1984

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The following extracts are from a two-volume study:

- R Buckminster Fuller.
*Synergetics*:*Explorations in the Geometry of Thinking*, 1975; and*Synergetics 2: Explorations in the Geometry of Thinking*, 1979. Macmillan,

0.1 "Synergetics is the geometry of thinking. How we think is epistemology, and epistemology is modelable; which is to say that knowledge organizes itself geometrically..." (I, 905.01)

0.2 "Any conceptual thought is a system and is structured tetrahedrally. This is because all conceptuality is polyhedral." (I, 501.101) "By tetrahedron, we mean the minimum thinkable set that would subdivide Universe and have interconnectedness where it comes back upon itself." (I, 620.03)

0.3 "All systems are polyhedra: All polyhedra are systems." (II, 400.56)

0.4 "Instead of starting with parts -- points, straight lines, and planes -- and then attempting to develop these inadequately definable parts into omnidirectional experience identities, we start with the whole system in which the initial "point" turned out to be self, which inherently embraced all of its parameters wrapped tightly in that initial underdeveloped, self-focused aspect of self..." (I, 488.00)

0.5 "I am confident that I have discovered and developed the conceptual insights governing the complete family of variables involved in realization by humanity of usable access to the ultimate computer.,.ultimate meaning here: the most comprehensive, incisive and swiftest possible information-storing, retrieving, and variable processing facility with the least possible physical involvement and the least possible investment of human initiative and cosmic energization." (I, 427.06)

0.6 "Human may be quite unconscious of their unavoidable employment of isotropic vector matrix fields of thought or of physical articulations....However, their physical brains...are always and only most economically interassociative, interactive, and intertransforming only in respect to the closest-packed isotropic vector matrix fields which altogether subconsciously accommodate the conceptual geometry picturing and memory storing of each individual's evolutionary accumulation of special-case experience happenings, which human inventories are accumulatingly stored isotropic-vector-matrix-wise in the brain..." (I, 426.473

0.7 "Human thoughts are always conceptually and definitively confined to system considerablility and comprehension....All systems are subject to comprehension (I, 400.07-20) "A considerable set is a locally definitive system within Universe that returns upon its considerability in all circumferential directions and therefore has an inherent withoutness and withinness..." (I. 509.02)

0.6 "Experience is inherently omnidirectional....Experience is inherently terminal, partial, and differentiable: the antithesis of eternal integrity. The difference between synergetics and conventional mathematics is that it is derived from experience and is always considerate of experience, whereas conventional mathematics is based upon "axioms"..." (I, 502.25-31)

0.9 "By synergic postulate intellect is irrevocable and irreversibly comprehensive -- both subjectively and objectively -- in respect to energy. Intellect mensurates and modulates relative energy events and event interrelationships." (II. 1075.21-22)

0.10 "The conceptual process is never static.".(I, 509,01) "Starting with whole Universe, we quickly reach any local system within the totality by differentiating it out temporarily from the whole for intimate consideration....Once you state what your realistic optimum recognition of totality consists of, then you find how many bits or subdivision stages it will take to isolate any items within that totality." (I, 522.32-34)

0.11 "When man employs nature's basic designing tools, he needs only generalized angles and special-case frequencies to describe any and all omnidirectional patterning experience subjectively conceived or objectively realized." (I, 515.12)

0.12 "Angle is independent of time. But angle is conceptual; angle is angle independent of the length of its edges....Time and size and special-case physical reality begin with frequency. Pre-time-size conceptuality is primitive conceptuality. Unfrequnced angular topology is primitive....Increase of relative size dimension is accomplished by multiplication of modular and cyclic frequencies, which is in turn accomplished only through subdividing a given system." (II, 1033.601-613)

0.13 "We have arrived at a new phase of comprehension in discovering that all of the physical cases experimentally demonstrable are only special cases of the generalized principles of the subfrequency, subtime, and subsize patterning integrity of the nucleus-containing, closest-packing isotropic vector matrix system." (II, 1006.34)

0,14 "We may hypothesize that information as it increases exponentially -- explodes. Conceptuality implodes, becoming increasingly more simplified," [I, 501.08)

0.15 "An elephant does walk, Elephants are successful designs. We have no evidence of biological species that are inherently incomplete designs." (l, 1050.13) "And all the categories of creatures act individually as special-case and may be linearly analyzed; retrospectively, it is discoverable that inadvertently they are all interaffecting one another synergetically as a spherical, interprecessionally regenerative, tensegrity spherical integrity. Goedesic spheres demonstrate the compressionally discontinuous -- tensionally continuous integrity. Ecology is tensegrity geodesic spherical programming." (l, 1005.53)

0.16 "Propostion to be proven: that structural systems are always special
case operational realizations in which there is a constant relative abundance
of all the topological and system characteristics, the only variable being
a quantity multiplier consisting of one of the first four prime numbers --
1, *2*, 3, and 5 -- or an intermultiplied plurality of the same first
four prime numbers." (II, 1072.10)

0.17 "Humans accommodated the primes 1, 2, 3, and 5 in the decimal and duodecimal systems....But so long as the comprehensive cyclic dividend fails to contain prime numbers which may occur in the data to be coped with, irrational numbers will build up or erode the processing numbers to produce irrational, ergo unnatural,results. We must therefore realize that the tables of the trigonometric functions include the first 15 primes..." (I, 1230.11)

0.18 "The first seven primes factorial is a sublimely rememberable number. It is a big number, yet rememberable. When nature gives us a number we can remember, she is putting us on notice that the cosmic communication circuits are open: you are connected through to many sublime truths." (I, 1234.02)

0.19 "Initial comprehension is holistic. The second stage is detailing differentiation. In the next stage the edges of the tetrahedron converge like petals through the vector-equilibrium stage. The transition stage of the icosahedron alone permits individuality in progression to the omni-triangulated spherical phase." (I, 1005.63)

1.1 "The sphere is maximal complex unity and the triangle is minimal simplex unity. This concept defines both the principles and the limits governing finite solution of all structural and general-system-theory problems." (I. 707.01)

1.2 "Unity, as represented by the internuclear vector modulus, is of necessity always of the value two; that is, unity is inherently two, for it represents a minimum of two energy centers." (I, 240.40)

2.1 "Where cyclic unity is taken as 360 degrees of central angle, the difference between infinity and finity is always exactly two, or 720 degrees." (I, 224.13)

2.2 "Defining frequency in terms of cycles requires a minimum of two cycles." (II, 1033.601)

2.3 "Frequency is plural unity. Frequency is a multicyclic fractionation of unity. A minimum of two cycles is essential to frequency fractionation." (, 515.30)

2.4 "Therefore, mathematically speaking, all defined conceptioning always equals finite Universe minus two. The indefinable quality of finite Universe inscrutability is exactly accountable by two." (I, 224.50)

2.5 "When man employ's nature's basic designing tools, he needs only generalized angles and special-case frequencies to describe any and all omnidirectional patterning experience subjectively conceived or objectively realized....There are only two possible covariables operative in all design in Universe: they are modifications of angle and frequency." (I, 515.12 and 516.01)

2.6 "Vectors and tensors constitute all elementary dimension. A vector represents an expelling force and a tensor an impelling force." (I, 521.101)

2.7 "No function exists by itself...Tension and compression are always and only interfunctioning covariables whose seemingly relative importance is a consequence of local pattern inspection." [I, 642.01)

