905.10 Doubleness of Unity
905.11 The prime number twoness of the octahedron always occurs in structuring doubled together as fouri.e.,
905.12 The three-great-circle model of the spherical octahedron only “seems” to be three; it is in fact “double”; it is only foldably produceable in unbroken (whole) great-circle sheets by edge-combining six hemicircularly folded whole great circles (see Sec. 850). Thus it is seen that the octahedron as in Iceland spar crystalsoccurs only doubly, i.e., omnicongruent with itself, which is “quadrivalent.”
905.13 Among the three possible omnisymmetrical prime structural systems the tetrahedron, octahedron, and icosahedrononly the tetrahedron has each of its vertexes diametrically opposite a triangular opening. (See Illus. 610.20.) In the octahedron and icosahedron, each vertex is opposite another vertex; and each of their vertexes is diametrically blocked against articulating a self-inside-outing transformation. In both the octahedron and the icosahedron, each of the vertexes is tense-vector-restrained from escaping outwardly by the convergent vectorial strength of the system’s other immediately surrounding at minimum threevertexial event neighbors. But contrariwise, each of the octahedron’s and icosahedron’s vertex events are constrainingly impulsed inwardly in an exact central-system direction and thence impelled toward diametric exit and inside-outing transformation; and their vertex events would do so were it not for their diametrically opposed vertexes, which are surroundingly tense-vector-restrained from permitting such outward egress.
905.14 As a consequence of its uniquely unopposed diametric vertexing ergo permitteddiametric exit, only the tetrahedron among all the symmetric polyhedra can turn itself pulsatingly inside-out, and can do so in eight different ways (see Sec. 624); and in each instance, as it does so, one- half of its combined concave-convex unity “twoness” is always inherently invisible.
905.15 The octahedron, however, restrainingly vector-blocked as described, can only infold itself pulsatingly to a condition of hemispherical congruence like a deflated basketball. Thus the octahedron’s concave-convex, unity-twoness state remains plurally obvious. You can see the concave infolded hemisphere nested into the as-yet outfolded convex hemisphere. Verifying the octahedron’s fourness as being an evolutionary transformation of the tetrahedron’s unity-twoness, we may take the four triangles of the tetrahedron which were edge-hinged together (bivalently) and reassemble them univalently (that is, corner-to-corner) and produce the octahedron, four of whose faces are triangular (ergo structurally stable) voids. This, incidentally, introduces the structural stability of the triangle as a visualizable yet physical nothingness.
905.16 The triangle is structure. Structure is spontaneous pattern stabilization of a complex of six individual events. Structure is an integral of six events. Structure is a pattern integrity. Pattern integrity is conceptual relationship independent of size. The integrity of the nuclear structuring of the atoms is conceptually thinkable, as are the associability and disassociability proclivities of chemistry, virology, biology, and all nonbiological structuring and mechanics.
905.17 Any and all of the icosahedron’s vertexes pulsate individually and independently from the convex to concave state only in the form of local dimpling, because each only-from-outward-motion-restrained vertex being free to articulate inwardly toward its system center, and having done so becomes abruptly five-vector-restrained by its immediate neighboring vertexial event convergences; and the abrupt halting of its inward travel occurs before it reaches the system center. This means that one vertex cannot pulse inwardly more deeply than a local dimple similar to the popping in of a derby hat.
905.18 Both the coexisting concave and convex aspects of the icosahedron like those of the octahedron, but unlike those of the unique case of the tetrahedron are always visually obvious on the inside and outside of the only locally dimpled-in, or nested-in, vertex. In both the octahedron and the icosahedron, the concave-convex, only inwardly pulsative self-transforming always produces visually asymmetrical transforming; whereas the tetrahedron’s permitted inside-outing pulsatively results only in a visible symmetry, the quasiasymmetry being invisibly polarized with the remainder of Universe outside the tetrahedron, which, being omniradially outward, is inferentiallybut not visuallysymmetrical; the only asymmetrical consideration of the tetrahedron’s inside-outing being that of an initial direction of vertexial exiting. Once exited, the visible remaining symmetrical tetrahedron is in verity the inside-outness of its previously visible aspects.
905.19 In either of the two sets of four each as alternatively described, one of the polar states is always visible and the other complementarily invisible. This is a dynamic relationship. Dynamically, all four of each of the two sets of the tetrahedral potential are co-occurrently permitted and are required by omni-action-reaction-resultant synergetics. The seeming significance of the separately considered asymmetries are cancelled by the omnidirectional symmetry.
905.20 The vertexes are the unique, individual, ergo in-time events; and the nonvertex voids are the outdividual, ergo out, timeless, sizeless non- events. The both outwardly and inwardly escaping nonevents complement the embryo, local-in-time, special-case, convergent-event, systemic pattern fixation of individual intercomplementary event identities.
905.21
905.22 So-called edges and vectors are inherently only convergent or divergent interrelationships between multiply-identifiable, point-to-able, vertex fixes.
905.23 Because each tetrahedron has both four vertexes and four subtending nonvertex voids, we can identify those four diametrically complementary sets of all minimal cosmic structural systems as the four visible vertexes and the four nonvisible nonvertexes, i.e., the triangularly symmetrical, peripheral voids. The tetrahedron thus introduces experientially the cosmic principle of the visible and invisible pairs or couples; with the nonvisible vertex as the inside-out vertex, which nonvertex is a nonconvergence of events; whereas the vertexes are visible event convergences.