Fig. 987.230 
987.230 Symmetries #1 & 3; Cleavages #1 & 2 
987.231 Of the seven equatorial symmetries first employed in the progression of self fractionations or cleavages, we use the tetrahedron's six midedge poles to serve as the three axes of spinnability. These three greatcircle spinnings delineate the succession of cleavages of the 12 edges of the tetracontained octahedron whose six vertexes are congruent with the regular tetrahedron's six midedge polar spin points. The octahedron resulting from the first cleavage has 12 edges; they produce the additional external surface lines necessary to describe the twofrequency, nontimesize subdividing of the primitive onefrequency tetrahedron. (See Sec. 526.23, which describes how four happenings' loci are required to produce and confirm a system discovery.) 
987.232 The midpoints of the 12 edges of the octahedron formed by the first cleavage provide the 12 poles for the further greatcircle spinning and Cleavage #2 of both the tetra and its contained octa by the six great circles of Symmetry #3. Cleavage #2 also locates the centerofvolume nucleus of the tetra and separates out the centerofvolume surrounding 24 A Quanta Modules of the tetra and the 48 B Quanta Modules of the two frequency, tetracontained octa. (See Sec. 942 for orientations of the A and B Quanta Modules.) 
Fig. 987.240 
987.240 Symmetry #3 and Cleavage #3 
Fig. 987.241 
987.241 Symmetry #3 and Cleavage #3 mutually employ the sixpolarpaired, 12 midedge points of the tetracontained octa to produce the six sets of greatcircle spinnabilities that in turn combine to define the two (one positive, one negative) tetrahedra that are intersymmetrically arrayed with the commonnuclearvertexed location of their eight equiinterdistanced, outwardly and symmetrically interarrayed vertexes of the "cube"^{__}the otherwise nonexistent, symmetric, squarewindowed hexahedron whose overall most economical intervertexial relationship lines are by themselves unstructurally (nontriangularly) stabilized. The positive and negative tetrahedra are internally trussed to form a stable eightcornered structure superficially delineating a "cube" by the most economical and intersymmetrical interrelationships of the eight vertexes involved. (See Fig. 987.240.) 
Fig. 987.242 
987.242 In this positivenegative superficial cube of tetravolume3 there is combined an eightfaceted, asymmetric hourglass polyhedron of tetravolumel½, which occurs interiorly of the interacting tetrahedra's edge lines, and a complex asymmetric doughnut cored hexahedron of tetravolume 1½, which surrounds the interior tetra's edge lines but occurs entirely inside and completely fills the space between the superficially described "cube" defined by the most economical interconnecting of the eight vertexes and the interior 1½tetravolume hourglass core. (See Fig. 987.242E987.242E.) 
987.243 An illustration of Symmetry #3 appears at Fig. 455.11A. 
987.250 Other Symmetries 
987.251 An example of Symmetry #4 appears at Fig. 450.10. An example of Symmetry #5 appears at Fig. 458.12B. An example of Symmetry #6 appears at Fig. 458.12A. An example of Symmetry #7 appears at Fig. 455.20. 
987.300 Interactions of Symmetries: Spheric Domains 
987.310 Irrationality of Nucleated and Nonnucleated Systems 
987.311 The six great circles of Symmetry #3 interact with the three great circles of Symmetry # 1 to produce the 48 similarsurface triangles ADH and AIH at Fig. 987.21ON. The 48 similar triangles (24 plus, 24 minus) are the surfacesystem set of the 48 similar asymmetric tetrahedra whose 48 central vertexes are congruent in the one^{__}VE's^{__}nuclear vertex's center of volume. 
Fig. 987.312 
987.312 These 48 asymmetric tetrahedra combine themselves into 12 sets of four asymmetric tetra each. These 12 sets of four similar (two positive, two negative) asymmetric tetrahedra combine to define the 12 diamond facets of the rhombic dodecahedron of tetravolume6. This rhombic dodecahedron's hierarchical significance is elsewhere identified as the allspacefilling domain of each closestpacked, unitradius sphere in all isotropic, closestpacked, unitradius sphere aggregates, as the rhombic dodecahedron's domain embraces both the unitradius sphere and that sphere's rationally and exactly equal share of the intervening intersphere space. 
987.316 With the nucleated set of 12 equiradius vertexial spheres all closest packed around one nuclear unitradius sphere, we found we had eight tetrahedra and six Half octahedra defined by this VE assembly, the total volume of which is 20. But all of the six Halfoctahedra are completely unstable as the 12 spheres cornering their six square windows try to contract to produce six diamonds or 12 equiangular triangles to ensure their interpatterning stability. (See Fig. 987.240.) 
987.324
When the tetrahedron is unity of tetravolume1 (see
Table
223.64), then (in
contradistinction to the vectorradiused VE of tetravolume20)

Fig. 987.326 
987.326
This positivenegative tetrahedron complex defines
a hexahedron of overall
volume3^{__}1½ inside and 1½ outside its intertrussed system's
insideandoutsidevertexdefined domain.

987.327 Repeating the foregoing more economically we may say that in this hierarchy of omnisymmetric primitive polyhedra ranging from I through 2, 2 , 3, 4, 5, and 6 tetravolumes, the rhombic dodecahedron's 12 diamondfacemidpoints occur at the points of intertangency of the 12 surrounding spheres. It is thus disclosed that the rhombic dodecahedron is not only the symmetric domain of both the sphere itself and the sphere's symmetric share of the space intervening between all closestpacked spheres and therefore also of the nuclear domains of all isotropic vector matrixes (Sec. 420), but the rhombic dodecahedron is also the maximumlimitvolumed primitive polyhedron of frequencyl. 
Next Section: 987.400 