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:

$$\begin
S^= 1.060660\ S^= 1.12487\ S^= 1.1931750\
\end$$

We learn that the sphere of radius 1 has a “cubical” volume of 4.188; corrected for tetrahedral value we have 4.188 X 1.193 =4.996 = 5 tetrahedra= 1 sphere. Applying the S^{2} to the area of a circle of radius 1, (pi = 3.14159) 3.14159 x 1.125 = 3.534 for the corrected “square” area.

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 .433. The constant pi 3.14159 \times 2.3094 = 7.254 = 774; thus 774 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, 774 \times 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\sfrac{1}{4} such triangles in the arc-chord zones equal the area of the circle: \frac{1.25}{ 6}= .208 zone arc-chord 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 5 = 4; S =4 for four great-circle areas of the surface of a sphere; therefore S for one great-circle area equals exactly one spherical triangle, since ¹²⁰Ai = 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 star-point-vertexed 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 high-frequency geodesic spheres, or irregular giraffe’s or crocodile’s chordally-interconnected, outermost-skin-points-defined, polyhedral, surface facets’ corner-angle summation.

985.06 Thus it become clear that S = 1 is the difference between the infinite frequency series’ perfect nuclear sphere of volume 5 and 120 quanta modules, and the four-whole-great-circle 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 120-equiplanar-faceted 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 integral-finite, 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 three-tetra sections of the tetrahelixthese 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 = Zero-one frequency = 774.

Table of whole triangles only with F = Even N, which is because Even N = closed wave circuit.

      $`pN`$                              

Open X
Close X
Open X (63 + 274)
Close X (112 + 4) (also surface of one sphere)
Open X (175 + 674)
Close X (252 + 9)

985.20 Spheric Experience: Experientially defined, the spheric experience, i.e., a sphere, is an aggregate of critical-proximity 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, four-dimensional vectors’ most critical proximity, convergently-divergently interpassing region, local, locus, terminal and macrocosmically the most complex of such point events are the celestial stars; i.e., the highest-speed, high-frequency energy event, importing-exporting exchange centers. Microcosmically the atoms are the inbound terminals of such omniorderly exchange systems.

985.21 Spheres are further cognizable as vertexial, star-point-defined, polyhedral, constellar systems structurally and locally subdividing Universe into insideness and outsideness, macrocosm-macrocosm.

985.22 Physically, spheres are high-frequency event arrays whose spheric complexity and polyhedral system unity consist structurally of discontinuously islanded, critical-proximity-event huddles, compressionally convergent events, only tensionally and omni-interattractively cohered. The pattern integrities of all spheres are the high-frequency, traffic-described subdivisionings of either tetrahedral, octahedral, or icosahedral angular interference, intertriangulating structures profiling one, many, or all of their respective great-circle orbiting and spinning event characteristics. All spheres are high-frequency geodesic spheres, i.e., triangular-faceted polyhedra, most frequently icosahedral because the icosasphere is the structurally most economical.