In an indefinite accretion of spheres, indiscriminite deep-nesting inevitably led to a breakdown of the cuboda’s intrinsic pattern. To some degree, rules for extending the pattern with additional spheres assured against such degradation – except for the ambiguity posed by 3-sphere triangular clusters, a quandary arising from the cuboda’s characteristic duality.
Such an attribute in the context of a pattern otherwise rife with unique qualities raises the question of whether a form lacking that particular trait can be grafted onto the cubodal cluster in such a way that the alternative universe offered complements the cuboda in a harmonious and useful manner.
Before proceeding with an alternative accretion, how a sphere may be centered on any intersection of lines extended from the cubodal pattern is revisited. No matter how isolated, a full cluster can be built around that sphere with only the lines for guidance. For such cluster to be aligned with the pattern from which the lines were extended, however, a 3rd line (not of the same plane) is required if the intersection is hexagonally angled.
Conversely, no extra line is needed if the 2 lines intersect orthogonally, and such is power of squareness. Indeed, its disappearance attended the growing entropic cancerous assemblage of ill-placed spheres, a clue that can help guide the first brave move.
So armed, together with the knowledge that any outer cubodal sphere may be regarded as the corner of a 4-sphere cluster, the cuboda below is oriented to face a vertex directly, with all spheres removed except for those comprising the square. Sphere 14 is then placed on one sphere such that right angles form mutually with the lines converging on that corner. Such placement represents a radical departure from previous sphere placements, much as sphere 5 was, except that sphere14 does not benefit from any nesting whatsoever, and stands on principle only.
If sphere 15 is placed similarly on an adjacent sphere of the same square cluster, orthogonal planes result. Sphere 16 following the trend creates 3 mutually orthogonal planes representative of the 3 spatial dimensions, and finally, sphere 17 completes the circuit with formation of an underlying cube.
Qualities unique to the cube are its most economical expression of 3 dimensions and the homogeneity of its intrinsic pattern. It can be extended infinitely or infinitessimally divided like that of the cuboda, but unlike that form the cube’s pattern can do so with one form only. Thus is the cube built on and from plane and principle provided by the cuboda; and thus is the cube the end result of a rational accretion of spheres, with the whole package constituting the structure of Geocentric Design Code’s conceptual model.
Although cube and cuboda complement each other perfectly, their totality can be characterized as asymmetric and their conjunctive relationship technically separated – not truly integrated. That such perfection is not possible in a pure and geometrically consistent way lies the limitation imposed on rational accretion and its resulting artifice, a circumstance reflective of existential limitations subject to the geometry of physical law. For the challenges of that reality, the complementary patterns are inter-penetrable and I would posit (being privvy to how it is applied) that overall the result is an optimal one.
. . And one that situates at a nexus wherefrom the shape of a sublime integration can best be suggested by the totality of ways in which the dynamism of the model’s geocentric context is spun by the code to guide its designs, a gestault led by the return of symmetry with the implied element of time – and the derivation of cube-based shelter whose form upholds the earth-sphere’s centrality amid the endless pattern, and in a sense houses it.