If an indefinite accretion of spheres were to follow the natural and rational accretion of such that formed the core cuboda underlying them, and was to do so with indiscriminate deep-nesting, a breakdown of the bode’s intrinsic pattern is inevitable.
To some degree, adhering to simple rules with additional sphere placements assures against the pattern degradation; but with the 3-sphere triangular clusters of an extended accretion, ambiguity arises stemming from the bode’s characteristic duality.
Fortunately, a form lacking that particular trait may be grafted onto the bodal cluster otherwise rife with unique qualities, and may be done so as to pose a kind of alternate universe that complements the bode in a harmonious and useful manner.
Before proceeding with the alternative accretion, how a sphere may be centered on any intersection of lines extended from the bode pattern is revisited. As it turns out, no matter how isolated, a full bode cluster can be built around that sphere with the lines making the intersection supplying guidance. However, for the cluster to be aligned with the pattern from which the lines are extended, a 3rd line (not of the same plane) is required – if the intersection is hexagonal.
On the other hand, if the lines intersect at a right angle, no extra guiding line is needed to assure the pattern’s extension. Such is the power of squareness. Indeed, its disappearance leads to the cancerous assemblage of ill-placed spheres, and as such a clue is given that can help guide the first brave move: placement of sphere 14 onto the 13-sphere cluster to build the complementary form alluded to above.
With the knowledge that any bode sphere partly functions as a corner sphere of a 4-sphere cluster, the bode is oriented to face a that sphere directly. Then, with all spheres removed except for those that comprise the square, sphere 14 is placed on the corner sphere such that right angles are formed with each of the lines converging on that corner. This placement is a radical departure from the rational accretion of spheres that built the bode.
Actually, placement of sphere 14 is much like that of sphere 5, except that it 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 that are representative of the 3 spatial dimensions; and finally, sphere 17 completes the circuit and 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 which may be extended infinitely or infinitessimally divided like that of the cuboda; but unlike the bode pattern, the cube’s does so with one form only.
Thus does the bode supply the plane and principle by which the cube is grafted onto it. In this context, the cube can be viewed as the end result of a rational accretion of spheres, and as such the whole package constitutes the structure of the code’s conceptual model.
Although the cube and cuboda complement each other handily and harmoniously, their totality can be characterized as asymmetric and their conjunctive relationship one of separation and not true integration. That such perfection is not possible in a pure, geometrically consistent way perhaps speaks to existential limitations subject to the geometry of physical law, but I would posit that the result arrived at is an optimal one.
. . And one in which the shape of a truly sublime integration can well be suggested via the totality of ways in which the model’s geocentric context is spun by the code and applied to its design guidances, a kind of gestault led by the return of symmetry with the implied element of time best exemplified by the cube-based abode based on the earth-sphere centering the endless pattern, and in a sense housing it.