986.470 Geodesic Modular Subdivisioning

986.471 A series of considerations leads to the definition of the most spherical-appearing limit of triangular subdivisioning:

1. recalling that the experimentally demonstrable “most spherically-appearing” structure is always in primitive reality a polyhedron;

2. recalling that the higher the modular frequency of a system the more spheric it appears, though it is always polyhedral and approaching not a “true sphere” limit but an unlimited multiplication of its polyhedral facetings;

3. recalling that the 120 outer surface triangles of the icosahedron’s 15 great circles constitute the cosmic maximum limit of system-surface omni-triangular-self-subdivisioning into centrally collected tetrahedron components; and

4. recalling that the icosahedron’s 10- and 6-great-circle equators of spin further subdivide the 15 great circles’ outer 120 LCD triangles into four different right triangles, ADC, CDE, CFE, and EFB (see Fig. 901.03),

then it becomes evident that the icosahedron’s three sets of symmetrical great-circle spinnabilitiesi.e., 6+10+15 (which totals 31 great circle self-halvings)generate a total of 242 unit-radius, external vertexes, 480 external triangles, and 720 internal triangles (which may be considered as two congruent internal triangles, each being one of the internal triangular faces of the 480 tetrahedra whose 480 external triangular faces are showing in which case there are 1440 internal triangles). The 480 tetrahedra consist of 120 OCAD, Y2.0OCDE, 120 OCEF, and 120 OFEB tetrahedra. (See Fig. 986.471.) The 480 internal face-congruent tetrahedra therefore constitute the “most spheric-appearing” of all the hemispheric equators’ self-spun, surface-subdividing entirely into triangles of all the great circles of all the primitive hierarchy of symmetric polyhedra.

986.472 In case one thinks that the four symmetrical sets of the great circles of the spherical VE (which total 25 great circles in all) might omnisubdivide the system surface exclusively into a greater number of triangles, we note that some of the subdivision areas of the 25 great circles are not triangles (see quadrant BCEF in Fig. 453.01third printing of Synergetics 1of which quadrangles there are a total of 48 in the system); and note that the total number of triangles in the 25-great-circle system is 288 ergo, far less than the 31 great circles’ 480 spherical right triangles; ergo, we become satisfied that the icosahedron’s set of 480 is indeed the cosmic maximum-limit case of system-self-spun subdivisioning of its self into tetrahedra, which 480 consist of four sets of 120 similar tetrahedra each.

986.473 It then became evident (as structurally demonstrated in reality by my mathematically close-toleranced geodesic domes) that the spherical trigonometry calculations’ multifrequenced modular subdividing of only one of the icosahedron’s 120 spherical right triangles would suffice to provide all the basic trigonometric data for any one and all of the unit-radius vertex locations and their uniform interspacings and interangulations for any and all frequencies of modular subdividings of the most symmetrical and most economically chorded systems’ structuring of Universe, the only variable of which is the special case, time-sized radius of the special-case system being considered.

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Fig. 986.471 Modular Subdivisioning of Icosahedron as Maximum Limit Case: The 120 outer surface right spherical triangles of the icosahedron’s 6, 10, and 15 great circles generate a total of 242 external vertexes, 480 external triangles, and 480 internal face-congruent tetrahedra, constituting the maximum limit of regular spherical system surface omnitriangular self-subdivisioning into centrally collected tetrahedral components.

986.474 This surmise regarding nature’s most-economical, least-effort design strategy has been further verified by nature’s own use of the same geodesics mathematics as that which I discovered and employed in my domes. Nature has been using these mathematical principles for eternity. Humans were unaware of that fact. I discovered these design strategies only as heretofore related, as an inadvertent by-product of my deliberately undertaking to find nature’s coordination system. That nature was manifesting icosahedral and VE coordinate patterning was only discovered by other scientists after I had found and demonstrated geodesic structuring, which employed the synergetics’ coordinate-system strategies. This discovery by others that my discovery of geodesic mathematics was also the coordinate system being manifest by nature occurred after I had built hundreds of geodesic structures around the world and their pictures were widely published. Scientists studying X-ray diffraction patterns of protein shells of viruses in 1959 found that those shells disclosed the same patterns as those of my widely publicized geodesic domes. When Dr. Aaron Klug of the University of Londonwho was the one who made this discoverycommunicated with me, I was able to send him the mathematical formulae for describing them. Klug explained to me that my geodesic structures are being used by nature in providing the “spherical” enclosures of her own most critical design-controlling programming devices for realizing all the unique biochemical structurings of all biologywhich device is the DNA helix.

986.475 The structuring of biochemistry is epitomized in the structuring of the protein shells of all the viruses. They are indeed all icosahedral geodesic structures. They embracingly guard all the DNA-RNA codified programming of all the angle-and-frequency designing of all the biological, life-accommodating, life-articulating structures. We find nature employing synergetics geometry, and in particular the high-frequency geodesic “spheres,” in many marine organisms such as the radiolaria and diatoms, and in structuring such vital organs as the male testes, the human brain, and the eyeball. All of these are among many manifests of nature’s employment on her most critically strategic occasions of the most cosmically economical, structurally effective and efficient enclosures, which we find are always mathematically based on multifrequency and three-way-triangular gridding of the “spherical” because high-frequencedicosahedron, octahedron, or tetrahedron.

986.476 Comparing the icosahedron, octahedron, and tetrahedronthe isosahedron gives the most volume per unit weight of material investment in its structuring; the high-frequency tetrahedron gives the greatest strength per unit weight of material invested; and the octahedron affords a happybut not as stablemix of the two extremes, for the octahedron consists of the prime number 2, 2^{2}= 4; whereas the tetrahedron is the odd prime number 1 and the icosahedron is the odd prime number 5. Gear trains of even number reciprocate, whereas gear trains of an odd number of gears always lock; ergo, the tetrahedral and icosahedral geodesic systems lock-fasten all their structural systems, and the octahedron’s compromise, middle-position structuring tends to yield transformingly toward either the tetra or the icosa locked-limit capabilitieseither of which tendencies is pulsatively propagative.