2.8 "The word line was nondefinable: infinite. It is the axis of intertangency of unity as plural and minimum two. Awareness begins with two. This is where epistemology comes in. The "line" becomes the axis of spin....A line is directional experience." (I. 521.21-22)

2.9 "There are no specific directions or localities in Universe that may be opposingly designated as up or down. In their place, we must use the words out and in. We move in toward various individual energy-event concentrations, or we move out from them....In is temporal; out is eternal." (I, 524.101-14)

2.10 "There are in all systems the additive twoness of the poles and the multiplicative twoness of the coexistent concavity and convexity of the system's insideness and outsideness." (I, 400.46)

2.11 "The number of vertexes of every omnitriangulated structural system is rationally and differentially accountable:

- first, by selecting and separating out the always additive two polar vertexes that must accommodate the neutral axis of spin inherent in all individual structural systems to permit and account for their independent motional freedom relationship from the rest of the universe...
- second, we identify the always multiplicative duality factor of two characterizing the always coexistent insideness-outsideness of systems....' (I. 223.01)

2.12 "In an omnitriangulated structural system, the number of vertexes is always evenly divisible by two." (I, 223.19)

2.13 "Because people thought of the nucleus only as oneness, they for long missed the significant twoness of spherical unity as manifest in the atomic weights in the Periodic Table of Elements. When they finally learned that that inventory of data required the isolation of the neutron, they were isolating the concave. When they isolated the proton, they isolated the convex." (I. 415.10-113

2.14 "Recalling that the (24) vector edges of the vector equilibrium (25 great cicles) exactly equal the radial explosive forces, while the [31 great circle) icosahedron's 30 external edges are longer and more powerful than its 30 radial vectors, yet each has an excess of one great circle, which great circle must have two polar vertexes of spin, we encounter once more the excess two polar vertexes characterizing all topological systems, and witness the excess of embracingly cohering forces in contradistinction to the explosively disintegrative forces of universe." (I, 1052.31]

2.15 "Although not enumerated topologically...there are -- in addition to the nine aspects and 32 cases -- two additional ultimate conceptual aspects of the complementary macro- and micro-remainder of the physical Universe: all the as-yet-undiscovered...special cases as an epistemographic complementary to all the as-yet-undiscovered...general principles." (II, 1044.13)

3.1 "You can have no sense of awareness of shape with just one otherness or two othernesses. Shape awareness commences only with three othernesses where the relationship of the three as a triangle has finite closure. Shape is what you see areally..." (I, 1023.18)

3.2 "When three or more "lines", "vectors", or trajectories each cross two others, we have an opening." (I, 524.32)

3.3 "This triadic concept is exclusively planar -- ergo, nonexistent. What is inadvertently omitted is the observer of the planar triad, whose observer position marks the fourth corner of the tetrahedron, the minimum system." (II. 542.01)

3.4 "Every physical event in nonsimultaneous scenario Universe is characteried by three multidimensionally interlinked vectors that intersect precessionally, i,e, at angles other than 180 degrees to one another....The open-ended tripartite spiral can be considered as one energy event consisting of an action, reaction, and resultant. Two such tripartite-vectored "spirals", one negative and one positive, combine to form the terahedron." (I, 511.02-10)

3.5 "Vectors always and only coexist with two other vectors, whether or not expressed; i.e. every event has its nonsimultaneous action, reaction, and resultant. But every event has a cosmic complementary; ergo, every vector's action, reaction, and resultant have their cosmic tripartite complementarles." EI, 521.053

3.6 "The threeness of the quarks shows up at the three minimum convergent lines around each vertex of the minimum system consisting of only six lines." (II, 1052.355)

3.7 "Defining frequency in terms of interval requires a minimum of three intervals between four similar events." (II, 1033.601)

3.8 "While great circles are the shortest distance around spheres, a single great-circle band around a sphere will readily slide off....Not until we have three noncommonly polarized, great-circle bands providing omnitriangulation as in a spherical octahedron, do we have the great circles acting structurally to self-interstabilize their respective spherical postionings by finitely intertriangulating fixed points less than 180 degrees apart." (I, 706.20)

3.9 "There are only three possible cases of fundamental omnisymmetrical, omnitrianguläted, least-effort structural systems in nature: the tetrahedron, with three triangles at each vertexi the octahedron, with four triangles at each vertex; and the icosahedron, with five triangles at each vertex." (I, 532.41)

3.10 "Design strategies: All the calculations required for the design of geodesic domes may be derived from the three basic triangles of the three basic structural systems:

- the 120 right spherical triangles of the icosahedron
- the 48 right spherical triangles of the octahedron, and
- the 24 right spherical triangles of the tetrahedron.

All the great-circle behaviors occurring around the whole sphere take place within just one of those three basic right triangles and repeat themselves in all others." (II, 795.08)

3.11 "At the minimum limit of the hierarchy (of primitive structural systems] are the separate A,B and T Quanta Modules, and at the minimum limit of allspace filling -- ergo, of all Universe structuring -- we have the three-module mites consisting each of two A and one B Modules, The mites are the quarks. The two energy-holding A Quanta Modules and the one energy-dispersing B Quanta Module of which the mite is composited exactly correspond with the plus-two, minus-one characteristics model of the three-separate-entity functions of the quark." (II, 1052.362-363)

3.12 Symmetrical structural omnitriangulated systems contain three principal topological aspects: vertexes, faces and edges. "In respect to the original base number of nonpolar vertexes, there will always be twice as many openings ("faces") and three times as many vector edges of the symmetrical structural system..." (I, 223.01)

3.13 Symmetrical structural omnitriangulated systems give rise to three systems of great circles(according to whether opposite vertexes, face midpoints, or edge midpoints are used as the axis), or to more if faces differ. (I, 1042.05)

3.14 "There are three classes of closest packing of unit radius spheres:

- Systematic: symmetrical omnidirectional closest packing...
- Asymmetrical: closest packing conglomerates...
- Volumetric: symmetrical closest packing..." (I. 222.50)

3.15 The relationships between three or fewer events are always equal in number to the number of events. (I, 227.02)

4.1 "As minimum or prime systems consist of four event foci...unity is inherently plural." (I, 400.08)

4.2 "Systems are aggregates of four or more critically contiguous relevant events having neither solidity nor surface or linear continuity." (I. 400.26)

4.3 "At minimum, it takes four triangular planes having Inherent fourness of vertexes to constitute differential withinness and withoutness." (I. 400.05)

4.4 "Dimension begins at four. Four-dimensionality is primitive and exclusively within the primitive system's relative topological abundances and relative interangualr proportionment. Four-dimensionality is eternal, generalized, sizeless, unfrequenced." (II, 1033.611)

4.5 "All conceptually thinkable, exclusively metaphysical experlencings are fourfoldedly characterized...(systematically, topologically, angularly, numerically). All generalized principles are conceptually thinkable and fourfoldedly definable. Generalization is conceptually (i.e. systematically) imaginable independent of (5) frequency." (II, 1072.22)

4.6 "By tetrahedron, we mean the minimum thinkable set that would subdivide Universe and have interconnectedness where it comes back upon itself." (I, 620.03)

4.7 "All systems are polyhedra: All polyhedra are systems....we find that the first four prime numbers - 1, 2, 3, and 5 - are the only variables present in the Eulerean topological inventorying of all the omnitriangularly, nonredundantly stabilized, symmetrical polyhedra." (II, 400.56 and 1074.31)

4.8 "The relationship between four or more events are always greater in number than the number of events." (I, 227.02)

