1050.00 Synergetic Hierarchy
1050.10 Synergy of Synergies: We have the concept of synergy of synergies. Precession is not predicted by mass attraction. Chemical compounds are not prophesied by the atoms. Biological protoplasm is not predicted by the chemical compounds. The design of the elephant or the tree and their unwitting essential respiratory-gas conversion interexchanging is not predicted by the protoplasm. There is nothing about an elephant that predicts islanded star galaxies. As we get into larger and larger systems, the total system is never predicted by its lesser system’s components.
1050.11 We know that there is DNA and RNA, any one genetic code of which dictates both a species and within it an individual or special-case formulation. DNA-RNA codes do not explain why the protoplasm could produce either an elephant, pine tree, or daisy. They elucidate only how. What we call viral steerability as produced by the DNA-RNA codes is simply our familiar and generalized angle-frequency design control.
1050.12 DNA-RNA angle- and frequency-modulated designs are composed exclusively of four unique chemical constituents that operate as guanidine and cytosine; and as thiamine and adenine: inseparable but reversible tandem pairs. The first pair occur as GC or CG. The second pair occur as TA or AT. The DNA-RNA codes may be read in any sequence of those constituents, for instance, as CG - CG - CG - GC - TA - AT - GC - TA - TA - TA - AT - CG - CG - GC, etc. (See Sec. 932.)
1050.13 We know the codes, but we do not know the “how come” of their producing an elephant. The complementarity of the holisticness of these special-case individuals balances out. An elephant does walk. Elephants are successful designs. We have no evidence of biological species that are inherently incomplete designs. In the hierarchy of hierarchies of synergies, Universe is the unpredicted behavior of any of its sublevel synergetics. We must start our synergetic analysis at the level of Universe and thereafter with the known behavior of the greatest whole and the known behavior of some of its parts, then proceed as permitted mathematically to discover its unknown parts. We have the Greek triangle with its known 180 degrees of angle; which together with the knowledge of the magnitude of any two sides and their included angle, or of any two angles and their included side, etc., permit us to discover the magnitude of the balance of the triangle’s six parts. Or, using trigonometry, if we know the magnitude of any two parts, we can ferret out the others.
1050.20 Trigonometry: The way we were taught in school about fractions leads to inconsistency. We were taught that fractions can be multiplied, divided, added, and subtracted only when the fractions consisted of identical entities. We could not divide three elephants by four oranges. However, trigonometry introduced functionswhich are fractions or ratios, e.g., the sine, cosine, tangent, cotangent, and so forth. Contradicting our earlier lessons about fractions, these trigonometric fractions do mix together angles and edges of spherical triangles. This inconsistency could have been avoided by starting our geometry with spherical trigonometry. We would recognize that what we call a great circle arc or “edge” is indeed a central angle of the sphere. We would learn that we have central and external angles. We would spontaneously see that plane geometry derived from solid geometry and is an oversimplification of localized and superficial aspects of systems. This brings us back to angle and frequency modulation, i.e., outward, inward, and circumferentially around, complementary angle and frequency oscillations and pulsations and the congruence of the linear and angular frequency modulations. By teaching children plane geometry before teaching them spherical trigonometry, society became harnessed with a mathematical contradiction wherein trigonometry deliberately ratioed edge lines with angles which clearly seemed to be forbidden by arithmetical fractions’ law. Single lines are seemingly very different from angles, because angles involve two (convergent) lines. If, however, instead of starting elementary education with unrealistic, linear, one-dimensional arithmetic; and then going on to two-dimensional plane geometry; and thence to three-dimensional cubes; and thence to spherical trigonometry …if we instead start synergetically with whole systems such as spherical trigonometry, we altogether avoid the concept of an edge and instead learn that the arc-defined edges of spherical triangles are the central angles of the sphere; wherefore both the arc edges and corners are angles, ergo ratioable. Now, having both surface angles and central angles, we discover that spherical trigonometry is always dealing with whole tetrahedra whose interior apexes are always at the center of the spherical system; and three of whose triangular faces are the great-circle plane triangles hidden within the spheric system; and whose fourth triangular face is always the arcchord surface triangle of the sphere. These central- and surface-angle understandings are fundamental to transformational thinking, which deals with the falling-inward and precessing-outward proclivities.
1050.30 Simplest Trigonometric Solutions: Sequence to Accompany Poster in color plate 1.
1050.31 Stones may be broken into ever smaller stones, but they cannot be broken into no stones. They may be broken into gravel and the gravel into dust and the dust separated into crystals that are too small to be seen except through a lensed microscope; or they may be further broken apart into atoms that can be seen only through electron field microscopes. But the stones cannot be broken into nothingnesses—only into somethings. And somethings are always systems[^49].
1050.32 As the stones break, they have cleavage faces. They break into irregular pqlyhedra that are complex or simplex geometrically definable systems, each of which always has an inherent insideness and outsideness. The number of faces—hedra—of polyhedra cannot be reduced to less than four: the tetrahedron. The number of sides—gons—of a polygon cannot be reduced to less than three: the triangle. The minimum polyhedron of Universe is the tetrahedron, which requires a minimum of three triangles surrounding each of its four corners, whose four corners are omniinterconnected with a minimum of six edges that discretely outline the four triangular (minimum polygon) faces.
