982.81 We recall also that both Newton and Leibnitz in evolving the calculus thought in terms of a circle as consisting of an infinite number of short chords. We are therefore only modifying their thinking to accommodate the manifest discontinuity of all physical phenomena as described by modern physics when we explore the concept of a circle as an aggregate of short eventvectors^{__}tangents (instead of Newtonian short chords) whose tangential overall length must be greater than that of the circumference of the theoretical circle inscribed within those tangent eventvectors^{__}just as Newton's chords were shorter than the circle encompassing them. 
982.82 If this is logical, experimentally informed thinking, we can also consider the closesttangentialpacking of circles on a plane that produces a nonallareafilling pattern with concave triangles occurring between the circles. Supposing we allowed the perimeters of the circles to yield bendingly outward from the circular centers and we crowded the circles together while keeping themselves as omniintegrally, symmetrically, and aggregatedly together, interpatterned on the plane with their areal centers always equidistantly apart; we would find then^{__}as floortile makers learned long ago^{__}that when closest packed with perimeters congruent, they would take on any one of three and only three possible polygonal shapes: the hexagon, the square, or the triangle^{__}closestpacked hexagons, whose perimeters are exactly three times their diameters. Hexagons are, of course, cross sections through the vector equilibrium. The hexagon's six radial vectors exactly equal the six chordal sections of its perimeter. 
982.83
Assuming the vector equilibrium hexagon to be the
relaxed, cosmic, neutral,
zero energyevents state, we will have the flexible
but not stretchable hexagonal perimeter
spun rapidly so that all of its chords are centrifugally
expelled into arcs and the whole
perimeter becomes a circle with its radius necessarily
contracted to allow for the bending
of the chords. It is this circle with its perimeter
equalling six that we will now convert, first
into a square of perimeter six and then into a triangle
of perimeter six with the following
results:

982.84 In accomplishing these transformations of the uniformlyperimetered symmetrical shapes, it is also of significance that the area of six equiangular, uniform edged triangles is reduced to four such triangles. Therefore, it would take more equiperimeter triangular tiles or squares to pave a given large floor area than it would using equiperimetered hexagons. We thus discover that the hexagon becomes in fact the densestpacked patterning of the circles; as did the rhombic dodecahedron become the minimal limit case of selfpacking allspacefilling in isometric domain form in the synergetical fromwholetoparticular strategy of discovery; while the rhombic dodecahedron is the sixdimensional state of omnidensestpacked, nuclear field domains; as did the twofrequency cube become the maximum subfrequency selfpacking, allspace filling symmetrical domain, nuclearuniqueness, expandability and omni intertransformable, intersymmetrical, polyhedral evolvement field; as did the limitof nuclearuniqueness, minimally at threefrequency complexity, selfpacking, allspacefilling, semiasymmetric octahedron of Critchlow; and maximally by the threefrequency, four dimensional, selfpacking, allspacefilling tetrakaidecahedron: these two, together with the cube and the rhombic dodecahedron constitute the onlyfouristhelimitsystem set of selfpacking, allspacefilling, symmetrical polyhedra. These symmetrical realizations approach a neatness of cosmic order. 
983.00 Spheres and Interstitial Spaces 
983.01
Frequency: In synergetics, F =

983.02
Sphere Layers: The numbers of separate spheres in
each outer layer of
concentric spherical layers of the vector equilibrium
grows at a rate:
= 10r^{2} + 2, or 10F^{2} + 2.

983.03
Whereas the space between any two concentrically parallel
vector equilibria
whose concentric outer planar surfaces are defined by
the spheric centers of any two
concentric sphere layers, is always

983.04 The difference is the nonsphere interstitial space occurring uniformly between the closestpacked spheres, which is always 6  5 = 1 tetrahedron. 
984.00 Rhombic Dodecahedron 
984.10 The rhombic dodecahedron is symmetrically at the heart of the vector equilibrium. The vector equilibrium is the everregenerative, palpitatable heart of all the omniresonant physicalenergy hearts of Universe. 
985.00 Synergetics Rational Constant Formulas for Area of a Circle and Area and Volume of a Sphere 
985.01
We employ the synergetics constant "S," for correcting
the cubical XYZ
coordinate inputs to the tetrahedral inputs of synergetics:

