986.310 Strategic Use of Min-max Cosmic System Limits

986.311 The maximum limit set of identical facets into which any system can be divided consists of 120 similar spherical right triangles ACB whose three corners are 60 degrees at A, 90 degrees at C, and 36 degrees at B. Sixty of these right spherical triangles are positive (active), and 60 are negative (passive). (See Sec. 901.)

Section thru \cosbhebron , BOQECRHEDRON, RHOMBIC TRI SCON T A HEBRON , 120 TRIANGLES ENENICONTAHEDRC.N, USING, ONE qRERT CIRCLE OF THE 15 QRERT CIRCLES OF THE IcosrheoROn

image

986.312 These 120 right spherical surface triangles are described by three different central angles of 37.37736814 degrees for arc AB, 31.71747441 degrees for arc BC, and 20.90515745 degrees for arc AC which three central-angle arcs total exactly 90 degrees. These 120 spherical right triangles are self-patterned into producing 30 identical spherical diamond groups bounded by the same central angles and having corresponding flat-faceted diamond groups consisting of four of the 120 angularly identical (60 positive, 60 nega tive) triangles. Their three surface corners are 90 degrees at C, 31.71747441 degrees at B, and 58.2825256 degrees at A (See Fig. 986.502.)

986.313 These diamonds, like all diamonds, are rhombic forms. The 30-symmetrical-diamond system is called the rhombic triacontahedron, its 30 mid-diamond faces (right-angle cross points) are approximately tangent to the unit-vector-radius sphere when the volume of the rhombic triacontahedron is exactly tetravolume-5. (See Fig. 986.314.)

986.314 I therefore asked Robert Grip and Chris Kitrick to prepare a graphic comparison of the various radii and their respective polyhedral profiles of all the symmetric polyhedra of tetravolume 5 (or close to 5) existing within the primitive cosmic hierarchy (Sec. 982.62)i.e. other than those of tetravolumes 1, 2, 3, 4, and 6which carefully drafted drawing of the tetravolume-5 polyhedra (and those polyhedra “approximately” tetravolume 5) my colleagues did prepare (see Fig. 986.314). These exactly tetravolume-5 polyhedra are, for example

1. the icosahedron with outer edges of unit vector length;

2. the icosahedron of outer vertex radius of unit vector length;

3. the regular dodecahedron of unit vector edge; and

4. the regular dodecahedron of unit vector radiusall of which show that they have only a slightly greater radius length than that of the prime vector.

lines bo not converge

Tcosrheoron

Rhombic Triacontahedron

120 Triangles

ENENICONTAHEORON

SPHERE OF RROIUS 1

dodecahedron

RADIUS = I.OHOIU37M2

H5

b fCtd: 50

TETRA EbqE

120

BO NOT CONVERGE

the Volume of EACH Polyhedron is equal to 5 REGULAR TETRAHEBRH OF E DQE LENGTH EQUAL TO 2 OR A TOTAL VOLUfTIE OF-

3.300

b = 31 .717

C - 37.377

771 802

3GS

20.905 157

90 -b

b = 90 -C

C = 90-d

Fig. 986.314 Polyhedral Profiles of Selected Polyhedra of Tetravolume-5 and Approximately Tetravolume-5: A graphic display of the radial proximity to one another of exact and neighboring tetravolume-5 polyhedra, showing central angles and ratios to prime vector.

^D5 0 997 020

^D2 . 113 23l»
^D3 . 191
12O„₅ Ott
• RT-, 999
RT_S . 179 9t3
rt₃ . Ot9 19H
. 095
^X3

^E2 E39
^E5
^E3 . 112
[^e9] b9S

DISTANCES TO C-3

986.315 The chart of the polyhedral profiles (Fig. 986.314) shows the triacontahedron of tetravolume 5 having its mid-diamond-face point C at a distance outward radially from the volumetric center that approximately equals the relative length of the prime vector. I say “approximately” because the trigonometrically calculated value is .999483332 instead of 1, a 0.0005166676 radial difference, whichthough possibly caused in some very meager degree by the lack of absolute resolvability of trigonometric calculations themselves is on careful mathematical review so close to correct as to be unalterable by any known conventional trigonometric error allowance. It is also so correct as to hold historical significance, as we shall soon discover. Such a discrepancy is so meager in relation, for instance, to planet Earth’s spheric diameter of approximately 8,000 miles that the spherical surface aberration would be approximately the same as that existing between sea level and the height of Mount Fuji, which is only half the altitude of Mount Everest. And even Mount Everest is invisible on the Earth’s profile when the Earth is photographed from outer space. The mathematical detection of such meager relative proportioning differences has time and again proven to be of inestimable value to science in first detecting and then discovering cosmically profound phenomena. In such a context my “spherical energy content” of 4.99, instead of exactly 5, became a thought-provoking difference to be importantly remembered.

986.316 By careful study of the Grip-Kitrick drawings of tetravolume-5 polyhedra it is discovered that the graphically displayed zones of radial proximity to one another of all the tetravolume-5 symmetric polyhedra (Fig. 986.314) describe such meager radial differences at their respective systems outermost points as to suggest that their circumferential zone enclosed between the most extremely varied and the most inwardly radiused of all their axially spun vertexes of the exact tetravolume-5 polyhedra may altogether be assumed to constitute the zone of limit cases of radiantly swept-out and pulsating tetravolume-5 kinetic systems.

986.317 Recognizing that polyhedra are closed systems and that there are only seven cases of symmetrical subdivisioning of systems by the most economical great-circle spinnings (and most economically by the chords of the great-circle arcs), we discover and prove structurally that the maximum-limit abundance of a unit-symmetrical-polyhedral-system’s identical facetings is the rhombic triacontahedron, each of whose 30 symmetrical diamond planar faces may be symmetrically subdivided into four identical right triangles (30 \times 4 = 120), and we find that the triacontahedron’s 120-spherical-right-triangled frame of system reference is the maximum-limit case of identical faceting of any and all symmetrical polyhedral systems in Universe. This maximum-limit-system structuring proof is accomplished by the physically permitted, great-circle-spun, hemispherical self-halvings, as permitted by any and all of the seven cosmic limit cases of symmetric systems’ being spun-defined around all the respective system’s geometrically definitive (ergo, inherent) axes of symmetrical spinnability. It is thus that we learn experimentally how all the symmetric systems of Universe self-fractionate their initial system unities into the maximum number of omniangularly identical surface triangles outwardly defining their respective internal-structure tetrahedra whose anglescentral or surface are always independent of a system’s time-size considerations. And because they are independent of time-size considerations, such minimum-maximum limit-case ranges embrace all the symmetrical polyhedral systems’ generalized-primitive-conceptuality phenomena.

image

image

Fig. 986.405 Respective Subdivision of Rhombic Dodecahedron (A) and Rhombic Triacontahedron (B) into Diamond-faced Pentahedra: O is at the respective volumetric centers of the two polyhedra, with the short axes A-A and the long axes B-B (diagrams on the right). The central and surface angles of the two pentahedra differ as shown.

image

image