requires time. Time is a cosmically designed consequence of humanity’s having been endowed with innate slowness of apprehension and comprehension, which lags induce time-lapse-altered concepts. (Compare Sec. 529.09.)

986.814

Time-lapsed apprehension of any and all energy-generated, human-sense- reported, human-brain-image-coordinated, angular-directional realization of any physical experiences, produces (swing-through-zero) momentums of misapprehending, which pulsatingly unbalances the otherwise equilibrious, dimensionless, timeless, zero-error, cosmic intellect perfection thereby only inferentially identified to human apprehending differentiates the conceptioning of all the special case manifests of the generalized laws experienced by each and every human individual.

986.815

Academic thought, overwhelmed by the admitted observational inexactitude of special case human-brain-sense experiences, in developing the particular logic of academic geometry (Euclidean or non-Euclidean), finds the term “identical” to be logically prohibited and adopts the word “similar” to identify like geometrical entities. In synergetics, because of its clearly defined differences between generalized primitive conceptuality and special-case time-size realizations, the word “identical” becomes logically permitted. This is brought about by the difference between the operational procedures of synergetics and the abstract procedures of all branches of conventional geometry, where the word “abstract” deliberately means “nonoperational,” because only axiomatic and non-physically-demonstrable.

986.816

Circular transclusion detected: Extras/figure-and-table-pages/Fig.-986.816

In conventional geometry the linear characteristics and the relative sizes of lines dominate the conceptioning and its nomenclature-as, for instance, using the term “equiangular” triangle because only lengths or sizes of lines vary in time. Lines are unlimited in size and can be infinitely extended, whereas angles are discrete fractions of a discrete whole circle. Angles are angles independently of the lengths of their edges. (See Sec. 515.10.) Lengths are always special time-size cases: angles are eternally generalized… We can say with scientific accuracy: “identical equiangular triangles.” (See Fig. 986.816.)

986.817

In summary, lines are “size” phenomena and are unlimited in length. Size measuring requires “time.” Primitive synergetics deals only in angles, which are inherently whole fractions of whole circular azimuths.

986.818

Angles are angles independent of the length of their edges. Triangles are triangles independent of their size. Time is cyclic. Lacking one cycle there is no time sense. Angle is only a fraction of one cycle.

986.819

Synergetics procedure is always from a given whole to the particular fractional angles of the whole system considered. Synergetics employs multiplication only by division… only by division of finite but non-unitarily-conceptual Scenario Universe, subdivided into initially whole primitive systems that divide whole Universe into all the Universe outside the system, all the Universe inside the system, and the little bit of Universe that provides the relevant set of special case stars of experience that illuminatingly define the vertexes of the considered primitive generalized system of consideration. (See Sec. 509.) Conventional geometry “abstracts” by employment of nonexistent—ergo, nondemonstrable—parts, and it compounds a plurality of those nonexistents to arrive at supposedly real objects.

986.820

Because the proofs in conventional geometry depend on a plurality of divider-stepped-off lengths between scribed, punched, or pricked indefinably sized point- speck holes, and because the lengths of the straightedge-drawn lines are extendible without limit, conventional geometry has to assume that any two entities will never be exactly the same. Primitive synergetics has only one length: that of the prime unit vector of the VE and of the isotropic vector matrix.

986.821

Synergetics identifies all of its primitive hierarchy and their holistic subdivisions only by their timeless-sizeless relative angular fractional subdivisions of six equiangular triangles surrounding a point, which hexagonal array equals 360 degrees, if we assume that the three angles of the equiangular triangle always add up to 180 degrees. Synergetics conducts all of its calculations by spherical trigonometry and deals always with the central and surface angles of the primitive hierarchy of pre-time-size relationships of the symmetrically concentric systems around any nucleus of Universe—and their seven great-circle symmetries of the 25 and 31 great-circle systems (Sec. 1040). The foldability of the four great-circle planes demonstrates the four sets of hexagons omnisurrounding the cosmic nucleus in omni-60-degree angular symmetry. This we call the VE. (See Sec. 840.) Angular identities may be operationally assumed to be identical: There is only one equiangular triangle, all of its angles being 60 degrees. The 60-ness comes from the 60 positive and 60 negative, maximum number of surface triangles or T Quanta Modules per cosmic system into which convergent-divergent nuclear unity may be subdivided. The triangle, as physically demonstrated by the tube necklace polygons (Sec. 608), is the only self-stabilizing structure, and the equiangular triangle is the most stable of all triangular structures. Equiangular triangles may be calculatingly employed on an “identical” basis.

