1010.00 Prime Volumes
1010.01 A prime volume has unique domains but does not have a nucleus.
1010.02 A prime volume is different from a generalized regenerative system. Generalized regenerative systems have nuclei; generalized prime volumes do not.
1010.03 There are only three prime volumes: tetrahedron, octahedron, and icosahedron. Prime volumes are characterized exclusively by external structural stability.
1010.10 Domain and Quantum: The unique insideness domain of a prime system is, in turn, a prime volumetric domain, which is always conceptually defined by the system’s topological vertex-interconnecting lines and the areas finitely enclosed by those lines.
1010.11Prime volumetric domain and prime areal domain together provide space conceptuality independent of size, just as the tetrahedron provides prime structural system conceptuality independent of size.
1010.12 Complex bubble aggregates are partitioned into prime volumetric domains by interiorly subdividing prime areal domains as flat drawn membranes.
1010.13A prime volumetric domain has no volumetric nucleus. A prime areal domain has no planar nucleus. So we have prime system volumetric domains and prime system areal domains and linear interconnections of all vertexes extemdash all with complete topological conceptual interpatterning integrity utterly independent of size.
1010.14 This frees conceptual-integrity comprehending and all the prime constituents of prime-pattern integrity, such as “volume,” “area,” and “line,” from any special-case quantation. All the prime conceptuality of omnitopology is manifest as being a priori and eternally generalized phenomena. Thus quantum as prime-structural-system volume is eternally generalized, ergo, transcends any particulate, special-case, physical-energy quantation. Generalized quanta are finitely independent because their prime volumetric- domain-defining lines do not intertouch.
1010.20 Nonnuclear Prime Structural Systems: The domain of the tetrahedron is the tetrahedron as defined by four spheres in a tetrahedral, omniembracing, closest-packed tangency network. The domain of an octahedron is vertexially defined by six spheres closest packed in omnitriangular symmetry. The domain of an icosahedron is vertexially defined by 12 spheres om- nicircumferentially intertriangulated and only circumferentially symmetrically triangulated in closest packing without a nucleus (in contradistinction to the center sphere of the vector equilibrium, whose 12 outer sphere centers define the vector equilibrium’s 12 vertexes; all 13 of the vector equilibrium’s spheres are intersymmetrically closest packed both radially and circumferentially).
1010.21 All of the three foregoing non-nuclear-containing domains of the tetrahedron, octahedron, and icosahedron are defined by the four spheres, six spheres, and 12 spheres, respectively, which we have defined elsewhere (see Sec. 610.20, “Omnitriangular Symmetry: Three Prime Structural Systems”) as omnitriangulated systems or as prime structural systems and as prime volumetric domains. There are no other symmetrical, non-nuclear-containing domains of closest-packed, volume-embracing, unit-radius sphere agglomerations.
1010.22 While other total closest-packed-sphere embracements, or agglomerations, may be symmetrical or superficially asymmetrical in the form of crocodiles, alligators, pears, or billiard balls, they constitute complexedly bonded associations of prime structural systems. Only the tetrahedral, octahedral, and icosahedral domains are basic structural systems without nuclei. AU the Platonic polyhedra and many other more complex, multidimensional symmetries of sphere groupings can occur. None other than the three-and-only prime structural systems, the tetrahedron, the octahedron, and the icosahedron, can be symmetrically produced by closest-packed spheres without any interiorally contained, i.e., nuclear, sphere.