4.9 "Life consists of alternate observing and articulating interspersed with variable-recall rates of "retrieved observations" and variable rates of their reconsideration to the degrees of "understandability" (namely 4 processes). Resonantly propogated evolution oscillatingly induces tetrahedral quanta -- both metaphysical and physical -- formulated vectorially between four "star-event" phases:

(1) observation or (1) recall (2) consideration (2) reconsideration t3) understanding, and (3) understanding (4) articulation (4) articulation" (I, 513.06-07)

4.10 "The fact that unity is two means that an observer is at a minimum two, but realistically four, because the observer is a system; and the observed is at a minimum two, but being a system, is realistically four. A range-finder is inherently tetrahedral." (II, 267.04)

4.11 "There are four kinds of positive and negative." (II, 223.06)

4.12 "Topologically the additive twoness identifies the opposite poles of spinnability of all systems! the multiplicative twoness identifies the concave- insideness and convex-outsideness of all systems: these four are the four unique twonesses." (II, 223.09)

4.13 "The four closest packed spheres make a closest packing array. They are mass-attractively locked together as a tetrahedron. No further interrotation is possible (as it is with three or less)." (I, 411.08)

4.14 "Only four primary systems or contours can be developed by closest packing of spheres in omnisymmetrical concentric layers: tetrahedron, octahedron, cube. cuboctahedron."(I, 223.20)

4.15 "In an omnitriangulated system the number of faces is always evenly divisible by four." (I, 223.19)

4.16 "Four sets of axes of spin....These four sets of (50) unique topological aspects of the vector equilibrium provide four different sets of symmetrically positioned polar axes of spin to generate 25 great circles of the vector equilibrium."(I. 450.11) (Compare with 3 sets of 62 features of icosahedron generating 31 great circles)

5.1 "If the system is frequnced, it is at a minimum linearly five-dimensional.... Size is special case, temporal, terminal, and more than four-dimensional." (II, 1033.61-2)

5.2 "Generalized principles have topological defineability of angle, number and constancy. Special cases have unique frequency dimenslonability. Wherefore we propose that all recallably thinkable experiencings, physical and metaphysical, are fivefoldedly characterized: (1) systematically, (2) topologically... (3) angularly...(4) numerically...(5) frequency definable...All conceptually thinkable, exclusively metaphysical experiencings are fourfoldedly characterized as above. All generalized principles are conceptually thinkable and fourfoldedly definable. Generalization is conceptually (i.e. systematically) imaginable independent of (5) frequency. The fifth characteristic, frequency, is the unique special case variable. Physicalexperiencings are dependent not only on the four generalizable characteristics, but also on the fifth, frequency (i.e. size)" (II, 1072.21-23)

5.3 "Going beyond the original formulation of the four-sphere-vertexed minimum structural system, we observe that the addition of a fifth spherical otherness to the four-ball structural system's symmetry brings about a polarized-system condition....Due to this inherent individual differentiability, the fivefoldness constitutes a self-exciting, pulsating-propagating system."(I,981.03)

5.4 "We discover that an icosahedron is the first degree of contraction of the vector equilibrium. We never catch the vector equilibrium in its true existence in reality: it is always going one way or the other. When we go to the icosahedron, we get to great realities. In the icosahedron, we get to a very prominent fiveness: around every vertex you can always count five." (I, 461.03)

5.5 "The fundamental fiveness of the icosahedron is split two ways,
with *2 1/2*going one way (the outside-out way) and 2 1/2 going
the other way (the inside-out way)... It is then in transforming from a positive
two-and-one-halfness to a negative two-and-one-halfness that the intertransformable
vector-equilibrium-to- icosahedron, icosahedron-to-vector-equilibrium, equilibrious-to-disequilibriousness
attains sumtotally and only dynamically a spherical fiveness."(II,
1053.12-15)

5.6 "All the phenomena external to -- and more complex than -- the five-frequency vector equilibria relate to chemical compounds. Anythings internal to -- or less complex than -- the five-frequency vector equilibrium relates principally to single atoms. Single atoms maintain onisymmetries! whereas chemical compounds may associate as polarized and asymmetrical chain systems."(I, 440.03)

6.1 "It takes a minimum of six interweaving trajectories to isolate insideness from outsideness, ergo, to divide all Universe systematically into two parts

-- macrocosm and microcosm." (I, 240.32)

6.2 "A system is the first subdivision of Universe. It divides all the Universe into six parts:

- first, all the universal events occurring geometrically outside the systems
- second, all the universal events occurring geometrically inside the system:
- third, all the universal events occurring nonsimultaneously. remotely, and unrelatedly prior to the system events;
- fourth, the Universe events occurring nonsimultaneously, remotely, and
- unrelatedly subsequent to the system events;
- fifth, all the geometrically arrayed set of events constituting the system itself; and
- sixth, all the Universe events occurring synchronously and or coincidentally to and with the systematic set of events uniquely considered."(I. 400.011)

6.3 "There are six uniquely differentiable components of all systems and of all thoughts: CD insideness; (2) outsideness; and (3) (4) (5) (6) the four star events (see 4.9 above) that do the systemic defining of the insideness and outsideness..." (II. 400.71)

6.4 "Shape requires six restraints. Exactly six system Intel-restraints produce structure. Six restraints are essential to structure and pattern stability." (II, 400.664)

6.5 "The definition of structure as the pattern of self-stabilization of a complex of events with a minimum of six functions as three edges and three vertexes, speaking both vectorially and topologically." (I, 251.11)

6.6 "You have six vectors or none for every energy event....When we bring together these two sets of three vectors each, they integrate as six vectors and coincidentally also make one tetrahedron (of six vector edges). The tetrahedron is the veritably conceptualizable unit of one energy quantum." (I. 511.03-04)

6.7 "There are six basic motions in Universe. All of them are positive and negative: active and passive. The first five are individually experiencable and popularly familiar; the last -- precession -- is less popularly familiar:

- spin... - expansion-contraction... - orbit... - torque... - inside-out... - precession..." (II, 400.654)

6.8 "There is a unique and limited set of angle and magnitude consequences of interfering events....There are six fundamentally unique patterns of the resultants of interferences....

tangential avoidance - refraction - modulated noninterference - smash-up (explosion) (frequency modulation) - critical proximity (the minimum knot) - reflection (I. 517.05, 101-12)

6.9 "In an omnitriangulated system the number of edges is always evenly divisible by six" (I. 223.19)

6.10 "Not until a six otherness appears remotely, approaches, and associates with the fivefold system can the latter learn from the newcomer of its remote witnessing that the fivefold system has indeed been rotating axially." (I. 981.07)

6.11 "The exterior and separate set of polar vertexes is the additive twoness of systems, and the congruent exterior-interior set is the multiplicative twoness of all systems. The interior-exterior differentiating fourness has an interrelated sixness that differentiates as a unique third kind of twoness of unique interrelatedness of all systems." (II, 1073.22)

6.12 "If we assume that the vertexes are points of discharge, then we see how the six great circles (of the 31) of the icosahedron -- which never get near its own vertexes -- may represent the way residual (energy) charge will always remain held on the surface of the icosahedron....The icosahedron makes it possible to have individuality in Universe. The vector equilibrium never pauses at equilibrium, but our consciousness is caught in the icosahedron when mind closes the switch. The icosahedron's function in Universe may be to throw the switch of cosmic energy into a local shunting circuit. In the icosahedron energy gets itself locked up even more by the six great circles..." (I, 458.05-11)

