454.00 Vector Equilibrium: Spherical Polyhedra Described by Great Circles

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 equilibrium’s is 20 (see drawings section).

454.02 This is the hierarchy of rational energy quanta values in synergetics, which the author discovered in his youth when he first sought for an omnirational coordinate system of Universe in equilibrium against which to measure the relative degrees of orderly asymmetries consequent to the cosmic myriad of pulsatively propagated energetic transactions and transformations of eternally conserving evolutionary events. Though almost all the involved geometries were long well known, they had always been quantized in terms of the cube as volumetric unity and its edges as linear unity; when employed in evaluating the other polyhedra, this method produced such a disarray of irrational fraction values as to imply that the other polyhedra were only side-show geometric freaks or, at best, “interesting aesthetic objets d’art.” That second-powering exists today in academic brains only as “squaring” and third-powering only as cubing is manifest in any scientific blackboard discourse, as the scientists always speak of the*² Change: rix footnote they have just used as “x squared” and likewise always account x³ as “x cubed” (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 (see drawings section).

454.04 The spherical octahedron is composed of eight spherical triangles, each consisting of six basic-vector-equilibrium, least-common-denominator triangles of the 25 great-circle, spherical-grid triangles.

454.05 The spherical cube is composed of six spherical squares with corners of 120each, each consisting of eight basic-vector-equilibrium, least-common-denominator triangles of the 25 great-circle spherical-grid triangles.

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.

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Fig. 454.06 Definition of Spherical Polyhedra in 25-Great-Circle Vector Equilibrium System: The 25 great circles of the spherical vector equilibrium provide all the spherical edges for four spherical polyhedra in addition to the vector equilibrium whose edges are shown here as heavy lines. The shading indicates a typical face of each as follows:

1. The edges of one of the spherical tetrahedron’s four spherical triangles consists of 12 VE basic LCD triangles.

2. The edges of one of the spherical octahedron’s eight spherical triangles consists of six VE basic LCD triangles.

3. The edges of one of the spherical cube’s six spherical squares consists of eight VE basic LCD triangles.

4. The edges of one of the spherical rhombic dodecahedron’s 12 spherical rhombic faces consists of four VE basic LCD triangles.