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. 
1010.12 Complex bubble aggregates are partitioned into prime volumetric domains by interiorly subdividing prime areal domains as flat drawn membranes. 
1010.21 All of the three foregoing nonnuclearcontaining domains of the tetrahedron, octahedron, and icosahedron are defined by the four spheres, six spheres, and twelve 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, nonnuclearcontaining domains of closestpacked, volumeembracing, unitradius sphere agglomerations. 
1010.22 While other total closestpackedsphere 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. All the Platonic polyhedra and many other more complex, multidimensional symmetries of sphere groupings can occur. None other than the three andonly prime structural systems, the tetrahedron, octahedron, and icosahedron, can be symmetrically produced by closestpacked spheres without any interioral, i.e., nuclear, sphere. (See Secs. 532.40, 610.20, 1010.20 and 1011.30.) 
1011.00 Omnitopology of Prime Volumes 
1011.33 Special case always has frequency and sizetime. 
1011.34 Generalization is independent of size and time, but the generalization principle must be present in every special case of whatever magnitude of size or time. 
1011.41 The phenomenon "congruence of vectors" occurs many times in nature's coordinate structuring, destructuring, and other intertransformings, doubling again sometimes with four vectors congruent, and even doubling the latter once again to produce eight congruent vectors in limittransformation cases, as when all eight tetrahedra of the vector equilibrium become congruent with one another. (See Sec. 461.08.) This phenomenon often misleads the uninformed observer. 
1011.51
The prime vector equilibrium has a nucleus surrounded,
closepackingly and
symmetrically, by 12 uniradius spheres. (See Illus.
222.01.)
As we add unit radius sphere
layers to the prime vector equilibrium, the 12 balls
of the first, or prime, outer layer
become symmetrically enclosed by a second closestpacked,
unit radius layer of 42 balls
circumferentially closest packed. This initiates a vector
equilibrium with modular edge and
radius intervals that introduce system frequency at
its minimum of two.^{2}
(Footnote 2: The number of balls in the outer shell of the vector equilibrium = 10 F^{2} + 2. The number 42, i.e., F^{2}, i.e., 2^{2} = 4, multiplied by 10 with the additive 2 = 42.) 
1011.52 The edge frequency of two intervals between three balls of each of the vector equilibrium's 24 outer edges identifies the edges of the eight outer facet triangles of the vector equilibrium's eight edgebonded (i.e., doublebonded) tetrahedra, whose common internal vertex is congruent with the vector equilibrium's nuclear sphere. In each of the vector equilibrium's square faces, you will see nine spheres in planar arrays, having one ball at the center of the eight (see Illus. 222.01), each of whose eight edge spheres belong equally to the adjacent tetrahedra's outwardly displayed triangular faces. This single ball at the center of each of the six square faces provides the sixth sphere to stabilize each of the original six halfoctahedra formed by the nuclear ball of the vector equilibrium common with the six halfoctahedra's common central vertex around the six fourball square groups showing on the prime vector equilibrium's surface. This second layer of 42 spheres thus provides the sixth and outermost ball to complete the sixball group of a prime octahedron, thus introducing structural stability increasing at a fourthpower rate to the vector equilibrium. 
1011.57 But at F^{3} we still have only one true nuclear ball situated symmetrically at the volumetric center of three layers: the first of 12, the next of 42, and the outer layer of 92 balls. There is only one ball in the symmetrical center of the system. This threelayer aggregate has a total of 146 balls; as noted elsewhere (see Sec. 419.05) this relates to the number of neutrons in Uranium Element #92. 
1011.61 At the F^{4}, 162ball layer, the eight potential nuclei occurring in the mid triangle faces of the F^{3} layer are now omnisurrounded, but as we have seen, this means that each has as yet only the 12 balls around it of the F^{0} nucleardevelopment phase. Not until the F^{5}, 252ball layer occurs do the eight potential secondgeneration nuclei become structurally enclosed by the 42ball layer, which has as yet no new potential nuclei showing on its surface^{__}ergo, even at the F^{5} level, the original prime nucleus considered and enclosingly developed have not become fullfledged, independently qualified, regenerative nuclei. Not until F^{6} and the 362ball layer has been concentrically completed do we now have eight operatively new, regenerative, nuclear systems operating in partnership with the original nucleus. That is, the first generation of omnisymmetrical, concentric, vector equilibrium shells has a total of nine in full, active, operational condition. These nine, 8 + 1, may have prime identification with the eight operationally intereffective integers of arithmetic and the ninth integer's zero functioning in the prime behaviors of eternally selfregenerative Universe. We may also recall that the full family of Magic Numbers of the atomic isotopes modeled tetrahedrally occurs at the sixth frequency (see Sec. 995). 
1012.00 Nucleus as Nine = None = Nothing 
1012.01 Nucleus as nine; i.e., non (Latin); i.e., none (English); i.e., nein (German); i.e., neuf (French); i.e., nothing; i.e., interval integrity; i.e., the integrity of absolute generalized octaval cosmic discontinuity accommodating all specialcase "space" of space time reality. (See Secs. 415.43 and 445.10.) 
1012.13
As shown in Numerology (Sec.
1223), when we begin to
follow through the
sequences of wave patterning, we discover this frequency
modulation capability
permeating the "Indig's" octave system of four positive,
four negative, and zero nine. (See
drawings section.)
Indigs of Numerology:

Fig. 1012.14A Fig. 1012.14B 
1012.14 Applying the IndigNumerology to the multiplication tables, this wave phenomenon reappears dramatically, with each integer having a unique operational effect on other integers. For instance, you look at the total multiplication patterns of the prime numbers three and five and find that they make a regular X. The foumess ( = + 4) and the fiveness ( =  4) are at the positivenegative oscillation center; they decrease and then increase on the other side where the two triangles come together with a common center in bowtie form. You find that the sequences of octaves are so arranged that the common ball can be either number eight or it could be zero or it could be one. That is, it makes it possible for waves to run through waves without having interference of waves. (See drawings section.) 
Fig. 1012.15 
1012.15 Each ball can always have a neutral function among these aggregates. It is a nuclear ball whether it is in a planar array or in an omnidirectional array. It has a function in each of the two adjacent systems which performs like bonding. This is the single energy transformative effect on closestpacked spheres which, with the arhythmical sphere space space sphere space space^{__}suggests identity with the neutronproton interchangeable functioning. 
Next Section: 1012.30 