Spheres play a role in all transport template linking schemes, even if they are reduced to circles by their plane bisections. Aside from the h-expansion, the sphere can be at least co-equal to a cube link when its radius is one edge length. The sphere’s role is elevated still further by the allowance of any whole number multiple of unit cube lengths in all 3 dimensions, in which case the unit cube could be dwarfed.
In employing the tetrahedron as a link, recall that this simplest of polyhedra was the first arrived at with the pre (natural) rational accretion of spheres. If utilized as a link, the tetrahedron’s vertex-centered spheres can be sized to dominate the form so much as to virtually obscure recognition of the form. This possibility is allowed provided spheres do not overlap.
The most extreme instance of an overwhelming role is observed when the sphere radius equals the tetrahedron’s edge length, in which case, the tetrahedron is totally immersed.
Planes either adjacent to or opposite of the vertex/sphere center are segmented to interface bodal planes, as in joining triangle and square-up bodal constructs.
The sphere can also be the whole link in 2 quite ordinary instances. Most familiarly, it can serve as a ball joint between cylindrical tubes or solid rods aligned to the horizontally and vertically biased hexagonal orientations, not to mention within one structure’s orientation.
A bicycle frame is the prime example of such usage. To avoid overlapping tubes, they and spheres can be sized such that their interfaces are 120°, 150°, or 180°. To attain such melding, sphere/tube radius ratios are crafted with the simple equation indicated.
The ultimate use of the sphere as a link has for its base the fact that any intersection within the bode pattern – which can consist of a line and a potential line – can be regarded as the center of an intrinsic sphere, i.e. the sphere can virtually be anywhere, if minimally connected.
So conceptualized, the inside of the sphere (and it’s cylindrical extensions) can not only interface a bodal construct of any of an infinite possible orientations beyond the 4 prime orientations, the inside of a sphere can also interface a non-code construct.
In the latter case, the spherical link should be labled with a skull and crossbones. Just kidding! For what its worth, the code’s author, as much as he believes in the ability of the code to solve problems and improve the lot of humankind, does not view non-code as something bad. Non code is simply non code, period. I can only advocate for code and non code constructs being interfaced cleanly so that in their being distinguished, they also complement each other.
The idea of spheres serving as internal links is turned inside out in Ground Design, which involves the integration of stationary constructs with the code’s largest employed sphere, the earth.