6.13 "The rational spherical excess of six degrees (of the icosahedron's
120 -- 60 plus and 60 minus -- similar tetrahedral components) is symmetrically
distributed to each of the three central and three surface angles *of *each
of the 120 tetrahedral components of the spherical icosahedron. This sixness
phenomenon tantalizingly suggests its being the same transformative sixness
as that which is manifest in the cosmically constant sixfoldness of vectors
of all the topological accountings..." (II, 1043.01-02)

6.14 "The total available energy of a system is related to its surface area, involving the second power of the radius....And since surface functions as the electromagnetic-energy carrier, and since the energy relayed to their surfaces alternates from the positive to the negative tetrahedron...the rotation of the triangle within the cube passes through the common energy centres of the two tetrahedra and delivers its content to the other base surface, after which it pulses through center delivery of the opposite charge to the other surface...The six cube-edge travelings of the triangle's vertexes accomplished with each cycle of the triangle-in-cube shuttle coincides in number and is akin to the six vector edges comprising one tetrahedral quantum; the sixness of wavilinear and sometimes reangularly redirected travelings employs also the six basic degress of freedom articulated by each and every one cosmic event." (I, 464, 07)

7.1 "Since synergetics geometry embraces nuclear and angular topology, it adds four more minimum aspects to Euler's inventory of three:

Euler (vertexes, faces, edges) - Synergetics (angles, insideness/outsideness, convexity/concavity, axis of spin)" (II, 1044.01)

7.2 "The seven unique cosmic axes of symmetry describe all of crystallography....

Vector equilibrium: Icosahedron: 1. Square face centrepoints 3 5. Triangular face centrepoints 10 2. Triangular face " 4 6. Midedges 15 3. Vertexes 6 4. Midedges 12 7. Vertexes 6 Total great circles 25 Total great circles 31 Grand total (*25) 56 Axes of symmetry 1 through 6 result in great circles through the same 12 vertexes. Great circles generated by. the 7th go through no vertexes (I, 1042.05)

7.3 "The prime generation of the seven axes of symmetry are the seven unique perpendiculars to the faces of the seven possible truncations of the tetrahedron." (II. 104.11)

7.4 "The trigonometric identification of the great-circle trajectories of the seven axes of symmetry with the 120 basic disequilibrium LCD triangles of the spherical icosahedron." (II, 251.29)

8.1 "The four cosmically complementary twonesses and the four local system twonesses altogether eternally regenerate the scientific generalization known as complementarity. Complementarity is sum-totally eightfoldedly operative..." (II, 223.08)

8.2 "Spin twoness is additive. Duality twoness (concave-convex) is multiplicative. The spin twoness and duality together comprise a third relationship twoness. The fourth twoness is comprised of the macro-micro (insideness and outsideness) twoness.... Integral Universe is fourfold twoness -- 8 " (II, 1073.13-14)

8.3 "The fourness of self and the fourness of otherness " comprehension. Comprehension involves tuned-in octave resonance as well as omnidirectional and local angualr integrity." (II. 268.02)

8.4 "There is always a total of eight (four positive, four negative) unique interpermutative, intertransformative, interequatable, omniembracing phases of all cyclically described symmetrical systems, within any one octave of which all the intercovariable ranging complementations of number occur. For instance, in a system such as spherical trigonometry, consisting of 360 degrees per circle or cycle, all the numerical intervariabilities occur within the first 45 degrees, (therefore) 45x8 - 360" (II, 1238.28)

8.5 "Cosmic hierarchy of primitive polyhedral systems: The constant octave system interrelationship is tunable to an infinity of different frequency keys." (II, 1033.20)

8.6 "The conceptual sequence....illustrates the basic octave behavior of structural transformations. The first three figures -- tetra, octa, icosa -- represent the positive outside-out set of primitive structural systems....As the transformation sequence changes from divergent evolution to convergent evolution, from five, then four, then three equiangular triangles around each corner, it produces successively the inside-out icosa, octa, and tetra, until the convergent involutional contraction attains the phase of maximum nothingness," (II. 610.31)

8.7 "The begetted "eightness" as the system-limit number of nuclear uniqueness of self-regenerative symmetrical growth may well account for the fundamental octave of unique interpermutative integer effects..." (I, 415.40)

8,6 "Pulsative octave: The interaction of all numbers other than nine creates the wave phenomenon described, i.e., the self-invertible, self-inside-outable octave increasing and decreasing pulsatively, fourfoldedly, and tetrahedrally. No matter how complex a number-aggregating sequence of events and conditions may be, this same number behavior is all that ever happens. There is thus a primitively comprehensive, isotropically distributive, carrier-wave order omniaccommodatingly permeating and embracing all phenomena." (I, 1223.10)

8.9 "The vector equilibrium is a tetrahedron exploding itself, turning itself inside out in four possible directions. So we get eight: inside and outside in four directions. The vector equilibrium is all of the eight potentials." (I. 441.02)

8.10 "The tetrahedron discovers that each of its four vertexes can be plungepassed through its innards to be extended on the opposite side of its four triangular faces. This automatically develops eight tetrahedral, self-transformation awarenesses and produces eight common nuclear-vertex tetrahedra of the vector equilibrium." (I, 482.00)

8.11 "The vector equilibrium has four inside-out and four outside-out self-inter-cancelation, eight-conguent, zerovolume tetrahedra, as well as eight centrally single-bonded tetrahedra of maximum zerovolume expansion: both invoke the cosmically intolerable vacuum voids of macro-micro-nothingness essential to the spontaneous capture of one quantum's six vectors..." (II. 1033.653)

8.12 "Eight spheres will not close-pack as a cube and are utterly unstable." (I, Fig 416.01)

8.13 "In the octahedron as the maximum conservation and quantum-annihilabillty model of substance the processing vector edge of the entropie octahedron drops out 1 tetra; 1 tera=6 vectors * 1 quantum of energy which...may be effortlessly reformed by reentering the vector equilibrium to produce the icosahedron and thus to form new substance or matter....Icosahedron is the least dense of all matter." (II, 1033.655-6)

9.1 "Every tetrahedron, every prime structural system in Universe, has nine separate and unique states of existence: four positive, four negative, plus one schematic unfolded nothingness, unfolded to an infinite, planar, neither- one-nor-the-other equilibrious state....Thus we find the vector equilibrium to be the inherent zero-nineness of fundamental number." (II, 1013.41-42)

9.2 "Planar and nonpolar-vertex four-dimensionality accommodates and imposes four positive, four negative, and neutral (nineness) of the operational interweave behavior of number. (I, 527.31)

9.3 "The None or Noneness/Nineness permits wave frequency propagation cessation. The Nineness/Zeroness becomes a shutoff valve. The Zero/Nineness provides the number logic to account for the differential between potential and kinetic energy." (I, 415.43)

9.4 "Four fixes/define an insideness-including and outsideness-excluding tetrahedror which is the minimum cosmic system and which cannot have less than 32 unique : and differentially describable generalized cases of the nine irreducible-in- number unique topological aspects of the minimum system, but which in special frequenced cases may have more." (II, 1044.12)

9.5 "Synergetics further augments Euier's inventory of three topological aspects (14 cases) with six additional and primitively constant topological aspects: 4th aspect (12 cases): the 12 unique...angles of the minimum system; 5th aspect [two cases): ...the insideness...integral otherness, and the outsideness...irrelevance otherness; 6th aspect (two cases): ...co-occurring concavity and convexity of all systems, and...inseparable convergence to and divergence from system center; 7th aspect [two cases): the additive twoness of the two vertexes always extracted from the system's total inventory of vertexes to serve as the poles of the system's neutral axis of spin; 8th aspect: the sum of the angles externally surrounding the vertexes of any system will always equal 720 degrees less than the number of external vertexes multiplies by 360 degrees; 9th aspect: the sum of the angles around all the external vertexes of any system will always be evenly divided by 720 degrees, which is the angular description of one tetrahedron." (I, 1044.06)

10.1 "In omnidirectional closest packing of spheres, symmetrically concentric shells have a number of spheres per layer governed by 10F "2" (I, 222.00) Where F = frequency.

