453.00 Vector Equilibrium: Basic Equilibrium LCD Triangle |
Fig. 453.01 |
453.01 The system of 25 great circles of the vector equilibrium defines its own lowest common multiple spherical triangle, whose surface is exactly 1/48th of the entire sphere's surface. Within each of these l/4sth-sphere triangles and their boundary arcs are contained and repeated each time all of the unique interpatterning relationships of the 25 great circles. Twenty-four of the 48 triangles' patternings are "positive" and 24 are "negative," i.e., mirrorimages of one another, which condition is more accurately defined as "inside out" of one another. This inside-outing of the big triangles and each of their contained triangles is experimentally demonstrable by opening any triangle at any one of its vertexes and holding one of its edges while sweeping the other two in a 360-degree circling around the fixed edge to rejoin the triangle with its previous outsideness now inside of it. This is the basic equilibrium LCD triangle; for a discussion of the basic disequilibrium LCD triangle, see Sec. 905. |
Fig. 453.02 |
453.02 Inside-Outing of Triangle: The inside-outing transformation of a triangle is usually misidentified as "left vs. right," or "positive and negative," or as "existence vs. annihilation" in physics. |
453.03 The inside-outing is four-dimensional and often complex. It functions as complex intro-extroverting. |
454.00 Vector Equilibrium: Spherical Polyhedra Described by Great Circles |
Fig. 454.01A Fig. 454.01B Fig. 454.01C |
454.01 The 25 great circles of the spherical vector equilibrium provide all the spherical edges for five spherical polyhedra: the tetrahedron, octahedron, cube, rhombic dodecahedron, and vector equilibrium, whose corresponding planar-faceted polyhedra are all volumetrically rational, even multiples of the tetrahedron. For instance, if the tetrahedron's volume is taken as unity, the octahedron's volume is four, the cube's volume is three, the rhombic dodecahedron's is six, and the vector cquilibrium's is 20 (see drawings section). |
454.03 The spherical tetrahedron is composed of four spherical triangles, each consisting of 12 basic, least-common-denominator spherical triangles of vector equilibrium. |
Fig. 454.06 |
454.06 The spherical rhombic dodecahedron is composed of 12 spherical diamond- rhombic faces, each composed of four basic-vector-equilibrium, least-common- denominator triangles of the 25 great-circle, spherical-grid triangles. |
455.00 Great-Circle Foldabilities of Vector Equilibrium |
Fig. 455.11 |
455.11 In the vector equilibrium's six great-circle bow ties, all the internal, i.e., central angles of 70° 32' and 54° 44', are those of the surface angles of the vector equilibrium's four great-circle bow ties, and vice versa. This phenomenon of turning the inside central angles outwardly and the outside surface angles inwardly, with various fractionations and additions, characterizes the progressive transformations of the vector equilibrium from one greatcircle foldable group into another, into its successive stages of the spherical cube and octahedron with all of their central and surface angles being both 90 degrees even. |
Fig. 455.20 |
455.20 Foldability of 12 Great Circles into Vector Equilibrium: We can take a disc of paper, which is inherently of 360 degrees, and having calculated with spherical trigonometry all the surface and central angles of both the associated and separate groups of 3^{__} 4^{__} 6^{__} 12 great circles of the vector equilibrium's 25 great circles, we can lay out the spherical arcs which always subtend the central angles. The 25 great circles interfere with and in effect "bounce off" or penetrate one another in an omnitriangulated, nonredundant spherical triangle grid. Knowing the central angles, we can lay them out and describe foldable triangles in such a way that they make a plurality of tetrahedra that permit and accommodate fastening together edge-to-edge with no edge duplication or overlap. When each set, 312, of the vector equilibrium is completed, its components may be associated with one another to produce complete spheres with their respective great- circle, 360-degree integrity reestablished by their arc increment association. |
Next Section: 456.00 |