BUCKYVERSE

411.00 Four Spheres as Minimum System

411.01 The South Seas islander piling his coconuts, the fruit dealer selling oranges, and the cannoneer stacking equiradius cannonballs, or the much earlier round-rock-slinging soldiers were probably the first to learn about the closest packing of spheres. The stacking of balls in symmetrical rows and layers leads inevitably to a stable pyramidal aggregate.

411.02 Closest packing of spheres does not begin with a nucleus. Closest packing begins with two balls coming together.

411.03 One ball cannot zoom around alone in Universe. Without otherness, there is no consciousness and no direction. If there were only one entitysay it is a sphere called *“me”*there would be no Universe: no otherness: no awareness: no consciousness: no direction. Once another entity let’s say a sphereis sighted, there is awareness and direction. There is no way to tell how far away the other sphere may be or what its size may be. Size sense comes only with a plurality of comparative experiences.

411.04 As a single sphere, now aware of an otherness sphere somewhere out there, "me” has to rotate about without restraint and can observe its rotation in relation to the otherness, but could misassume the otherness sphere to be zooming around it, as there are no third, or more, othernesses by which to judge.

411.05 When moved unknowingly toward another by mass attraction, the “other” ball and the "me” ball, either or both of which could have been rotating as they approach one another, each misassumes the other to be growing bigger, until finally they touch each other. Now they can roll around upon one another, and they might be cotraveling together, but there is not as yet anywhere to go because there is no otherness than their joint selves relative to which to travel. They can’t go through each other and they can’t get away from each other. They are only free to rotate upon each other, and because of friction they must do so cooperatively.

411.06 When a third ball looms into sight, providing a sense of direction for the tangently rolling-upon-each-other first couple, the third one and the first two are mass-attracted toward each other and finally make contact. The newcorner third sphere may roll around on one of the first couple until it rolls into the valley between the first two. The third ball then gets locked into the valley between the first two by double mass attraction, and now becomes tangent to both of the first two.

411.07 The three balls, each one tangent to both of the others, now form an equiangular triangular group with a small opening at their center. The friction of each of their double contacts with the other two gives them a geartooth interaction effect. With two gears, one can turn clockwise and the other can tum counterclockwise. Even numbers of gears reciprocate; odd numbers of gears block one another. Thus our three balls can no longer roll circumferentially around each other; they can only rotate cooperatively on the three axes formed between each of their two tangent contact points. The friction between their surface contacts forces all three to rotate unidirectionally about their respective contact axes, which are parallel to the edges of the equilateral triangle defined by the three sphere centers; i.e., the three spheres can now only co-rotate over and into the hole at the center of the triangle, and out and away upwardly again from the bottom of the center hole. Thus the three balls can involute or evolute axially, like a rubber doughnut in respect to the hole at their triangular center, but they cannot rotate circumferentially.

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Fig. 411.05 Four Spheres Lock as Tetrahedron:

  1. A single sphere is free to rotate in any direction.

  2. Two tangent spheres although free to rotate in any direction must do so cooperatively.

  3. Three spheres can rotate cooperatively only about respective axes which are parallel to the edges of the equilateral triangle defined by joining the sphere centers, i.e. each sphere rotates toward the center of the triangle.

  4. Four spheres lock together. No rotation is possible, making the minimum stable closest-packed-sphere system: the tetrahedron.

411.08 Finally, a fourth ball appears in Universe and is mass-attracted to tangency with one of the three previously triangulated spheres, then rolls on one of the joined balls until it falls into the valley formed by the hole at the top center of the triangulated three; but being of equal diameter with the other three balls, it cannot fall through that hole whose radius is less than those of the associated spheres. Thus nested in the central valley, the fourth sphere now touches each of the other three and vice versa. The four closest-packed spheres make a closest-packing array. They are mass-attractively locked together as a tetrahedron.[^8] No further interrotation is possible. As a tetrahedron, they form the minimum stable structural system and provide the nuclear matrix for further mass attraction and closest-packed growth of additional spheres falling into their four-surface triangular nests. This produces increasing numbers of closest-packed nests. Thus do atoms agglomerate in closest packing in tetrahedrally conformed arrays, often truncated asymmetrically at corners and along edges to obscure the tetrahedral origin of the collection.

411.10 Unpredicted Degrees of Freedom: Reviewing the history of self-discoveries of restraints progressively and mutually interimposed upon one another by the arrival of and association with successive othernesses, self may discover through progressive retrospections and appreciate the significance of theretofore unrecognized and unrealized degrees of cosmic freedom successively and inadvertently deducted from the original total inventory of unexpended self-potentials with which the individual is initially, always and only, endowed. Only with the progressive retrospection inventoryings induced only by otherness-developed experience of awareness does the loss of another degree of freedom become consciously subtracted from the previous experience inventory of now consciously multiplying rememberable events.

411.11 With the discovery of principles through progressive deprivations dawns new awareness of the elective employability of the principles initially separated out from their special-case experiencing to self-control more and more of the pattern of events.

411.12 Only through relationships with otherness can self learn of principles; only by discovery of the relationships existing between self and othernesses does inspiration to employ principles objectively occur. There is nothing in self per se, or in otherness per se, that predicts the interrelatedness behaviors and their successively unique characteristics. Only from realization of the significance of otherness it can be learned further that only by earnest commitment to others does self become inadvertently behaviorally advantaged to effect even greater commitment to others, while on the other hand all self-seeking induces only ever greater self-loss.

