987.240 Symmetry #3 and Cleavage #3

987.241 Symmetry #3 and Clea vage #3 mutually employ the six-polar-paired, 12 mid-edge points of the tetra-co ntained octa to produce the six sets of great-circle spinnabilities that in turn combine to define the two (one positive, one negative) tetrahedra that are inters ymmetrically arrayed with the common-nuclear-vertexed location of their eight equ i-interdistanced, outwardly and symmetrically interarrayed vertexes of the “cube ”the otherwise nonexistent, symmetric, square-windowed hexahedron wh ose overall most economical intervertexial relationship lines are by themselves u nstructurally (non-triangularly) stabilized. The positive and negative tetrahedra are internally trussed to form a stable eight-cornered structure superficially de lineating a “cube” by the most economical and intersymmetrical interrelationshi ps of the eight vertexes involved. (See Figs. 987.240and 241.) image image Fig. 987.241 Subdivision of Tetrahedral Unity: Octet: Duo-Tet Cube: Rhombic Dodecahedron: 987.242 In this positive-negative superficial cube of tetravolu me-3 there is combined an eight-faceted, asymmetric hourglasspolyhedron of tetravolume-1 \sfrac{1}{2}, which occurs interiorly of the interacting tetrahed ra s edge lines, and a complex asymmetric doughnutcored hexahedron of tetr avolume-1 \sfrac{1}{2}, which surrounds the interior tetra’s edge lines but occ urs entirely inside and completely fills the space between the superficially desc ribed “cube defined by the most economical interconnecting of the eight vertexes and the interior1\sfrac{1}{2}-tetravolume hourglasscore. (See Fig. 987.242E.) 987.243 An illustration of Symmetry #3 appears at Fig. 455.11A.

image image **E.**Fig. 987.230 Subdivision of Tetrahedral Unity: Symmetry #3:

1. The large triangle is the tetrahedron face. The smaller inscribed triangle is formed by connecting the mid-points of the tetra edges and represents the octa face congruent with the plane of the tetra face.

2. Connecting the midpoints of the opposite pairs of the internal octahedron’s 12 edges provides the six axes of spin for the six great circle system of Symmetry #3. The perpendicular bisectors at A and B are projections resulting from the great circle spinning. B also shows an oblique view of the half-Tetra or “Chef’s Caps” separated by the implied square. (For other views of Chef’s Caps compare Figs. B. 100.103B and 527.08A&B.)

3. The six great circle fractionations subdivide the tetrahedron into 24 A Quanta Modules.

4. Exploded view of the tetrahedron’s 24 A Quanta Modules.

5. Further explosion of tetrahedron’s A Quanta Modules.

image Fig. 987.240 Subdivision of Tetrahedral Unity: Symmetry #3: Subdivision of Internal Octahedron:

1. Bisection of tetrahedron face edges describes a congruent octahedron face.

2. The spinning of the internal octahedron on axes through the opposite mid-edges generates the six great circle system of Symmetry #3.

3. The six great circle fractionations subdivide the octahedron into 48 Asymmetric Tetrahedra; each such Asymmetric Tetrahedron is comprised of one A Quanta Module and one B quanta Module.

4. Exploded view of octahedron’s 48 Asymmetric Tetrahedra.

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