540.10 Prime Vector

540.11 All structural accounting of nature is accomplished with rational quantities of tetrahedra. The XYZ coordinates may be employed to describe the arrangements, but only in awkward irrationality, because the edge of the cube is inherently irrational in respect to the cube’s facial diagonal. The hypotenuses actually function only as the edges of the positive and negative tetrahedra, which alone permit the cube to exist as a structure. The hypotenuses connect the sphere centers at the cube corners; they function concurrently and simultaneously as the natural structuring of tetrahedra edges in the omnidirectional isotropic vector matrix; as either hypotenuse or tetra edge they are prime vectors.

540.12 Of the eight corners of the cube only four coincide with the sphere centers of closest-packed, unit-radius spheres; therefore only the cube’s facial diagonals can interconnect closest-packed spheres. One closed set of six cubeface diagonals can interconnect only four sphere-center corners of the prime tetrahedron, which alone provides the structural stability of the cube, whose eight-cornered, structural-stability completeness requires the saturation of the alternate set of six diagonals in each of the cube’s six faces. This alternate set of six diagonals intertriangulates the other four sphere centers of the cube’s eight corners. The cube diagonals and the edges of the tetrahedra structuring the cube are two aspects of the same phenomenon. The tetra-edge, cube-face diagonals connecting the two sets, of four corners each, of the cube’s total of eight corners are the prime vectors of the vector equilibrium and of the isotropic vector matrix.

540.13 The second power of the length of the prime vector that constitutes the diagonal of the cube’s face equals the sum of the second powers of any two edges of the cube. Because these two edges converge at the cube’s corner to form one standing wave that may be multifrequenced to apparently coincide with the cube’s facial diagonal, we discover that this relationship is what we are talking about in the deliberately nonstraight line. It is the same mathematical relationship demonstrated in the ancients’ proof of the Pythagorean theorem, wherein the square of the hypotenuse is proven to be equal to the sum of the squares of the triangle’s two legs. Thus the deliberately nonstraight line displays an evolutionary transformation from coincidence with the two sides of the parallelogram to coincidence with the seemingly straight, wavilinear diagonal of the parallelogram.

540.14 Prime vector may be considered variously as:

the axis of intertangency (Secs. 521.21 and 537.22);

the control line of nature (Sec. 982.21);

the deliberately nonstraight line (Sec. 522);

the diagonal of the cube (Sec. 463);

diametric unity (Sec. 986.160 and Fig. 986.161);

half-vectors (Sec. 537.21);

the hypotenuse (Sec. 825.26);

the internuclear vector modulus (Sec. 240.40);

the line of interrelationship (Secs. 505.74 and 505.82);

the line between two sphere centers (Sec. 537.21);

linear mensuration unity (Sec. 982.51);

the radial line (Sec. 537.21);

the T Quanta Module edge (Sec. 1033.130);

the tetra edge (Sec. 982.53);

unit radius (Sec. 1106.23).