987.230 Symmetries #1 & 3; Cleavages #1 <& 2
987.231 Of th e seven equatorial symmetries first employed in the progression of self-fractiona tions or cleavages, we use the tetrahedron s six mid-edge poles to serve as the t hree axes of spinnability. These three great-circle spinnings delineate the succe ssion of cleavages of the 12 edges of the tetracontained octahedron whose six ver texes are congruent with the regular tetrahedron’s six mid-edge polar spin points . The octahedron resulting from the first cleavage has 12 edges; they produce the additional external surface lines necessary to describe the two-frequency, non-ti me-size subdividing of the primitive one-frequency tetrahedron. (See Sec. *526.23 ,*which describes how four happenings’ loci are required to produce and confirm a system discovery.) Fig. 987.210 Subdivision of Tetrahedral Unity: Symmetry #1:
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Initial tetrahedron at two-frequency stage.
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Tetrahedron is truncated: four regular corner tetra surround a central octa. The truncations are not produced by great-circle cleavages. C, D, and E show great-circle cleavages of the central octahedron. (For clarity, the four corner tetra are not shown.) Three successive great-circle cleavages of the tetrahedron are spun by the three axes connecting the midpoints of opposite pairs of the tetras six edges.
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First great-circle cleavage produces two Half-Octa.
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Second great-circle cleavage produces a further subdivision into four irregular tetra called “Icebergs.”
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Third great-circle cleavage produces the eight Eighth-Octahedra of the original octa.
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Eight Eighth-Octa and four corner tetras reassembled as initial tetrahedron.
image Fig. 987.221 Four-great-circle Systems of Octahedron and Vector Equilibrium : Symmetry #2:
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Six-great-circle fractionation of octahedron (as shown in Figs. 987.240B and C) defines centers of octa faces; interconnecting the pairs of opposite octa faces provides the octahedron’s four axes of symmetryhere shown extended.
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Four mid-face-connected spin axes of octahedron generate four great circle trajectories.
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Octahedron removed to reveal inadvertent definition of vector equilibrium by octahedron’s four great circles. The four great circles of the octahedron and the four great circles of the vector equilibrium are in coincidental congruence. (The vector equilibrium is a truncated octahedron; their triangular faces are in parallel planes.)
987.232 The midpoints of the 12 edges of the octahedron formed by the first c leavage provide the 12 poles for the further great-circle spinning and Cleavage # 2 of both the tetra and its contained octa by the six great circles of Symmetr y #3. Cleavage #2 also locates the center-of-volume nucleus of the tetra and s ep arates out the center-of-volume-surrounding 24