1050.33 Make the “V for Victory” sign with two adjacent fingers. The V is visual. The V is a specifically visible angle. The angle is an angle independent of the length of the fingers—that is, independent of the length of the sides of the angle. Angles, triangles, and tetrahedra are conceptual pattern integrities independent of size. Angles are always and only fractional parts of whole circles (of 360 degrees). Likewise, triangles are always and only components of a priori whole physical polyhedral systems (or of a plurality of whole polyhedral systems) each of 720 degrees (or whole multiples of 720 degrees) of angles surrounding all the external vertexes describing those systems. Only triangles produce structural stability. Only triangles produce pattern stability. The omnitriangulated tetrahedron is the minimum structural system of Universe.
1050.34 Drawing or scribing are physical operations executed upon a physical system. Triangles can be drawn or scribed or traced or trajectoried only upon or within an a priori physical system, or defined by a constellation of three physical systems within a greater a priori system.
1050.35 There are six and only six different but always orderly inter-covarying geometrical characteristics or integral parts of all triangles: three surface-angle corners A, B, and C, and three sides a, b, and c. In reality these sides are always the central angles of the scribed-upon system and they are only evidenced by their surface-arc lines.
1050.36 Individual angular values or the relative interrelationships or interratios or functions of these parts hold true independent of the size of the triangle. This is to say that an equiangular triangle is equiangular and humanly conceptual independent of the size of any of our special case triangular experiences. The four most useful of these functions and their symbols are:
sine = sin tangent = tan
cosine = cos cotangent = cot
1050.37 The science that measures the respective angle magnitudes of the six ever-orderly intercovarying angles of triangles is called trigonometry. All of the geometrical interrelationships of all triangles—spherical or planar—are discoveringly calculated by the same trigonometry because plane triangles are always very small spherical triangles on very large spheric systems such as high-frequency symmetric polyhedra. A circle is a spherical triangle each of whose three comer angles is 180 degrees.
1050.38 To find the value of all the central angles (sides) and surface (corner) angles of any spherical triangle, we can always start by dropping a perpendicular from any vertex of that triangle upon its opposite side—making it into two “right” triangles. In order to discover all six angular values of a given triangle it is necessary to know—in addition to knowing the 90-degree comer—the surface- or central-angular values of any other two of the to-be- solved triangle’s five other parts: A, B, a, b, c. Many mathematicians have devised strategic formulas for coping with trigonometric solutions, most of them involving plus or minus quadrant symbols that invite errors of calculation.
1050.39 To make the trigonometry of the sea captain’s celestial navigation as simple and foolproof as possible the mathematician Lord Napier (1550-1667)[^50] evolved the following diagrams and procedures. To avoid what is known in navigation as “the 180-degree error”—going in exactly the opposite direction from that which will get you where you want to go—Napier arranged the five non-90-degree “parts” of a triangle in a five-segment “clock.”
1050.40 Napier had two equally simple ways to solve trigonometric problems without plus or minus symbols, provided that any two of the non- 90-degree angles are known at the outset. His superscript c means that/c, cc, Bc are the 180-degree complements of A c, B. For instance, Ac + A = 180, wherefore sin Ac = cos A; or tan cc = cot c, etc.
1050.41 First we check-mark the two “known-in-advance” non-90-degree parts on Napier’s five-segment clocklike pattern. It is clear that the two already-knowns are always either divided from one another or are side by side. In Napier’s Case One the two knowns are side by side in the clock: Napier calls this the case of Opposites. Opposite Case see Rule 1 chosen unknown for first solution.
1050.42 In Case Two the two knowns are separated from one another in the clock: Napier calls this the case of Adjacents. Adjacent Case see Rule 2 first unknown to be solved.
1050.43 Napier’s two easily remembered rules are:
Rule 1. The sine of any unknown part theta is equal to the product of the cosines of the two known opposite parts. This is written as:
unknown’s angle thetas sin = cos-cos of its two known opposite parts.
Rule 2. The sine of any unknown part is equal to the product of the tangents of its two known adjacent parts. This is written as:
unknown angle theta’s sin =tan tan of its two known adjacent parts.
1050.44 Next we employ the appropriate formula with the known cosine or tangent values. Next we must remove the superscript c of the complementaries, if any, by substituting cosines for sines, sines for cosines, tangents for cotangents, and cotangents for tangents.
Example: When the equation as first written is
sin b = cos cc • cos bc
the equation must be rewritten
sin b = sin c • sin b;
or if the equation first reads
sin Ac = tanc • tan b,
it must be rewritten as
cos A = cot c • tan b
before going on to intermultiply the functions of the two knowns whose product will be the function value of the previously unknown angle theta. The angle values of the new]y found knowns may be located in any table of trigonometric functions or may be “remembered” by computers. When the value is found for an angle’s function (sin, cos, tan, cot), its specific angular value may also be read out of the tables.