The (Accreted) Cube (Topic 1 of Cube-based Shelter)
After building the cubodal pattern by rational accretion in Part I, potential haphazardness lurking in the pattern's continued generation is solved by posing the alternatively unambiguous pattern of a familiar form.
The potential problem with the cubodal pattern is apparent upon nesting sphere 14 into a square cluster to thereby complete an octahedron which, with the tetrahedron, comprises the cubodal pattern. Although both forms pose seemingly identical triangles (in the same plane), only 1 may accept an additional sphere.
To place a sphere in the wrong triangle breaks the pattern and, unless this is done for a definite purpose with sound reasoning, a disordered mess ensues. With only few spheres in play, this scenario is avoidable.
But with lots of spheres, continuing the pattern becomes a 50/50 proposition - a pitfall originating in the cuboda's intrinsic planar duality.
An alternative pattern free of this ambiguity is posed by first focusing on the cuboda's orientation in which a vertex is to the fore. Then extraneous spheres are removed to accentuate a square cluster.
Rather than nest sphere 14 into the square, it takes the right-angle precedent of rational accretion a step further by forming not 1, but 2 mutual right angles as it perches on one of the square's corner spheres.
This placement is kind of radical in that the support of planar nesting is not involved and the sphere sets on principle only. Similar placements of spheres 15, 16, and 17 on remaining square corners form yet newer right angle types between planes, and finally the symmetric orthogonal parallelism of the hexahedron, or cube.
By its innate pattern, the cube potentially fills space with a oneness unique among all forms. Spheres cannot truly fill space, but that which does most simply is the end result of their rational accretion - crowning the cuboda that provided the plane and principle necessary to resolve the problem it posed.
In the context of earth-centered co-axiality, the cuboda also serves to orient a pair of parallel co-cubes whose patterns alight by way of Celestial Projection, topic 2 of Cube-based Shelter.
