260.50 Precession of Two Sets of 10 Closest-Packed Spheres
260.51 Two identical sets of 10 spheres in closest packing precess in 90-degree action to form a prime, nonnucleated, four-ball-to-the-edge tetrahedron with a total of 20 spheres. Each of the two sets of 10 balls consists of a line of four balls arranged in a tangentially cohered row nested in the long valley of a rect angle consisting of three pairs of balls tangentially cohered to one another in a parallel array, with two balls on one end and three balls on the other end. Cohering the four-ball row tangentially to the valley of the six-ball quadrangle produces a 10-ball aggregate. When brought together, these two 10-ball assemblies produce the prime, four-ball-edge tetrahedron of 20 balls, the largest single-shell tetrahedron without a nuclear ball. (This 20-ball tetrahedron is at the heart of the tetrahedral assembly of 120 balls comprised of two sets of 60 closest-packed spheres see Sec. 417.) To bring them into tetrahedral symmetry of assembly, each four-ball edge of the two separate assemblies must be precessed (turned at right angles) to the other’s four-ball edge. In these conditions the two-ball ed ges of the six-ball rectangle are now addressing the three-ball edg es of the oth er quadrangle. To the trained eye and rationale of rectilinear coordination it seems illogical to address two balls to three balls or three balls t o two balls. In matching such assemblies people think of doing so only in parallels or perpendiculars. (See Sec. 527.08.)
260.52 In universally convergent-divergent coordinate growth or shrinking, each row is greater (or lesser) by one than the next. Three automatically goes to two in a convergent, planar-arrayed, structurally stable system and two automatically goes to three in a divergent, planar-arrayed, structurally stable system. Tetrahedral expansion or contraction produces a structurally stable systematic model of universal behavior. In tetrahedral growth one goes to three and three goes to six and six goes to 10 (See Sec. 415.5 5 and Fig. 415.55A). Tetrahedral growth from unity is special-case angularly directional. Vector equilibrium growth from unity is nuclear-divergent at a growth rate of ten times frequency to the second power plus two:
260.53 A tetrahedron has threeand only three inherent polar symmetries; their axes run between the midpoints of the tetrahedron’s three pairs of opposite edges. (See Sec. 622.) These midpoints are in edges that are oriented at 90 degrees to one another.