11.1 "The primary and secondary icosa symmetries altogether comprise 121 - 11 great circles." (II, 987.132)

11.2 "The toal of the secondary great circles of the VE is 96 new great circles (which with the 25 primary circles makes 121 - 11 )" (II, 987.137)

12.1 "In omnidirectional closest packing of spheres, the first layer around the nuclear sphere contains 12 spheres," (I, 222.22)

12.2 "The network of vectorial lines most economically interconnecting the volumetric centers of 12 spheres closest packed around one nuclear sphere of the same radius describes not only the 24 external chords and 12 radii of the vector equilibrium but further outward extensions of the system by closest packing of additional uniform-radius spheres..." (I, 451.03]

12.3 "System awareness begins when we find the otherness surrounding us, when we are omnidirectionally enclosed." (I. 1023.20)

12.4 "It takes six external push vectors and six external pull vectors to define the minimum nuclear structural system: a primary subdivision of Universe." (I, 401.06)

12.5 "Redundancy can be neither predicted nor predetermined by observation of either the integral constraints or external freedoms of energetic behaviors of single struts, or beams, or columns, or any one chain link of a series that is less than 12 in number, i.e. six positive vectors and six negative tensors. Of these 12, six are open-endedly uncoordinate, disintegrative farces that are always omnicohered by six integrative forces in finitely closed coordination." (I. 721.02)

12.6 "With each and every event in Universe -- no matter how frequently recurrent -- there are always 12 unique, equieconomical, omnidirectionally operative, alternate-action options, which 12 occur as four sets of three always interdependent and concurrent actions, reactions, and resultants." (II, 537.O6)

12.7 "A vector is one-twelfth of relevant system potential." (I, 521.09)

12.8 "Synergetics discovers that whatever is rigidly related to anything else discloses 12 restraints. There are a minimum of 12 restraints in developing anything we might call a rigidly related set of events....The game of Universe is like chess with 92 unique men, each of which has four frequencies available, and it works on 12 degrees of freedom instead of a planar checkerboard." [I, 537.01-03)

12.9 "Experience is inherently omnidirectional; ergo, there is not just one "other". There are always at least twelve "others". The connection between the six degrees of freedom and omnidirectionality is, of course, the vector equilibrium. Pulsation in the vector equilibrium is the nearest thing we will ever know to eternity and god: the zerophase of conceptual integrity inherent in the positive and negative asymmetries that propagate the problems of consciousness evolution. Our inherently limited perceptivity requires these definitions of the asymmetric emphasis of experience." (I, 502.25)

12.10 "The definition of gravity as a spherical circumferential force whose effectiveness has a constant advantage ratio of 12 to 1 over radial inward mass- attraction." (I, 251.07)

12.11 "We may think independently of size in respect to tetrahedra, which consist of 12 separate angles." (I, 515.11)

13.1 "In omnidirectional closest packing of equiradius spheres around a nuclear sphere, 12 spheres will always symmetrically and intertangentially surround one sphere with each sphere tangent to its immediate neighbours. We may then close-pack another symmetrical layer of identical spheres surrounding the original 13." (I, 413.01)

13.2 "When...(a powered product of the primes up to 13) is substituted as a comprehensive dividend for 360 00' 00" to express cyclic unity in increments equal to one second of arc. while recalculating the tables of trigonometric functions, it is probable that many, if not most, and possibly all the function fractions will be expressible as whole rational numbers."(I, 1234.04)

14.1 "Yet the fundamental interrelatedness of the seemingly disorderly subdividing of bubble aggregates is elegantly identified with the absolute order of the isotropic vector matrix, in that all the internal polyhedre manifest 14 facets each, though a variety of polygonal shapes end sizes. This 14-ness is also manifest in the closest packing of biological cells." (I. 1025.13)

14.2 "The dynamics of symmetry may employ any seven sets of the 56 foldable great- circle variations of planar orientation. Thus it follows that both the biological cell arrays and the bubble arrays display vast varieties of asymmetries in their 14 enclosing planes, so much so that their interidentiflability with the 14 topological characteristics of the tetrahedron... has gone unnoticed until now." (II, 1041.13)

14.3 "The seven unique axes of the three unique sets (4+4+6) producing the 14 planes of the truncated tetrahedron are also identifiable with:

- the 14 planes that bound and enclosingly separate all biological cells!
- the 14 facets interbonding all bubbles in the bubble complexes; and
- the 25 and 31 unique planes generated by the seven sets of foldable great circles..." (II, 1041.12)

14.4 "In this composite spherical matrix we see all the 25 primary vector equilibrium great circles and two sets skewed-positive and negative of the icosahedron 31 great-circle sets. (31 x 2 - 62. 62 + 25 "87. But 14 of the 87 are redundant ....Thus seven positive are redundant and seven negative are redundant.)" (II, Fig. 1132.018)

14.5 "The geometric form most compactly developed from the closest packing of spheres around one nuclear sphere is not that of a composite sphere, but is always a polyhedra of 14 faces composed of six squares and eight triangles, with 12 vertexes extending in tangential radius from the original 12 spheres surrounding the nuclear sphere. It is called the vector equilibrium because the radiais and the circumierentials are all of the same dimension and the tendencies to both explode and implode are symmetrical....In terms of vectorial dynamics, the outward radial thrust of the vector equilibrium is exactly balanced by the circumferentially restraining chordal forces..." (I, 430.01-03)

14.6 "A system consists at minimum of four star events (vertexes) with four nothingness window facets and six lines of unique four-star interrelationships. As in synergetics' 14 truncation faces. Euler's three aspects result in 14 cases: 4 vertexes + 4 faces + 6 edges - 14 cases." (II, 1044.05)

15.1 "Tetrahedron has the extraordinary capability of remaining symmetrically coordinate and entertaining 15 pairs of completely disparate rates of change of three different classes of energy behaviours in respect to the rest of Universe and not changing its size. As such, it becomes a universal Joint to couple disparate actions in Universe....Thus the tetrahedron becomes the only exchange agent of Universe that is not itself altered by the exchange of accommodation." (I, 623.13-14)

15.2 "The 15 (of 31) great circles of the icosahedron interact to produce 15 "chains' of three varieties of four corner-to-corner, sausage-linked, right triangles, with four triangles in each chain." (I, 456.05)

15.3 "The Basic Disequilibrium 1ZO LCD Spherical Triangle of synergetics is derived from the 15 great-circle, symmetric, three-way grid of the spherical icosahedron (which has a total of 31 great circles). It is the lowest common denominator of a sphere's surface, being precisely 1/120th of that surface as described by the icosahedron's 15 great circles." (I, 901.02)

