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Letter on Tensegrity from Buckminster Fuller:
Section 3

As we have shown in Synergetics,  621.20, and Synergetics, vol. 2,  621.30-34, three vector legs of the tetrahedron AB, AC, and AD, like a camera tripod, when energy loaded, are only held together by one tetrahedron drawn as tripod with triangle restraining the tripod legsvertex A and feet of their legs, and, as with the camera tripod, try to slide apart, but are prevented from doing so by the closed triangle BCD, the sum of all three of whose energy vectors BC, CD, and DB exactly equal the sum of the energy of the three leg vectors AB, AC, and AD, but the integrated-into-a triangle three have both ends of their vectors tied to their adjacent vectors in a closed circuit wherefore they are more effectively operative as an integrative force in Universe than the only-at-one-end of each fastened together, disintegrative AB, AC, AD set of vectors.

The AB, AC, AD set of vectors represent the sum of all disintegrative radiation forces of Universe and the BC, CD, DB set of three vectors represent the sum of all gravitation forces of Universe and, while the radiation and gravitation forces of Universe are equal, the gravity forces are twice as effectively employed as are the radiation forces, wherefor the gravitational forces will always predominate no matter how much time it may take. This predominance of gravity over radiation is even more effectively demonstrated by the vector equilibrium (see Synergetics).

We next discover that if we take thirteen spheres of unit-radius, we can exactly, tangentially surround one such sphere with twelve others. And, as the following illustrations show, we discover that if we interconnect the centers of the twelve-around-one with one another and with the center sphere we will have Produced a geometrical system of twelve external vertexes, six superficial squares, and eight superficial triangles symmetrically surrounding a central point. This geometrical figure (F) shows twelve radii and 24 externally embracing lines (chords) all of the same length.

Figure 14 Vector Equilibrium: Onnidirectional Closest Packing of Unit-Radius Spheres around a Nucleus

transparent vector equilibrium with radii drawn in

The reason this polyvertexion is called the Vector Equilibrium (symbol vector equilibrium symbol (capital V and E joined together at the top)) follows:

If we look at this configuration carefully we will discover that it has four hexagonally divided interior planes which are the same four-dimensional planes of the four-vertexion (look at the eight little four-vertexia). We then discover that two of each of the interior hexagonal planes' radii are congruent with one another where the hex planes cross and the four hexagons' six each radii produce a collection of twenty-four radii exactly equalling the twenty-four equi-lengthed chords of the system.

Since the lines represent the forces structuring the system we call this figure the vector equilibrium in which we have twenty-four disintegrative, interrepulsive vectors radiating from one another and the system center which radii are fastened to one another only at their inner ends. The fastened together ends are at the center of the radially disintegrating system. In contradistinction we have twenty-four chordally interattractive vectors fastened to one another at both of each of their ends, thus finitely embracing and closing the system. Since disintegrating radii vectors represent the radiation behaviors of Universe and the integrating chords represent the gravity of Universe we find that total Universe's gravity is twice as efficient as the total radiation of Universe in their respective uses of their energy force vectors. This guarantees the integrity of ever transforming, only overlappingly episoded, eternally regenerative, scenario Universe.

If we now apply another layer of equi-radii, closest-packed spheres around the twelve-around-one grouping we will find we have forty-two additional spheres.

We can completely surround this group with another layer of ninety-two equi-radii, closes-packed together spheres. See illustration.

Next we find experimentally that we can continue infinitely to completely surround this system arrangement with additional layers each of which layer's number of spheres will always be the second power of the number of layers multiplied by ten plus the number two. See illustration.

Figure 15 Equation for Omnidirectional Closest Packing of Spheres
(Similar to Fig. 222.01 from Synergetics)

If we interconnect each of these unit-radius sphere's centers with each of their respective tangentially surrounding neighboring spheres' centers within a many layered system, we will find that except for the outermost layer we have produced a linear matrix within which each tetrahedron is completely surrounded by octahedra and each octahedron is completely surrounded by tetrahedra, with all the lines of interconnection of equal length -- which length is twice that of the unit radius of all the spheres. This because the radii reach only the point of tangency between adjacent closest packed spheres.

If we now identify each of all these equi-length lines of interrelationship as being lines of the spheres' interpositioning push-pull forces, we realize it produces an energy model of what is quite clearly a complex of balance of alternate forces most closely adjacent to equilibrium: the total model of all these lines becomes what is called an isotropic vector matrix, isotropic meaning everywhere the same.

(Illustrations of isotropic vector matrix showing closest packing layers of 12, 42, 92, 162 spheres, i.e. showing matrix up to 4 frequency.)

