961.00 Unitary Quantation of Tetrahedron
961.01 The area of a triangle is arrived at by multiplying the length of the baseline by one-half of the triangle’s apex altitude.
961.02 The volume of a tetrahedron is the product of the area of the base and one-third of its altitude.
961.03 A minimum garland of “granteds” combines only synergetically to disclose the following:
961.10 Granted: A Slidable Model of Constant Volume: Granted any point A that is movable limitlessly anywhere within one of two planes parallel to one another at a given perpendicular distance X from one another, and, cogliding anywhere within the other parallel plane, two parallel lines lying at a given perpendicular distance Y from one another, and a point B that is slidable anywhere along one of the parallel lines, along the other line of which (two parallel lines) is a slidable pair of points, C and D, always slidable only at a constant and given distance Z from one another; it will be found that the vast variety of tetrahedra to be formed by interconnecting these four points (two independently variable and two only covariable) will always enclose the same volume. (See Sec. 923 and Illus. 923.10D.)
961.11 Provided the relationship between X, Y, and Z remain constant as described, and the distances X and Y in respect to the “constructed” distance are always such that
T=VZ2-(1/2 Z)2
and
X = VZ2 - (2/3 Y)2
then, by varying Z to correspond with the distance between two experiential event foci, all the other vertexial positions of all the tetrahedra of equal volume can be described by revolution of the constantly cohered tetrahedral system around the axial line running through the two points C andZ). This axial line may itself be angularly reoriented to aim the tetrahedral system by combining and interconnecting circuitry closing in any direction, thereby to reach any other two points in Universe to be tetrahedrally inteijoined in unitary quantation.
961.12 With the observer-articulator’s experientially initiated and interpositioned two control points C and D, these uniformly quantated observer-articulator variable initial “tunings,” accomplished exclusively by frequency and angle modulations, may “bring in” subjectively-objectively, receivingly-and-transmittingly, omnicosmic events occurring remotely in nonsimultaneously evolving and only otherness-generated self-awareness and deliberately thinkable self-conceptioning of progressively omnicomprehending embracements and penetrations. This in turn enables the conscious designing capabilities to be realized by these omnicosmic reaching tetrahedral coordinations (which are resolvable into generalized, quantum-regularized sets, consisting of only two human individual mentalities’ predeterminable variables consisting exclusively of frequency and angle modulations identified only with the self-and-other, C- and Z)-defined, selectable wavelength Z and theZ axis’ angular aiming and reorientation regulatability of the Z axis’ ever-constant reorientations of itsX-and /-dependent coordinates in exclusively angle-and frequency-determined invariant relationship), all of whose synergetic integrities’ intersignificance realizations are eternally interaccommodated by the tetrahedral structural system’s prime conceptual initiations.
961.20 Granted: A Model of Comprehensive Covariability: Granted that a tetrahedron of given altitude X, with a base triangle of given altitude Y and given baseline edge length Z, is volumetrically constant independent of the omnivariable interangling of its four vertexes, and five variabie- length edges, and four variable triangular faces, whose comprehensive covariability can altogether accommodate any symmetric or asymmetric aspect transformability to correspond exactly in all its interangular face relationships and relative edge lengths with any tetrahedron to be formed by interconnecting any four points in Universe, provided the relative values of X and Y in respect to Z (which is the only experientially known distance) are always such that:
y=VZ2-C/2Z)2
and,
X= Vz2-(2/3 /)2
As the values only of Z are altered, the respective value of the uniformly volumed tetrahedra will vary at a rate of the third power of Z’s linear change.
961.30 Granted: A Model for Third-Power Rate of Variation: Granted that there is then in respect to any two points in Universe a tetrahedron that can be given any symmetrical or asymmetrical tetrahedral shape, any of whose volumes will remain uniform or will vary uniformly at a third-power rate in respect to any alteration of the distance between the two initial control points on the axial control line; then, any four points in Universe, provided one is not in the plane of the other three, can be interconnected by varying the angular orientation of the control-line axis and the distance between the two central control points.
961.31 Being generalized, these three relative distance-control coordinates X, Y, and Z are, of course, also present in the special-case, omnirectilinear XYZc.g,.s. coordinate system. That the most economical time distances between the two parallel planes and two parallel lines are coincident with perpendiculars to those parallel planes and lines does not impose any rectilinear profiling or structuring of the tetrahedron, which is a unique, four-planes-of-symmetry, self-structuring system, as the three-plane-defined cube of basic reference is not.