16.1 "The 25 great circles of the vector equilibrium are the only omni-intersystem- connecting "railroad tracks" of energy in Universe. When an energy entity holding locally on a local sphere gets a green light to get back on the grand-omni-intersphere system tracks, it can do so by crossing over one of the 12 inter-atomic-sphere bridges....Because only 12 of 16 station transfers (opportunities per each spheric moment) can be accommodated, each sphere has inherently eternal retention of four energy entities. Four energy entities comprise the minimum system-defining constellation." (II, 1132.12-13)

17.1 "We find that we can get along without multirepowerings (of the smaller prime numbers) after the second repowering of the prime number 17. The prime number 17 is all that is needed to accommodate both the positive and negative octave system and their additional zero-nineness. You have to have the zero-nine to accommodate the noninterfered passage between octave waves by waves of the same frequency. The prime number 17 accommodates all the positive-negative, quanta-wave primes up to and including the number 18, which in turn accommodates the two nines of the invisible twoness of all systems." (II. 1236.42-43)

17.2 The powered product of primes up to 17 "may have an important function in atomic nucleation, since it accommodates all the prime numbers involved in the succession periods." (I, 1232.25)

18.1 "The total of nine minimum topological aspects of three from Euler (14 cases) plus synergetics' inventory of six additional aspects, with 12 angular cases and six nuclear cases for a total of 18 synergetics cases. The 14 Euler cases and the 18 synergetics cases provide a total of 32 minimum topological cases." (II, 1044.07)

18.2 "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..:(see 24) (II, 986.854)

19.1 (This is the smallest prime number on which Fuller has no remarks)

20.1 "The three-frequency tetrahedron is the highest frequency single-layer, closest-packed sphere shell without a nuclear sphere. This three-frequency, 20-sphere, empty, or nonsphere nucleated, tetrahedron may be enclosed by an additional shell of 100 balls...Note that the 20-ball empty set...consists of five sets of four-ball simplest tetrahedra...(and has)...four 10-ball tetrahedral faces." (I, 415.57-59)

20.2 "In connection with the (living organism's) 20-amino-acid system's indestructibility, we intuitively sense the necessity to consider the possible interrelations of all of the 20 amino acids' indestructible pattern integrities with other twentyfoldnesses. The number 20 is particularly significant in a plurality of nature's most elementary aspects ....It is further relevant to these considerations that the DNA-RNA code consists always and only of the four chemical compounds - guanidine, cytosine, adenine, and thiamine - and that the helix that they generate consists entirely of tetrahedra whose four constituents in all vast variety of combinations will always be the same tetrahelixes." (I, 1055.02-08)

20.3 "Vector equilibrium twentyness: Twentyness is significant as the inherent minimum twentfoldness of time-space, energy-mass, volume potential of the subfrequency vector equilibrium as quantized by using as unity the geometric volume of the minimum structural system of Universe: the tetrahedron, whose fractional integrity subdivided by the complex of A and B nodule reorientation! is in the high order number of magnitude of the amino acid'ss interrelationship permutations."(I. 1055.05)

20.4 "If the geometric volume of one of the uniform tetrahedra...is taken as volumetric unity, then the volume of the vector equilibrium will be 20" (I, 222.30)

20.5 "Equilibrious unity is 20; its
minimum frequency state is 160=2^{5}x5. This
is one of the properties of 60-degree coordination." (I. 431.04)

20.6 "There is, for instance, the minimum twentyfoldness of the icosahedron's 20 equiangular, triangular (ergo, structural) facets, which constitute the highest common unit-angle, unit-edge, and unit-vertex structural denominator of universal structural systems. The icosahedron encloses the most volume with the least energy investment as matter or work." (I, 1055.03)

20.7 "The jitterbug articulation (of the vector equilibrium) turns around at the negative tetrahedron to reexpand therefrom, returning through all of those volumetric stages to its original 20-volume integrity, to be alternately recontracted through all the 20-to-l and l-tetravolume-to-20 without any break ever occurring in the circuitry integrity of the vector-chord closures and intertriangulations in the four planes of the four-dimensional symmetry." (II, 541.19)

24.1 "To summarize, the celestial tetrahedron has six positive and six negative internal vectors and six positive and six negative external vectors."(I, 401.01

24.2 "The inherent subdivision of any tetrahedron, regular or irregular, into 24 equal modules."(II, 1224.21)

24.3 "...24 energy modules of one whole tetrahedral event, which is the quantum in nuclear physics." (I, 464.05)

24.4 "Experience is inherently omnidirectional; ergo, there is always a minimum of 12 "others" in respect to the nuclear observing self. The 24-positive and 24-negative-vectored vector equilibrium demonstrates an initially frequenced, tetrahedrally quantized unity of 20; ergo, the Universe, as an aggregate of all humanity's apprehended and comprehended experiences is at a minimum a plurality of 24 vectors." (II, 537.131)

24.5 "The 24 circumferential vectors of the vector equilibrium close back upon themselves and are united in the interconnections of their ends, providing integrative effectiveness of the circumferential vectors vs the individually acting, disintegrative abandonment of the total associative effectiveness of the 24 radial vectors of the vector equilibrium. These represent respectively, the total gravitational forces of Universe and the total radiational forces of Universe, rendering the total integrative force to be inherently more efficient that the total disintegrative forces of Universe." (II, 541.17)

24.6 "There are 24 positive and 24 negative unique numbers that reverse themselves between o and 50. This reflects three positive and three negative octaves with turnaround terminal zero accommodation." (II, 1224.33)

24.7 "Thesecond powering of numbers apparently involves a 24-positive and 24 negative resonance phasing." (II, 1224.11]

24.8 "The powerful 24-ness number behavior with its great-circle congruences and three-octave harmonics may have significant ramifications embracing the unique frequencies of the chemical compoundings as well as the nuclear geometry elucidated elsewhere in this work." (II, 1224.31)

24.9 "This 24-ness is also a requisite of three number behavior requirements as disclosed in the min-max variabilities of octave harmonics in tetrahedral and VE cumulative closest-packing agglomerations at holistic shell levels as well as in all second-powering "surface" shell growths..."(II, 1033.77)

24.10 "It requires 24-ness for the consideration of the total atomic behavior because the vector equilibrium is not allspace-fillingly complete in itself." (II, 1033.74)

24.11 "In addition to this 18-powered state (see 18.2 above) of the primitive hierarchy we discover the integrally potential six-way transformebilities of the primitive hierarchy, any one of which requires an additional powering factor, which brings us thus far to 24 powering states." (II, 966.854) (see also 33.1 i

24.12 "24 A Quanta Modules per regular tetrahedron" (II. 1224.21) (Contrast 24.2)

24.13 "24 modules of regular tetrahedron as cosmic bridge between equilibrious prime number one of metaphysics and disequilibrious prime number one of physical reality." (II, 1224.21)

24.14 "24 positive and negative triangles (basic equilibrium 48 LCD triangles) defined by the 25 great circles of the vector equilibrium." (II, 1224.21)

24.15 "24 spherical right triangles of the spherical tetrahedron's three-way great-circle grid." (II, 1224.21)

24.16 "24 highest common multiple of regular-tetrahedral-volume values of all congruently symmetric polyhedra of the hierarchy of concentric, symmetrical, rationally volumed geometries occurring within the isotropic vector matrix." (II, 1224.21)

24.17 "The cosmic hierarchy limit of 24 active tetravolumes per each sphere-into- space and each space-into-sphere intertransforming of the complex of jitterbugs." (II, 1224.21 and 1033.20)

25.1 "The 25 great circles of the vector equilibrium are the quators of spin of the 25 axes of the 50 unique symmetrically positioned topological aspects of the vector equilibrium." (I, 450.11)