[Editor's note: I found the above line on a page by itself without any illustrations. Maybe he's referring to something like Fig. 419.30 from Synergetics.]

Figure 16 Octet Truss
( Fig. 420.01 from Synergetics)
Figure 17 Isotropic Vector Matrix
( Fig. 420.02 (without caption) from Synergetics)

When the centers of equiradius spheres in closest packing are joined with lines, an isotropic vector matrix is formed. As with the tetrahedron, nature's simplest structural system, this isotropic vector matrix constitutes four-dimensionally an array of equilateral triangles which is seen as the comprehensive coordination frame of reference of nature's most economical, most comfortable structural interrelationships employing 60-degree association and disassociation. This provides an omnirational four-dimensional accounting system which, if arbitrarily accounted on a 90-degree, XYZ frame of reference, becomes inherently irrational in its volumetric, angular, and three-dimensional expressions.

All the celestially, everywhere and everywhen, ever more disorderly, multitudinous radiational broadcastings of all disintegrating star and galaxy systems of eternally regenerative Universe remotely and nonsimultaneously intermingle their differently accelerated and differently aimed cosmic radiation off-castings to produce variously distanced from one another entropic maelstroms.

In passing one another at a wide range of velocities and distances, these separate star- and novae-refuse entities intermingle in a wide variety of densities.

As their interpatternings now and then accidentally thicken, their relative interattractivenesses vary inversely as of Newton's law. The interattractivenesses are countered by their respective velocity momentums and various radiation winds of neighboring stars resulting in progressive veerings of the celestial courses of the individual items of cosmic trash.

When predominant, the interattractions produce internearings and progressively higher orders of gain and ever decreasing radii of individual orbiting.

Gradually entropy gives way here and there to syntropy as individual components from a myriad of different stars separate out from other cosmic "dust" to gravitationally integrate here and there as new individual cosmic clouds. within these clouds the process of orbital course veering into orbits of ever lesser radii continues and the cloud thickenings increase which thereby acquires and stores finally accumulating enough energy from its star (Sun) to itself transform into a radiantly entropic star.

The star's disorderly broadcast entities progressively intermingle in an ever more disorderly manner with entropically (disorderly) broadcast-off energetic entities of a myriad of other as yet entropically broadcasting or already energy-spent, "has been" stars. Those super-disorderly cosmic interminglings of the energetic rubbish of vast numbers of yesteryear's stars may persist for aeons.

Their component far-flung rubbish sometimes interapproach one another to such a degree of proximity that Newton's celestial-bodies-interattraction law progressively decreases the distances between them. This multiplies their interattraction at a second power rate of the arithmetical distances intervening. They palpitatingly interpulse this now way, now that way, finally assembling in xillions of initial trendings to symmetries and syntropic responses to the predominant forces of orderly convergence.

Gravitationally embraced, and sometimes impelled this-way-and-that-way by various star-radiation "winds," they begin interapproaching one another to become progressively electromagnetically and gravitationally self-sorting, size-wise, and in angular alignment to produce critical proximities. Here and there some attain closest proximities of symmetric interpositionings which when numerically and geometrically vector-balanced in sufficient degree finally attain the state of "matter."

In the same pattern of integrated interattracting and momentum veering of the individual cosmic trash particles' orbital courses, the separate individually thickening clouds progressively converge, here and there as larger and more complex clouds. These individual, larger clouds progressively thicken together. This process leads eventually to sufficient condensations here and there in Universe to produce asteroids, moons, and planets.

I will now repeat and amplify the foregoing as follows: The omniconserved, complexedly and nonsimultaneously intertransforming energies of eternally regenerative Universe consist most simply of two concurrent prime patterns. One is the energy convergent complex of ever less disorderly, cosmic omnicast offings. This group is more syntropic than entropic. The other is the multiplicatively, disintegrating, asymmetric aggregate of desystematically diverging cosmic omnicast offings. This group is more often entropic than syntropic. "With all the room in the world" to accommodate them, the convergent traffic and the divergent traffic omnidirectionally co-penetrate one another in a myriad of different speeds and orbits.

The organized and symmetrically converging, non-intertouching, intercoursing arrays ultimately assemble into asteroids, moons, and planets. Amongst the planets, some, like Earth, have the concentric, protective, and supportive surrounding ionosphere, atmosphere, biosphere, etc. providing the conditions that make possible biology's ecological regeneration and its vegetation's photosynthetic energy harvesting and fossilized storing system They are thereafter destined to agglomerate with other newly formed matter until a critical mass of matter is convened at which moment the matter again becomes a comet, asteroid, moon, or planet, and soon.