986.830 Unrealizability of Primitive Sphere

986.831

As is shown elsewhere (Sec. 1022.11), synergetics finds that the abstract Greek “sphere” does not exist; nor does the quasisphere—the sense-reported “spheric” experiencings of humans—exist at the primitive stage in company with the initial cosmic hierarchy of timeless-sizeless symmetric polyhedra as defined by the six positive and six negative cosmic degrees of freedom and their potential force vectors for adequately coping with all the conditions essential to maintain the individual integrity of min-max primitive, structural, presubdivision systems of Universe.

986.832

The sphere is only dynamically developed either by profiles of spin or by multiplication of uniformly radiused exterior vertexes of ever-higher frequency of modular subdivisioning of the primitive system’s initial symmetry of exterior topology. Such exclusively time-size events of sufficiently high frequency of modular subdivisioning, or high frequency of revolution, can transform any one of the primitive (eternal, sizeless, timeless) hierarchy of successive = 2½, 1, 2½, 3, 4, 5, 6-tetravolumed concentrically symmetric polyhedra into quasispherical appearances. In respect to each such ever-higher frequency of subdividing or revolving in time, each one of the primitive hierarchy polyhedra’s behavioral appearance becomes more spherical.

986.833

The volume of a static quasisphere of unit vector length (radius = l) is 4.188. Each quasisphere is subexistent because it is not as yet spun and there is as yet no time in which to spin it. Seeking to determine anticipatorily the volumetric value of the as-yet- only-potential sphere’s as-yet-to-be-spun domain (as recounted in Secs. 986.206-214), I converted my synergetics constant 1.0198255 to its ninth power, as already recounted and as intuitively motivated to accommodate the energetic factors involved, which gave me the number 1.192 (see Sec. 982.55), and with this ninth-powered constant multiplied the incipient sphere’s already-third-powered volume of 4.188, which produced the twelfth- powered value 4.99206, which seems to tell us that synergetics’ experimentally evidenceable only-by-high-frequency-spinning polyhedral sphere has an unattainable but ever-more-closely-approached limit tetravolume-5.000 (alpha) with however a physically imperceptible 0.007904 volumetric shortfall of tetravolume-5, the limit 4.99206 being the maximum attainable twelfth-powered dynamism—being a sphericity far more perfect than that of any of the planets or fruits or any other of nature’s myriads of quasispheres, which shortfallers are the rule and not the exceptions. The primitively nonconceptual, only- incipient sphere’s only-potentially-to-be-demonstrated domain, like the square root of minus one, is therefore a useful, approximate-magnitude, estimating tool, but it is not structurally demonstrable. The difference in magnitude is close to that of the T and E Quanta Modules.

986.834

Since structure means an interself-stabilized complex-of-events patterning (Sec. 600.01), the “spheric” phenomenon is conceptually—sensorially—experienceable only as a time-size high-frequency recurrence of events, an only-by-dynamic sweepout domain, whose complex of involved factors is describable only at the twelfth-power stage. Being nonstructural and involving a greater volumetric sweepout domain than that of their unrevolved structural polyhedral domains, all quasispheres are compressible.

986.835

Independently occurring single bubbles are dynamic and only superficially spherical. In closest packing all interior bubbles of the bubble aggregate become individual, 14-faceted, tension-membrane polyhedra, which are structured only by the interaction with their liquid monomer, closed-system membranes of all the trying-to- escape, kinetically accelerated, interior gas molecules—which interaction can also be described as an omniembracing restraint of the trying-to-escape gaseous molecules by the sum-total of interatomic, critical-proximity-interattracted structural cohesion of the tensile strength of the bubble’s double-molecule-layered (double-bonded) membranes, which comprehensive closed-system embracement is similar to the cosmically total, eternally integral, nonperiodic, omnicomprehensive embracement by gravitation of the always-and- only periodically occurring, differentiated, separate, and uniquely frequenced nonsimultaneous attempts to disintegratingly escape Universe enacted by the individually differentiated sum-total entities (photons) of radiation. Gravity is always generalized, comprehensive, and untunable. Radiation is always special case and tunable.

986.836

Bubbles in either their independent spherical shape or their aggregated polyhedral shapes are structural consequences of the omnidirectionally outward pressing (compression) of the kinetic complex of molecules in their gaseous, single-bonded, uncohered state as comprehensively embraced by molecules in their liquid, double-bonded, coherent state. In the gaseous state the molecules operate independently and disassociatively, like radiation quanta—ergo, less effective locally than in their double- bonded, integrated, gravity-like, liquid-state embracement.

986.840 Primitive Hierarchy as Physical and Metaphysical

986.841

A special case is time-size. Generalization is eternal and is independent of time-size “Spheres,” whether as independent bubbles, as highfrequency geodesic polyhedral structures, or as dynamically spun primitive polyhedra, are always and only special case time-size (frequency) physical phenomena. The omnirational primitive- numbered-tetravolume-interrelationships hierarchy of concentric symmetric polyhedra is the only generalized conceptuality that is both physical and metaphysical. This is to say that the prime number and relative abundance characteristics of the topology, angulation, and the relative tetravolume involvements of the primitive hierarchy are generalized, conceptual metaphysics. Physically evidenced phenomena are always special case, but in special cases are manifests of generalized principles, which generalized principles themselves are also always metaphysical.