25.2 "The 25 great circles of the spherical equilibrium provide all the spherical edges for five spherical polyhedra: the tetrahedron, the octahedron, cube, rhombic dodecahedron, and vector equilibrium, whose corresponding planar- faceted polyhedra are all volumetrically rational, even multiples of the tetrahedron. "(I, 454.01)

25.3 "Energy can and does travel from sphere to sphere of closest-packed sphere agglomerations only by following the 25 great circles of the vector equilibrium, always accomplishing the most economical travel distances through the only 12 points of closest-packed tangency." (I, 452.02)

25.4 "When the nucleus of the vector equilibrium is collapsed, or contracted, permitting the 12 vertexes to take the icosahedral conformation, the 12 points of contact of the system go out of register so that the 12 vertexes that accommodate the 25 (of the 31) great circles of the icosahedron no longer constitute the shortest routes of travel of the energy....The icosahedron, in fact, can only occur as a single shell of 12 vertexes remote from the vector equilibrium's multi-unlimited-frequency, concentric-layer growth...but it provides the most economical route between a chain of tangent icosahedra.., as well as for energy locked up on its surface to continue to make orbits of their own in local travel around that single sphere's surface."(I, 458.02-3!

30.1 "Both the positive and negative tetrahedra can locally accommodate the 45 [3 x 15) different energy exchange couplings and message contents, making 90 such accommodations all told. These accommodations would produce 3G different "apparent" tetrahedron position shifts, whose successive movements would always involve an angualr change of direction producing a helical trajectory." [I, 634.03)

30.2 "Not including the...additional intersystem- relationship powerings (see 33.1 below), beyond the 24 systematically integral powers (see 24.11) there are six additional...unique behavior potentials of all primitive hierarchy systems..."(II, 986.857) (See 33.1 below)

31.1 "We have now described altogether the 10 great circles generated by the 10 axes of symmetry occurring between the centers of area of the triangular faces; plus 15 axes from the mid-points of the edges; plus six axes from the vertexes....The:31 great circles of the spherical icosahedron provide spherical edges for three other polyhedra in addition to the icosahedron: the rhombic triacontahedron, the octahedron, and the pentagonal dodecahedron."(I, 457.30-40)

31.2 "The 31 great circles of the icosahedron always shunt the energies into local holding great-circle orbits, while the vector equilibrium (25 great circles) opens the switching to omniuniverse energy travel. The icosahedron is red-light, non-go; whereas vector equilibrium is green light, go. The six (of the 31) great circles of the icesahedron act as holding patterns for energies." (II, 1132.02)

31.3 "In this composite spherical matrix we see all the 25 primary vector equilibrium great circles and two sets skewed-positive and negative of the icosahedron's

31 great circles. (31 x 2 = 62. 62 + 25 = 87. But 14 of the 87 are redundant);" (II, Fig 1132.01B)

31.4 The powered product of the primes up to 31 "accommodates all trigonometric functions, spherical and planar, when unity is 60 degrees...It also accommodates the octave-nine-zero of the icosahedron's corner angles of 72 degrees..." (I. 1236.02)

32.1 "The total of nine minimum topological aspects consists of three from Euler Euler (14 cases) plus synergetics' inventory of six additional aspects, with 12 angular cases and six nuclear cases for a total of 18 synergetics cases. The 14 Euler cases and the 18 synergetics cases provide a total of 32 minimum topological cases." (II, 1044.07)

32.2 " 'Minimum somethings' consist altogether of a minimum of four corners, four faces, six edges, 12 angles, insideness, outsideness, concavity, convexity, and two poles of spinnability -- a minimum total of 32 unique geometrical features." (II, 100.201)

32.3 "Emptiness at the center: All four planes of the eight tetrahedra, i.e. 32 planes in all. are congrosnt in the four visible planes passing through their common vector equilibrium center." (I, 441.01)

33.1 "Not including the...additional intersystems-reletionship powerings, beyond the 24 systematically 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 can be comprehensively accounted for only by additional powerings. They are:

- 25th power axial roation 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-macro-isotropicity of VE-ness" (II, 9B6.857)

42.1 "In omnidirectional closest packing of spheres, the second tangential layer contains 42 spheres (around 12, around 1 central sphere)". (I, 222.23)

45.1 "Both the positive and negative tetrahedra can locally accommodate the 45 (3 x 15) different energy exchange couplings and message contents, making 90 such accommodations all told." (I, 624.03)

47.1 "The first prime number beyond the trigonometric limit (of 45) is 47. The number 47 may be a flying increment to fill allspace, to fill out the eight triangular facets of the non-allspace-filling vector equilibrium to form the allspace-filling first nuclear cube....It may be that 47 is the cosmic random element, the agent of infinite change. If all the trigonometric functions are reworked using...all prime numbers to SO, (each at leastto third power, employed as volumetric, cyclic unity, all functions will prove to be whole rational numbers as with the whole atomic populations." (II, 1238.51-52)

48.1 "Every "point" (event embryo) may articulate any one of its four event vector sets, each consisting of six positive and six negative vectors, but only one set may be operative at any one time; its alternate sets are momentarily potential." (I, 240.28)

48.2 "The system of 25 great circles of the vector equilibrium defines its own lowest common multiple spherical triangle, whose surface is exactly 1/48th of the entire sphere's surface....Twenty-four of the 48 triangles' patternings are "positive" and 24 are "negative", i.e. mirror-images of one another, which condition is more accurately described as "inside out" of one another." (I, 453.01)

48.3 "There are 48 spherical triangular tiles of the vector equilibrium nuclear sphere, which 48 triangles' pattern can be symmetrically divided into five sets of symmetrical interpatterning which coincide exactly with the projection outward onto a sphere of the five omnisymmetrical planar-defined Platonic polyhedra. whose linear edges are outlined by the respective chords of the congruent vector equilibrium's symmetrical 25-great-circle-grid and the icosahedron's 31-great-circle-grid." (I, 1053.20)

48.4 "The VE's involvement domain of 25 symmetrical, ellspace-filling tetravolumes represents only one of the two alternate intertrensformation domains of closest-packed, unit radius spheres transforming into spheres: ergo, it requires 46-tetra-volumes to accommodate this phenomenon." (II, 1033.741)

48.5 "The isosceles dodecahedron is composed of 48 blue A (Quanta) Modules, 24 of which are introverted....An additional 24 extroverted A (Quanta) nodules ...form the outermost shell..." (I, 942.21)

50.1 "The omni-equi-edged and radiused vector equilibrium is omnisymmetrical, having 12 vertexes, six square faces, eight triangular faces, and 24 edges for a total of 50 symmetrically positioned topological features." (I, 450.11)

50.2 "Six (of the seven) unique cosmic axes of symmetry give rise to 50 great circles passing through all 12 vertexes of the vector equilibrium or icosahedron." (I, 1042.05)

56.1 "The seven unique cosmic axes of symmetry describe all of crystallography (and give rise to 56 great circles)." (I, 1042.05)

56.2 "The 56 axes of cosmic symmetry interprecess successively to regenerate the centripetal-centrifugal inwardness, outwardness, and aroundnesses asthe omnipulsative cycling and omniinterresonated eternally regenerative Universe, always accommodated by the six positive and six negative alternately and maximally equieconomical degrees of freedom characterizing each and every event cycle of each and every unique frequency-quantum magnitude of the electromagnetic spectrum range." (II, 533.22)

