417.00 Precession of Two Sets of 60 Closest-Packed Spheres

417.01 Two identical sets of 60 spheres in closest packing precess in 90- degree action to form a seven-frequency, eight-ball-to-the-edge tetrahedron with a total of 120 spheres; exactly 100 spheres are on the outer shell, exactly 20 spheres are in the inner shell, and there is no sphere at the nucleus. This is the largest possible double-shelled tetrahedral aggregation of closest-packed spheres having no nuclear sphere. As long as it has the 20-sphere tetrahedron of the inner shell, it will never acquire a nucleus at any frequency.

417.02 The 120 spheres of this non-nuclear tetrahedron correspond to the 120 basic triangles that describe unity on a sphere. They correspond to the 120 identical right-spherical triangles that result from symmetrical subdividing of the 20 identical, equilateral, equiangular triangles of ei±er the spherical or planar-faceted icosahedron accomplished by the most economical connectors from the icosahedron’s 12 vertexes to the mid-edges of the opposite edges of their respective triangles, which connectors are inherently perpendicular to the edges and pass through one another at the equitriangles’ center and divide each of the equilaterals into six similar right triangles. These 120 triangles constitute the highest common multiple of system surface division by a single module unit area, as these 30 , 60, 90triangles are not further divisible into identical parts.

417.03 When we first look at the two unprecessed 60-ball halves of the 120-sphere tetrahedron, our eyes tend to be deceived. We tend to look at them “three-dimensionally,” i.e., in the terms of exclusively rectilinear and perpendicular symmetry of potential associability and closure upon one another. Thus we do not immediately see how we could bring two oblong quadrangular facets together with their long axes crossing one another at right angles.

417.04 Our sense of exclusively perpendicular approach to one another precludes our recognition that in 60-degree (versus 90-degree) coordination, these two sets precess in 60-degree angular convergence and not in parallel- edged congruence. This 60-degree convergence and divergence of mass- attracted associabilities is characteristic of the four-dimensional system.