411.20 Discovery as a Function of Loss: It is a basic principle that you only discover what you had had by virtue of losing it. Due to our subconscious organic coordination, you don’t know what you are losing until you lose it. Naught can be so advantageous as thoughtfully considered loss and resolve to employ the principles thereby discovered for others. You don’t know how much you have to give until you start trying to give. The more you try to give effectively to advantage others, the more you will possess to give, and vice versa.

411.21 Retrospective awareness of losses can bring preoccupation with self, blinding self to recognition of the synergetic gains that, by virtue of the second-power law, have brought group advantage gains in which the individual has attained fourth-power continuance potential often way offsetting individual freedom losses, particularly in view of the group’s discovery that as a group it can enjoy all the original freedoms individually lost but never realized by the individual to existergo, unemployable consciously by the individual, who was more of a victim of the unknown freedoms than an enjoyer.

411.22 Only as group structuring occurs do the discovered cosmic freedoms become consciously employableemployable effectively only for all and not for self. It is when this retrospective discovering is made by the grouped-in individual and he tries to employ the freedoms exclusively for self or exclusive subdivisions of the group, that his attempts become inherently unfulfillable and scheduled for ultimate failure.

411.23 Self-seeking brings a potential loss that engenders first caution, then fear: fear of change; change being inexorable, fear increases and freezes. Self-seeking always eventuates in self-destruction through inability to adapt.

411.30 Intergeared Mobility Freedoms: Only with the arrival of the second otherness do individuals become aware of the loss of mutual any where-around-one-another rollability, and then discover that they have also lost the ability to go in all directions, for they cannot go through each other. It is inferred that they haven’t lost mutually accomplishable omnidirectional mobility. (Here is an example of one of those comprehendings from an apprehending.) With the acquisition of the second otherness, self discovers what it has lost which self didn’t know it had, until the loss brought retrospective awareness of the lost freedom: an inter-anywhere-roll-aroundness with the first other.

411.31 With the mutually interattracted threeness, each having two contacts with their two otherness partners, they learn, as a fourth otherness nests into their triangular opening, that they have now lost a frictionally intergeared mutually evoluting-involuting rotational freedom (torus). Now blocked by the frictionally intergeared fourth otherness, the mutual omnidirectionality of the structural system so produced by that structural system can be discovered only by the self-observation of the realization of another structural system’s cumulative repetition repeating the evolutionary accumulation of its own fourfolding; observing that the other structural system can move omnidirectionally, their observed rotations and magnitude changes can be explained only by the omnidirectional freedom only mutually experienceable by the whole individual structural system as it had been originally and only subconsciously experienced by the individual self. Naught has been lost. Much has been mutually gained. Each can take off from and return to the others.

411.32 The variations of the features of second-structural-system otherness can be explained only by the self-structural system assumption of increasing distance of travel of the otherness to and away, or by the other system’s experiencing a freedom theretofore unself-realized: that of individual expansion and contraction. For any one of the four members of a structural-system team can expand and contract coordinatedly at individual rates, mutual rates, or interpaired rates, provided one does not become so small as to “fall through,” the triangular opening at its nest bottom. (If it fell through, one otherness would start rotating hingewise around the axis of tangent contact with the next-largest, without touching the fourth and next-smallest-to-self.)

411.33 Thus the self-structural system discloses by observation of otherness’s system changing features that its own system had been enjoying, as with freedoms of which it was previously unconscious.

411.34 Our inventory of intergeared mobility freedoms is fourfold. It is four-dimensional:

  1. omnidirectionality of united movement;

  2. roll-aroundness (orbiting);

  3. polarized evoluting-and-involuting, and polarized spin; and

  4. inward-outward expandability singly, doubly, three- or four-partite.

411.35 The inward-outward expandability is the basis of convergencedivergence and radiation-gravitation pulsationwhich seems furthest from man’s awareness. This is what science has discovered: a world of waves in which waves are interpenetrated by waves in frequency modulation. There is a systemic interrelationship of basic fourness always accompanied by a sixness of alternatives of freedoms.

411.36 When a sphere gets so small that it can roll through a hole between other close packed spheres, the omnidirectionality of any one individual would not be impeded under the following circumstances:

  1. The individual mass-attracted by any threeness being drawn through the hole of any other threeness.

  2. Where a fourth otherness could be attracted by a momentary critical- proximity threeness.

  3. We learn there is individuality and magnitude change; then we learn that, due to the energy losses and gains of systems occasioned by the continual variations of omnidirectional proximities and omnivariability of expansion-contraction system accumulating rates, there is a degree of freedom phenomena rate as well as a terminal condition.

411.37 Rate occurs only when there is terminal. Rate is a modulation between terminals. With termination, a system’s integrity is brought about by the individually covarying magnitudes and the omnidirectional experience pulls on the system.

411.38 The degree of freedom that is lost is discoverable only retrospectively by the very fact of the loss. It is an inverse synergetic behavior wherein no feature of the self part predicted the successive behaviors of the whole and where the individual part freedoms were only mutually disclosed by their subsequently realized loss.