986.850 Powerings as Systemic-integrity Factors

986.851

Synergetics is everywhere informed by and dependent on experimental evidence which is inherently witnessable—which means conceptual—and synergetics’ primitive structural polyhedra constitute an entire, infra-limit-to-ultra-limit, systemic, conceptual, metaphysical hierarchy whose entire interrelationship values are the generalizations of the integral and the “internal affairs” of all systems in Universe—both nucleated and nonnucleated. Bubbles and subatomic A, B, T, and E Quanta Modules are nonnucleated containment systems. Atoms are nucleated systems.

986.852

The systemically internal interrelationship values of the primitive cosmic hierarchy are all independent of time-size factorings, all of which generalized primitive polyhedra’s structurings are accommodated by and are governed by six positive and six negative degrees of freedom. There are 12 integrity factors that definitively cope with those 12 degrees of freedom to produce integral structural systems—both physical and metaphysical—which integrity factors we will henceforth identify as powerings.

986.853

That is, we are abandoning altogether the further employment of the word dimension, which suggests (a) special case time-size lengths, and (b) that some of the describable characteristics of systems can exist alone and not as part of a minimum system, which is always a part of a priori eternally regenerative Universe. In lieu of the no longer scientifically tenable concept of “dimension” we are adopting words to describe time-size realizations of generalized, timeless, primitive systems as event complexes, as structural selfstabilizations, and structural intertransformings as first, second, third, etc., local powering states and minimum local systemic involvement with conditions of the cosmic totality environment with its planetary, solar, galactic, complex-galactic, and supergalactic systems and their respective macro-micro isotropicities.

986.854

In addition to the 12-powered primitive structurings of the positive and negative primitive tetrahedron, the latter has its primitive hierarchy of six intertransformable, tetravolumed, symmetrical integrities which require six additional powerings to produce the six rational-valued, relative-volumetric domains. In addition to this 18-powered state of the primitive hierarchy we discover the integrally potential six- way intertransformabilities of the primitive hierarchy, any one of which requires an additional powering factor, which brings us thus far to 24 powering states. Realization of the intertransformings requires time-size, special case, physical transformation of the metaphysical, generalized, timeless-sizeless, primitive hierarchy potentials.

986.855

It is demonstrably evidenceable that the physically realized superimposed intertransformability potentials of the primitive hierarchy of systems are realizable only as observed from other systems. The transformability cannot be internally observed. All primitive systems have potential external observability by other systems. “Otherness” systems have their own inherent 24-powered constitutionings which are not additional powerings—just more of the same.

986.856

All systems have external relationships, any one of which constitutes an additional systemic complexity-comprehending-and-defining-and-replicating power factor. The number of additional powering factors involved in systemic self-systems and otherness systems is determined in the same manner as that of the fundamental interrelationships of self- and otherness systems, where the number of system interrelationships is

$\frac{{\mathrm{n}}^2-\mathrm{n}}{2}$

986.857

Not including the
$\frac{{\mathrm{n}}^2-\mathrm{n}}{2}$
additional intersystems-relationship powerings, beyond the 24 systemically integral powers, there are six additional, only- otherness-viewable (and in some cases only multi-otherness viewable and realizable), unique behavior potentials of all primitive hierarchy systems, each of which behaviors can be comprehensively accounted for only by additional powerings. They are:
25th-power = axial rotation of the system
26th-power = orbital travel of the system
27th-power = expansion-contraction of the system
28th-power = torque (axial twist) of the system
29th-power = inside-outing (involuting-evoluting) of the system
30th-power = intersystem precession (axial tilting) of the system
31st-power = external interprecessionings amongst a plurality of systems
32nd-power = self-steering of a system within the galaxy of systems (precessionally accomplished)
33rd-power = universal synergistic totality comprehensive of all intersystem effects and ultimate micro- and macroisotropicity of VE-ness

986.860 Rhombic Dodecahedron 6 Minus Polyhedron 5 Equals Unity

986.861

High-frequency, triangulated unit-radius-vertexed, geodesically interchorded, spherical polyhedral apparencies are also structural developments in time-size. There are therefore two kinds of spherics: the highfrequency-event-stabilized, geodesic, structural polyhedron and the dynamically spun, only superficially “apparent” spheres. The static, structural, multifaceted, polyhedral, geodesic sphere’s vertexes are uniformly radiused only by the generalized vector, whereas the only superficially spun and only apparently profiled spheres have a plurality of vertexial distances outward from their systemic center, some of which distances are greater than unit vector radius while some of the vertexes are at less than unit vector radius distance. (See Fig. 986.861.)