961.40 Granted: A Model for Six Degrees of Freedom: Granted the area of a triangle is base times one-half the altitude, with one given length of line AB marked on a flat plane and another infinitely extensible line number two lying in the same plane as short line AB, with line two parallel to AB; then connecting any point C on line two with both A and B will produce a constant-area triangle ABC. Holding AB fixed and moving only C in any direction on line two, the shape of triangle ABC will change, but its area will be constant. If we move C along line two in one direction the three edges will approach congruence with one another, appearing only as a line but being, in fact, a constant-area triangle.
961.41 Granted the volume of a tetrahedron is its base area times one-third of its altitude, we can now take the permitted, special condition discussed in Sec. 961.40 whereby C on line two is equidistant from both of line one’s terminal-defining points A and B. We may next take a fourth point D, lying in an infinitely extensible second plane which is parallel to the first infinitely extensible plane defined by points ABC. With/) equidistant from .4, B, and C, the volume of the regular tetrahedron ABCD will not be altered by letting D travel to any point in plane two while point C travels to any point on line two. Thus we learn that constant-volume tetrahedron ABCD might become so distended as to appear to be a line of no volume. Since there could be no volumeless line produced operationally, we may assume that all visible lines must be at minimum extended tetrahedra.
961.42 These variabilities of the constant-volume tetrahedron and its constant-area faces will permit congruence of the four vertexes of the tetrahedron with any four points of Universe by simply taking the initial distance AB to suit the task. This unit linear adjustment is a familiar wavelength tuning function. Here we have the six cosmic degrees of freedom (see Sec. 537.10); whereby we are free to choose the length of only one line to be held constant, while allowing the other five edge-lines of the tetrahedron to take any size. We can connect any four points in Universe and produce a tetrahedron that is matchable with whole, unit, rational-number, volume increments of the A and B Quanta Modules.
961.43 With large, clear plexiglass Models of the A and B Quanta Modules, we can easily see their clearly defined centers of volume. The centers of area of the triangular faces are arrived at by bisecting the edges and connecting the opposite angle. The center of volume of the tetrahedron is arrived at by interconnecting the four centers of triangular area with their opposite vertexes. These four lines constructed with fine, taut wires will converge to tangency atand then diverge away fromthe tetrahedron’s center of volume.
961.44 The lines defining the center of four triangles and the center of volume inherently divide the modules into 24 equal parts. The same progressive subdivisions of the last 24 can be continued indefinitely, but each time we do so the rational bits become more and more asymmetrical. They get thinner and thinner and become more and more like glass splinters. By varying the frequency we can make any shape tetrahedron from the regular to the most asymmetric.
961.45 The modules make all the geometriesall the crystallography. Any probabilities can be dealt with. With the two of Euler: and Gibbs the Me-and-Other-Awarenessthe beginning of time, if there is time. … It starts testing the special cases that have time. They are absolutely quantized. The As are blue and the Bsare red. The blues and reds intertransform. Every sphere becomes a space, and every space becomes a sphere, palpitating in the wire model of electromagnetic wave action.
961.46 The A and B Quanta Modules become linear, as did the progression of concentric, common-base, uniform, linear, frequencied, electric-impulse conductors (see Sec. 923.21); and as also did the concentric, annually-frequencied, common-base-into-cone-rotated tetrahedra (see Sec. 541.30); the free energy put in at the base electronically, when you close the circuit at the beginning of the wireyou get the same package out at the other end, the same quanta. The longer the wire gets or the tree growsas it approaches parallelism, the more the energy packages begin to precess and to branch out at right angles.
961.47 Fluorescing occurs until all the juice is finally dissipated off the wire or until all of this year’s additional frequency’s growth is realized in new branches, twigs, leaves and tetrahedrally-precessed buds. Birth: buds: A and B Modules; three-, four-, five-, and six-petallings: tetra, octa, icosa, rhombic-dodeca bud petals. The original inputthe six A Quanta Modules of the original base tetrahedronbecomes distributive at 90 degrees. Coaxial cables tend to divert the precessional distributives inwardly to reduce the loss.
961.48 When great electrostatic charges built into clouds become dischargingly grounded (to Earth) by the excellently-conducting water of rain, and lightning occurs; we see the Earthward, precessionally-branching lightning. In grounding with Earth, lightning often closes its circuit through the tree’s branches, whose liquid, water-filled, cell fibers are the most efficient conductors available in conducting the great electric charges inward to Earth through the trunk and the precessionally-distributive roots’ branchings. Lines are tetrahedra. Lines can wave-bounce in ribbons and beams: tetra, octa, and icosa energy lock-up systems. E = Mc^{2}. All tetra and only tetra are volumetric, i.e., quanta-immune to any and all transformation.