987.130 Primary and Secondary Great-circle Symmetries |
Fig. 987.132E Fig. 987.132F |
987.132 The primary and secondary icosa symmetries altogether comprise 121 = 11^{2} great circles. (See Fig. 987.132E.) |
987.133 The crossing of the primary 12 great circles of the VE at G (see Fig. 453.01, as revised in third printing) results in 12 new axes to generate 12 new great circles. (See color plate 12.) |
987.134 The crossing of the primary 12 great circles of the VE and the four great circles of the VE at C (Fig. 453.01) results in 24 new axes to generate 24 new great circles. (See color plate 13.) |
987.135 The crossing of the primary 12 great circles of the VE and the six great circles of the VE at E (Fig. 453.01) results in 12 new axes to generate 12 new great circles. (See color plate 14.) |
987.136 The remaining crossing of the primary 12 great circles of the VE at F (Fig. 453.01 results in 24 more axes to generate 24 new great circles. (See color plate 15.) |
Fig. 987.137B Fig. 987.137C |
987.137 The total of the above-mentioned secondary great circles of the VE is 96 new great circles (See Fig 987.137B.) |
987.200 Cleavagings Generate Polyhedral Resultants |
Fig. 987.210 |
987.210 Symmetry #1 and Cleavage #1 |
987.212 A simple example of Symmetry #1 appears at Fig. 835.11. Cleavage #1 is illustrated at Fig. 987.210E. |
987.213 Figs. 987.210A-E demonstrate Cleavage #1 in the following sequences: (1) The red great circling cleaves the tetrahedron into two asymmetric but identically formed and identically volumed "chef's hat" halves of the initial primitive tetrahedron (Fig. 987.210). (2) The blue great circling cleavage of each of the two "chef's hat" halves divides them into four identically formed and identically volumed "iceberg" asymmetrical quarterings of the initial primitive tetrahedron (Fig. 987.210B). (3) The yellow great circling cleavage of the four "icebergs" into two conformal types of equivolumed one- Eighthings of the initial primitive tetrahedron^{__}four of these one-Eighthings being regular tetra of half the vector-edge-length of the original tetra and four of these one-Eighthings being asymmetrical tetrahedra quarter octa with five of their six edges having a length of the unit vector = 1 and the sixth edge having a length of sqrt(2) = 1.414214. (Fig. 987.210C.) |
987.220 Symmetry #2 and Cleavage #4: |
Fig. 987.221 |
987.221 In Symmetry #2 and Cleavage #4 the four-great-circle cleavage of the octahedron is accomplished through spinning the four axes between the octahedron's eight midface polar points, which were produced by Cleavage #2. This symmetrical four-great- circle spinning introduces the nucleated 12 unit-radius spheres closest packed around one unit-radius sphere with the 24 equi-vector outer-edge-chorded and the 24 equi-vector- lengthed, congruently paired radii^{__}a system called the vector equilibrium. The VE has 12 external vertexes around one center-of-volume vertex, and altogether they locate the centers of volume of the 12 unit-radius spheres closest packed around one central or one nuclear event's locus-identifying, omnidirectionally tangent, unit-radius nuclear sphere. |
987.223 Symmetry #2 is illustrated at Fig. 841.15A. |
Next Section: 987.230 |