What is important to be ever and again concerned with is the most difficult to understand phase of maximum disorderliness-occurring as groups of mistily lesser or foggily greater densities of disorderly, every-which-way swiftly intercoursing individual star-dust entities. Here we find the multi-locally in Universe tentative-preclouds pulsating for aeons on the critical-thresholds between cosmic localizing or further dispersion between convergence and divergence between entropy and syntropy. If the former predominates, the total intercoursing becomes progressively denser.

Due to varying dominances of radiational and gravitational forces, the cosmic cloud formations may remain progressively disorderly for aeons. Vast quantities of energetic Universe are at all times preoccupied in these as yet kinetically disorderly phases. Much of Universe is at all times in this indeterminate energy-entity vacillating condition. Large quantities of cosmic energy tarry for long in this non-differentiable maelstrom. This maelstrom is not one of cosmic lawlessness. It is inherent in the principles of ceaseless motion and transformation and a myriad of swiftly shifting cosmic law predominances.

As we go from high frequency of complex intertransformings to low frequency of such we come to the human mind- and brain-tunability ranges. With frequency intertransforming into brain- and mind-tunability ranges we come to comprehendible entity-trendings realized as convergent gravitational collectings and divergent radiation broadcasting for speculatively predicatable formations or transformations.

My books Synergetics, volumes 1 and 2, tell you much more about the way energy quanta temporarily become vectorially lost to cosmic account yet are realistically recoverable by Universe (see Richter transformation, Synergetics, vol. 1). It is this pre-Magellanic-cloud, predust, pre-anything of the for-the-time-being inevident, non-conceptual, unimaginable, only potentially unlost, and only in pure principle recoverable (as physically demonstrated by the Richter transformation model) that renders this negative 0 = 2 phenomena.

This vastness of overlapping unaccountability is difficult for those unschooled in Synergetics to comprehend with its inverted energies, black-hole, nothingness phases of the ever regenerative Universe's multi-nonsimultaneous. only in overall eternity, regenerativity.

In the total scheme of Universal regenerativity, some entities remain longer than others as nonmembers of any cosmic group but Brouwer's theorem makes it mathematically possible that throughout all the transformings there is always one "something" point that has never lost central relationship in respect to all the other moving points. This persistently self-centralizing point, in respect to all the nonsimultaneous intertransformings of and re-interpositionings of our personally experienced, successively overlapping episodes of eternally regenerative Universe, always makes the observer, the individual life the center around which evolves the unique set of omnidirectionally experienced Universe.

Zerox of Octa Vol. 4 into Tetrahelix Vol 3 of Richter Transformation from B.F. Portfolio.

[Editor's note: The above line, this time in Bucky's handwriting instead of typed, is another one I found on a page by itself. He must be referring to the "Now You See It. Now You Don't." color plate from Synergetics 2.]

We next recall that because two lines cannot pass through the same point at the same time (see illustration repeated on following page) all the seemingly single lines seemingly interconnecting each of the vertexes of the isotropic vector matrix with its twelve surrounding vertexes are in fact twelve sets of two lines each with each of those two event histories approaching one another to a critical proximity distance then avoiding actual contact with one another at the vertexes (see "Degrees of Freedom" illustration on page 11).

Figure 18 Interference Phenomena: Lines Cannot Go Through the Same Point at the Same Time
(Fig. 6.1 from Cosmography which is an updated version of Fig. 517.10 from Synergetics)

Nature so abhors equilibrium that all the lines approach crossing one another at the vertexes but never touch one another always being shunted to one side or the other of the vertexes; each line getting only into what we call critical proximity of the other and then receding from one another. This interrepulsion produces interpulsing bulges which sum totally produce waves. This means that the word "vertex" which means "toward touching" is not inaccurate in respect to what actually transpires; that is, a vertex is an approached location. There is a nearest-to-but-not-actually-vertex-touching local of proximity which we will now symbolize as X. We discover that our four somethings (S of the individual minimum system corner location identification) are also the X locals in an isotropic push-pull vector matrix.

We next discover that all the hierarchy of primitive, pre-time and -size, geometrical configurations are describable by their one-side or the other equilibrium by-passing, push-pull vectors, non-touching, critical-proximity-located something events.

Much of the foregoing material is illustrated in much greater detail in the 1400 pages of volumes one and two of my book Synergetics.

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Copyright 1999, Estate of Buckminster Fuller, all rights reserved.