60.1 "The closest packing of unit radius spheres always associates in omnitriangulation of 60 degrees, whether in planar or in omnidirectional arrays." (I, 410.11

60.2 "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 per cosmic system into which convergent-divergent nuclear unity may be subdivided." (II, 986.821)

72.1 "The cube is composed of a total of 72 energy quanta modules, of which there are 48 blue A (Quanta) Modules and 24 red B (Quanta) Modules....Thus, as it is seen from the outside, the cube is an all-red tetrahedron, but its energy distributive surface layer of 24 red B Modules is tensively overpowered two- to-one and cohered as a cube by its 48 nuclear modules." (I, 942.40-42)

92.1 "In omnidirectional closest packing of spheres, the third tangential layer contains 92 spheres (around 42, around 12, around 1 nuclear sphere)." (I, 222.23)

92.2 "In closest packing of equiradius spheres, a nucleos by definition must be tangentially and symmetrically surrounded....Not until the third layer of 92 balls is added are all tangential spaces filled and all the optically direct angles of nuclear visibility intercepted. We then realize a nucleus." (I, 414.01)

92.3 "It will also be discovered that the third layer of 92 spheres contains eight new potential nuclei (at the center of each triangular face of the vector equilibrium so constituted)...This tells us that the nuclear group with 92 spheres in its outer, or third, layer is the limit of unique, closest-packed symmetrical assemblages of unit wave-length and frequency." (I, 414.02)

92.4 "Independent of their isotopal variations of neutron content, the 92 self- regenerative chemical elements belong to the basic inventory of cosmic absolutes. The family of prime elements consists of 92 unique sets of from one to 92 electron-proton counts inclusive, and no others." (I, 418.06)

92.5 "It is to be noted that the harmonics of the periodic table of elements add up to 92..." (II. 1238.43)

96.1 "There are seven other secondary symmetries based on the pairing into spin poles of vertexes produced by the complex secondary crossings of one another of the seven original great circle symmetries... .The total of the above-mentioned secondary great circles of the VE is 96 new great circles (making 121 with the 25 primary great circles)." (II, 987.131-137)

96.2 "The 96 secondary
great circles of the VE divide the charted edge of the VE...
into rational linear fractions of the (unit) edge length -- i.e. 1/2. 3/10.
1/4, 1/6, 1/10 -- end these fractions
embrace all the intercombinings of the
first four prime numbers 1, 2, *3
*and 5." (II, 987.411)

96.3 "24 x 4 = 96. But the number of self-regenerative chemical elements
is 92. What is missing betwen the VE 92 and the f^{2} Duo-tet cube's
96 is the fourness of the octahedron's function in the annihilation of energy:
92 + 4 = 24 x 4 = 96. The four is the disappearing octa set. " (II, 1033.704)

96.4 "To allow for each of these 48-teravolume domains to accommodate their respective active and passive phases, it requires 96-tetravolumes. F tetra- voluming, which is as yet primitive, introduces an allspace-filling, symmetrical cube of 192-tetravolumes as an essential theatre of omniatomic primitive interarrayings." (II. 1033.741)

96.5 "The octahedron...is composed
of 96 energy quanta modules of which 48 are
red B Quanta Modules and 48 blue A
Quanta Modules....The
octahedron is all
blue outside with *a *red nucleus." (I. 942.30)

120.1 "120 identical spherical triangles (although right- and left-handed) are the maximum number of like units which may be used to subdivide the sphere." (I, 459.02)

120.2 "The 15 great circles of the icosahedron interact to produce 15 "chains" of three varieties of four corner-to-corner, sausage-linked, right triangles, with four triangles in each chain. These 15 chains of 60 great-circle triangles are each interconnectible corner-to-corner to produce a total spherical surface subdivided into 120 similar spherical triangles....The 15 chains' 60 triangles' inadvertent formation of an additional 60 similar spherical triangles occurring between them...dramatically manifests the half-positive, half- negative. . .universal non-mirror-imaged complementarity inherently permeating all systems..." (I, 456.05)

120.3 "Two identical sets of 60 spheres in closest packing process in 90-degree action to form a seven-frequency, eight-ball-to-the-edge tetrahedron with a total of 120 spheres; exactly 100 are on the outer shell, exactly 20 are in the inner shell, and there is no sphere at the nucleus. This is the largest possible double-shelled tetrahedral aggregation of closest-packed spheres having no nuclear sphere....The 120 spheres of this non-nuclear tetrahedron correspond to the 120 basic triangles that describe unity on a sphere." (I, 417.01-02)

144.1 The rhombic dodecahedron is composed of 144 energy quanta modules....its two blue layers of 48 A (Quanta) Modules each on the outside enclosing its one nuclear layer of 48 red B (Quanta) Modules." (I. 942.50)

160.1 "In the generalized (subfrequency) nucleus-embracing, convergent-divergent,
bivalent tetravolume vector equilibrium of frequency two, its tetravolume
is 160. VE^{2} = 160." (II. 1053.846) (Compare VE^{1}
= 20)

192.1 "F^{2} tetravoluming. which is as yet primitive, introduces
an allspace-filllng,
symmetrical cube of 192-tetravolumes as an essential thatre of omniatomic
primitive interarrayings." (II, 1033.741)

242.1 "The 120 outersurface right spherical triangles of the icosahedron's...great circles generate a total of 242 external vertexes...constituting the maximum limit of regular spherical system surface omnitriangular self-subdivisioning into centrally collected tetrahedral components." (II, Fig. 986.471)

360.1 "Every triangle has two faces -- its obverse and reverse. Unityis
two. So we note that the angle of both faces of a triangle add up to 360^{0}
." (I, 224.06)

360.2 "The difference between the sum of all the angles around all the vertexes of any system and the total number of vertexes times 360° (as angular unity) is 720°, which equals two unities." (I, 224.04)

480.1 The vector equilibrium is composed of 336 blue A (Quanta) Modules and 144 red B (Quanta) Modules for a total of 480 energy quanta modules....An even number of 48 As and 48 Bs provide an equilibrious exterior shell..." (I, 942.60)

480.2 "The 120 outer surface right spherical triangles of the icosahedron's...great circles generate a total of 120 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." (II, Fig. 986.471)

480.3 "Looking at a vector equilibrium as unity, it is all the domain of a point with a volume of 480." (I, 536.41)

576.1 "The minimum cube that can be formed by closest packing of spheres [which are inherently stable,structurally speaking) is produced by nesting four balls in the triangular mid-face nests of the four faces of a three-layer, ten-ball tetrahedron, with no ball at its volumetric center....which consists of 14 balls in all...This complex cube has a total of 576 A and B (Quanta) Modules, in contradistinction to the simplest tetra-octe-produced cube constituted of 72 A and B Modules." (I, 942.43) (It is the F Double-têt cube)

720.1 "All local systems in Universe are always accomplished by nature through the elimination of 720 degrees of angle. This is the way in which nature takes two 360-degree angular tucks in the illusory infinity of a plane to render systems locally and visibly finite. The difference between visually finite system and illusory infinity is two cyclic unities." (I, 224.02)

720.2 "Each circle has 360^{0}; the two interference circles
that comprise the minimum knot always involve 720 degrees of angular change
in the hand-led pattern, just as the total angles of the four triangles of
a tetrahedron add up to 720 degrees. The hands describe circles nonsimultaneously;
the result is a progression. The knot is the same 720-degree angular value
of a minimum structural system in Universe, as is the tetrahedron." (I,
506.13)

720.3 "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 x 120 - 720....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." (II, 986.461)

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