985.02 We may also employ the XYZ to synergetics conversion factors between commonly based squares and equiangled triangles: from a square to a triangle the factor is 2.3094; from a triangle to a square the factor is 0.433. The constant pi 3.14159 × 2.3094 = 7.254 = 7 1/4; thus 7 1/4 triangles equal the area of a circle of radius 1. Since the circle of a sphere equals exactly four circular areas of the same radius, 7 1/4 × 4 = 29 = area of the surface of a sphere of radius 1. 
985.03 The area of a hexagon of radius 1 shows the hexagon with its vertexes lying equidistantly from one another in the circle of radius 1 and since the radii and chords of a hexagon are equal, then the six equilateral triangles in the hexagon plus 1 1/4 such triangles in the arcchord zones equal the area of the circle: 1.25/6 = 0.208 zone arcchord area. Wherefore the area of a circle of frequency 2 = 29 triangles and the surface of a sphere of radius 2 = 116 equilateral triangles. 
985.04 For the 120 LCD spherical triangles S = 4; S = 4 for four greatcircle areas of the surface of a sphere; therefore S for one greatcircle area equals exactly one spherical triangle, since 120/4 = 30 spherical triangles vs. 116/4 = 29 equilateral triangles. The S disparity of 1 is between a right spherical triangle and a planar equiangular triangle. Each of the 120 spherical LCD triangles has exactly six degrees of spherical excess, their three corners being 90 degrees, 60 degrees and 36 degrees vs. 90 degrees, 60 degrees, 30 degrees of their corresponding planar triangle. Therefore, 6 degrees per each spherical triangle times 120 spherical triangles amounts to a total of 720 degrees spherical excess, which equals exactly one tetrahedron, which exact excessiveness elucidates and elegantly agrees with previous discoveries (see Secs. 224.07, 224.10, and 224.20). 
985.05 The synergetical definition of an operational sphere (vs. that of the Greeks) finds the spheric experience to be operationally always a starpointvertexed polyhedron, and there is always a 720 degree (one tetrahedron) excess of the Greek's sphere's assumption of 360 degrees around each vertex vs. the operational sum of the external angles of any system, whether it be the very highest frequency (seemingly "pure" spherical) regular polyhedral system experience of the highfrequency geodesic spheres, or irregular giraffe's or crocodile's chordallyinterconnected, outermostskinpointsdefined, polyhedral, surface facets' cornerangle summation. 
985.06 Thus it becomes clear that S = 1 is the difference between the infinite frequency series' perfect nuclear sphere of volume S and 120 quanta modules, and the fourwholegreatcircle surface area of 116 equilateral triangles, which has an exact spherical excess of 720 degrees = one tetrahedron, the difference between the 120 spherical triangles and the 120 equilateral triangles of the 120equiplanarfaceted polyhedron. 
985.07 This is one more case of the one tetrahedron: one quantum jump involved between various stages of nuclear domain intertransformations, all the way from the difference between integralfinite, nonsimultaneous, scenario Universe, which is inherently nonunitarily conceptual, and the maximumminimum, conceptually thinkable, systemic subdivision of Universe into an omnirelevantly frequenced, tunable set which is always one positive tetrahedron (macro) and one negative tetrahedron (micro) less than Universe: the definitive conceptual vs. finite nonunitarily conceptual Universe (see Secs. 501.10 and 620.01). 
985.08 The difference of one between the spheric domain of the rhombic dodecahedron's six and the nuclear sphere's five^{__}or between the tetra volume of the octahedron and the threetetra sections of the tetrahelix^{__}these are the prime wave pulsation propagating quanta phenomena that account for local aberrations, twinkle angles, and unzipping angles manifest elsewhere and frequently in this book. 
985.10
Table: Triangular Area of a Circle of Radius 1
F^{1} = Zeroone frequency = 7 1/4.
Table of whole triangles only with F = Even N, which is because Even N = closed wave circuit.

985.20 Spheric Experience: Experientially defined, the spheric experience, i.e., a sphere, is an aggregate of criticalproximity event "points." Points are a multidimensional set of crossings of orbits: traceries, foci, fixes, vertexes coming cometlike almost within intertouchability and vertexing within cosmically remote regions. Each point consists of three or more vectorially convergent events approximately equidistant from one approximately locatable and as yet nondifferentially resolved, point; i.e., three or more visualizable, fourdimensional vectors' most critical proximity, convergentlydivergently interpassing region, local, locus, terminal and macrocosmically the most complex of such point events are the celestial stars; i.e., the highestspeed, highfrequency energy event, importingexporting exchange centers. Microcosmically the atoms are the inbound terminals of such omniorderly exchange systems. 
985.21 Spheres are further cognizable as vertexial, starpointdefined, polyhedral, constellar systems structurally and locally subdividing Universe into insideness and outsideness, microcosmmacrocosm. 
985.22 Physically, spheres are highfrequency event arrays whose spheric complexity and polyhedral system unity consist structurally of discontinuously islanded, criticalproximityevent huddles, compressionally convergent events, only tensionally and omniinterattractively cohered. The pattern integrities of all spheres are the high frequency, trafficdescribed subdivisionings of either tetrahedral, octahedral, or icosahedral angular interference, intertriangulating structures profiling one, many, or all of their respective greatcircle orbiting and spinning event characteristics. A11 spheres are highfrequency geodesic spheres, i.e., triangularfaceted polyhedra, most frequently icosahedral because the icosasphere is the structurally most economical. 
Next Section: 986.00 