410.00 Closest Packing of Spheres

410.01 Nature's Coordination
410.011 About 1917, I decided that nature did not have separate, independently operating departments of physics, chemistry, biology, mathematics, ethics, etc. Nature did not call a department heads' meeting when I threw a green apple into the pond, with the department heads having to make a decision about how to handle this biological encounter with chemistry's water and the unauthorized use of the physics department's waves. I decided that it didn't require a Ph.D. to discern that nature probably had only one department and only one coordinate, omnirational, mensuration system.
410.02 I determined then and there to seek out the comprehensive coordinate system employed by nature. The omnirational associating and disassociating of chemistry__always joining in whole low-order numbers, as for instance H2O and never HpiO__persuaded me that if I could discover nature's comprehensive coordination, it would prove to be omnirational despite academic geometry's fortuitous development and employment of transcendental irrational numbers and other "pure," nonexperimentally demonstrable, incommensurable integer relationships.
410.03 I was dissatisfied with abstract, weightless, unstable, ageless, temperatureless, straight-line-defined squares and cubes as models for calculating our omnicurvilinear experience. I was an early rebel against blockheads and squares. Reviewing the history of chemical science, I became intuitively aware that the clue to vectorial, volumetric, geometrical coordination with physical reality and all the fundamental energetic experiences of reality, such as temperature, time, and force, might be found in Avogadro's experimental proof of his earlier hypothesis, which stated that all gases under identical conditions of heat and pressure will always disclose the same number of molecules per given volume. Here was disclosed a "Grand Central Station" accommodating all comers; despite "fundamental" or elementarily unique differences of identity, all accommodated on a common volume (space)-to-number basis. One molecule of any element: One space. A cosmic democracy.
410.04 I felt intuitively that inasmuch as the variety of gases experimented with often consisted of only one unique chemical element, such as hydrogen or oxygen, and that inasmuch as these gases also could be liquefied, and also inasmuch as most of the elements are susceptible to some heat- or pressure-produced transformation between their liquid, crystalline, and vapor, or incandescent, states, it might also be hypothetically reasonable to further generalize Avogadro's hypothesis by assuming that, under identical energy conditions, all elements may disclose the same number of "somethings" per given volume. Such a generalized concept is not limited to pressure and heat: we wanted to be much more inclusive, so we said we assume that all the conditions of energy are identical; this includes not only the pressure and heat conditions of thermodynamics, which developed before electromagnetics became an applied realization, but also the conditions of electromagnetics as constant.
410.05 I went on from there to reason that vectors, being the product of physical energy constituents, are "real," having velocity multiplied by mass operating in a specific direction; velocity being a product of time and size; and mass being a volume-weight relationship. On impact, mass at velocity transforrns into heat and work. These energy factors can be translated not only into work, but into heat or into time as well. Furthermore, electromagnetic scientists had found that all their EMF (electromotive force) problems could be graphed vectorially; the fact that "graphable" or "modelable" vectors can interact modelably in real Universe space seemed to promise that the equations of nature's omnicoordinate transactions, expressed in omni-space-intruding vectorial models, might produce real models of reality of nature's Grand Central Station of omnicoordination.
410.06 So I then went on to say that, if all the energy conditions were everywhere the same, then all the vectors would be the same length and all of them would interact at the same angle. I then explored experimentally to discover whether this "isotropic vector matrix," as so employed in matrix calculus, played with empty sets of symbols on flat sheets of paper, could be realized in actual modeling. Employing equilength toothpicks and semi-dried peas, as I had been encouraged to do in kindergarten at the age of four (before receiving powerful eyeglasses and when I was unfamiliar with the rightangled structuring of buildings as were the children with normal vision), I fumbled tactilely with the toothpicks and peas until I could feel a stable structure, and thus assembled an omnitriangulated complex and so surprised the teachers that their exclamations made me remember the event in detail. I thus rediscovered the octet truss whose vertexes, or convergent foci, were all sixty-degree-angle interconnections, ergo omniequilateral, omniequiangled, and omni-intertriangulated; ergo, omnistructured. Being omnidirectionally equally interspaced from one another, this ornni-intertriangulation produced the isotropic matrix of foci for omni-closest-packed sphere centers. This opened the way to a combinatorial geometry of closest-packed spheres and equilength vectors.
410.07 Over and over again, we are confronted by nature obviously formulating her structures with beautiful spherical agglomerations. The piling of oranges, coconuts, and cannonballs in tetrahedral or half-octahedral pyramids has been used for centuries and possibly for ages. Almost a half-century ago, F. W. Aston, the British scientist first identified for physics the most economical uniradius spherical interagglomerations as the "closest packing of spheres," which had fresh interest for the physicist and crystallographer because of the then recently discovered microscopic realization that nature frequently employed omni-intertriangulated systems, which hold mathematical clues to the principles of symmetrical coordination governing natural structure, the dynamic vectorial geometry of the atomic nucleus, as well as of the atoms themselves.
410.10 Omnitriangulation of Sixty Degrees
410.11 The closest packing of unit radius spheres always associates in omnitriangulations of 60 degrees, whether in planar or omnidirectional arrays. Six unit radius spheres pack most tightly around one sphere on a billiard table. Twelve unit radius spheres compact around one sphere in omnidirectional closest packing.
410.12 If we take three billiard balls on a flat table, we find that they compact beautifully into a triangle. If we arrange four of them on a billiard table in a square, they tend to be restless and roll around each other. If compacted into a condition of stability, the four form a 60-degree-angled diamond shape made of two stable triangles.
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 entity__say 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 sphere__is 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.

Fig. 411.05
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 newcomer 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.
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.1 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.
(Footnote 1: There is an alternative sequence, which is perhaps more likely, in which the balls would first join as two pairs__like dumbbells "docking" in space. Then our old friend precession would cause them to form a momentary square, only to have one pair revolve until its axis is at 90 degrees to the other couple's axis, whereafter they dock to interlock as a tetrahedron.)
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 can it 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 selfseeking induces only ever greater self-loss. There is an alternate sequence, which is perhaps more likely, in which the balls would first join as two pairs__like dumbbells "docking" in space. Then our old friend precession would cause them to form a momentary square, only to have one pair revolve until its axis is at 90 degrees to the other couple's axis, whereafter they dock to interlock as a tetrahedron.
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 exist__ergo, 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 employable__employable 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 anywherearound-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 convergence-divergence and radiation-gravitation pulsation__which 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.

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Copyright © 1997 Estate of R. Buckminster Fuller