840.00 Foldability of Four Great Circles of Vector Equilibrium
841.00 Foldability Sequence |
841.11 Using the method of establishing perpendiculars produced by the overlapping of unit-radius circles in the first instance of the Greeks' exclusively one-planar initiation of their geometry (see Illus. 455.11), a diameter PP' perpendicular to the first straightedge constructed diameter DD' can be constructed. If we now fold the paper circles around DD' and PP', it will be found that every time the circles are folded, the points where the perpendicular to that diameter intercept the perimeter are inherently congruent with the same perpendicular's diametrically opposite end. |
841.13 We could also have constructed the same sphere by keeping point A of the dividers at one locus in Universe and swinging point B in a multiplicity of directions around A (see Illus. 841.15 ). We now know that every point on the surface of an approximate sphere is equidistant from the same center. We can now move point A of the dividers from the center of the constructed sphere to any point on the surface of the sphere, but preferably to point P perpendicular to an equatorially described plane as in |
841.11 and 841.12 . And we can swing the free point B to strike a circle on the surface of the sphere around point P. Every point in the spherical surface circle scribed by B is equidistant chordally from A, which is pivotally located at P, that is, as an apparently straight line from A passing into and through the inside of the spherical surface to emerge again exactly in the surface circle struck by B, which unitary chordal distance is, by construction, the same length as the radius of the sphere, for the opening of our divider's ends with which we constructed the sphere was the same when striking the surface circle around surface point A. |
Fig. 841.15A Fig. 841.15B |
841.15 We now take any point, J, on the spherical surface circle scribed by the divider's point B around its rotated point at P. We now know that K is equidistant chordally from P and from the center of the sphere. With point A of our dividers on J, we strike point K on the same surface circle as J, which makes J equidistant from K, P, and X, the center of the sphere. Now we know by construction integrity that the spherical radii XJ, XK, and XP are the same length as one another and as the spherical chords PK, JK, and JP. These six equilength lines interlink the four points X, P, J, and K to form the regular equiedged tetrahedron. We now take our straightedge and run it chordally from point J to another point on the same surface circle on which JK and K are situated, but diametrically opposite K. This diametric positioning is attained by having the chord- describing straightedge run inwardly of the sphere and pass through the axis PP', emerging from the sphere at the surface-greatcircle point R. With point A of the dividers on point R of the surface circle^{__}on which also lies diametrically point K^{__}we swing point B of the dividers to strike point S also on the same spherical surface circle around P, on which now lie also the points J, K and R, with points diametrically opposite J, as is known by construction derived information. Points R, S, P, and X now describe another regular tetrahedron equiedged with tetrahedron JKPX; there is one common edge, PX, of both tetrahedra. PX is the radius of the spherical, octahedrally constructed sphere on whose surface the circle was struck around one of its three perpendicularly intersectioned axes, and the three planes through them intersect congruently with the three axes by construction. PX is perpendicular to the equatorial plane passing through W, Y, W', Y' of the spherical octahedron's three axes PP', WW', and YY'. |
Fig. 841.22 |
841.22 Each of these paired bow-tie assemblies, the orange-green insiders and the violet-white insiders, may now be fastened bottom-to-bottom to each other at the four external fold ends of the fold cross on their bottoms, with those radial crosses inherently congruent. This will reestablish and manifest each of the four original circles of paper, for when assembled symmetrically around their common center, they will be seen to be constituted of four great circles intersecting each other through a common center in such a manner that only two circular planes come together at any other than their common center point and in such a manner that each great circle is divided entirely into six equilateral triangular areas, with all of the 12 radii of the system equilengthed to the 24 circumferential chords of the assembly. Inasmuch as each of the 12 radii is shared by two great circle planes, but their 24 external chords are independent of the others, the seeming loss of 12 radii of the original 24 is accounted for by the 12 sets of congruent pairs of radii of the respective four hexagonally subdivided great circles. This omniequal line and angle assembly, which is called the vector equilibrium, and its radii-chord vectors accommodate rationally and simultaneously all the angular and linear acceleration forces of physical Universe experiences. |
Fig. 841.30 |
841.30 Trisection by Inherent Axial Spin of Systems |
842.00 Generation of Bow Ties |
842.06 This sixness corresponds to our six quanta: our six vectors that make one quantum. |
Next Chapter: 900.00 |