986.862

Among the symmetrical polyhedra having a tetravolume of 5 and also having radii a little more or a little less than that of unit vector radius, are the icosahedron and the enenicontahedron whose mean radii of spherical profiling are less than four percent vector-aberrant. There is, however, one symmetrical primitive polyhedron with two sets of its vertexes at greater than unit radius distance outwardly from their system’s nucleic center; that is the rhombic dodecahedron, having, however, a tetravolume of 6. The rhombic dodecahedron’s tetravolume of 6 may account for the minimum intersystemness in pure principle, being the space between omni-closest-packed unit-radius spheres and the spheres themselves. And then there is one symmetric primitive polyhedron having a volume of exactly tetravolume 5 and an interpattern radius of 0.9995 of one unit vector; this is the T Quanta Module phase rhombic triacontahedron. There is also an additional rhombic triacontahedron of exact vector radius and a tetravolume of 5.007758031, which is just too much encroachment upon the rhombic dodecahedron 6 minus the triacontahedron 5 → 6 - 5 = 1, or one volumetric unit of unassigned cosmic “fail-safe space”: BANG—radiation-entropy and eventual turnaround precessional fallin to syntropic photosynthetic transformation into one of matter’s four states: plasmic, gaseous, liquid, crystalline.

986.863

All the hierarchy of primitive polyhedra were developed by progressive great-circle-spun hemispherical halvings of halvings and trisectings of halvings and quintasectings (see Sec. 100.1041) of halvings of the initial primitive tetrahedron itself. That the rhombic triacontahedron of contact-facet radius of unit vector length had a trigonometrically calculated volume of 4.998 proved in due course not to be a residual error but the “critical difference” between matter and radiation. This gives us delight in the truth whatever it may be, recalling that all the discoveries of this chronicle chapter were consequent only to just such faith in the truth, no matter how initially disturbing to misinformed and misconditioned reflexes it may be.

986.870 Nuclear and Nonnuclear Module Orientations

986.871

The rhombic triacontahedron may be fashioned of 120 trivalently bonded T Quanta Module tetrahedra, or of either 60 bivalently interbonded positive T Modules or of 60 bivalently interbonded negative T Modules. In the rhombic triacontahedron we have only radiantly arrayed basic energy modules, arrayed around a single spheric nuclear- inadequate volumetric domain with their acute “corners” pointed inwardly toward the system’s volumetric center, and their centers of mass arrayed outwardly of the system—ergo, prone to escape from the system.

986.872

In the tetrahedron constructed exclusively of 24 A Modules, and in the octahedron constructed of 48 A and 48 B Modules, the asymmetric tetrahedral modules are in radical groups, with their acute points arrayed outwardly of the system and their centers of mass arrayed inwardly of the system—ergo, prone to maintain their critical mass interattractive integrity. The outer sharp points of the A and B Modules are located at the centers of the four or six corner spheres defining the tetrahedron and octahedron, respectively. The fact that the tetrahedron’s and octahedron’s A and B Modules have their massive centers of volume pointing inwardly of the system all jointly interarrayed in the concentric layers of the VE, whereas in the rhombic triacontahedron (and even more so in the half-Couplers of the rhombic dodecahedron) we have the opposite condition—which facts powerfully suggest that the triacontahedron, like its congruent icosahedron’s nonnuclear closest-possible-packed omniarray, presents the exclusively radiational aspect of a “one” or of a “no” nuclear-sphere-centered and isolated most “spheric” polyhedral system to be uniquely identified with the nonnuclear bubble, the one-molecule-deep, kinetically-escape-prone, gas-molecules-containing bubble.

986.8721

In the case of the rhombic dodecahedra we find that the centers of volume of their half-Couplers’ A and B Modules occur almost congruently with their respective closest-packed, unit-radius sphere’s outward ends and thereby concentrate their energies at several spherical-radius levels in respect to a common nuclear-volume-adequate center—all of which suggests some significant relationship of this condition with the various spherical-radius levels of the electron “shells.”

986.873

The tetrahedron and octahedron present the “gravitational” model of self- and-otherness interattractive systems which inherently provide witnessable evidence of the systems’ combined massive considerations or constellations of their interbindings.

986.874

The highly varied alternate A and B Module groupings permitted within the same primitive rhombic dodecahedron, vector equilibrium, and in the Couplers, permit us to consider a wide spectrum of complexedly reorientable potentials and realizations of intermodular behavioral proclivities Lying in proximity to one another between the extreme radiational or gravitational proclivities, and all the reorientabilities operative within the same superficially observed space (Sec. 954). All these large numbers of potential alternatives of behavioral proclivities may be circumferentially, embracingly arrayed entirely within the same superficially observed isotropic field.

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