Constructing a conceptual model from scratch began by designating the elemental sphere as the model’s building unit. Radii from all-important center-points were specified to be equal for all spheres, and to be equal to the distance to the surface og any neighboring sphere as well, i.e., spheres contact at one point.

So specified, dimensional formation proceeded with a line between 2 spheres’ center-points; an (equilateral triangle) plane between 3 center-points; and a solid (tetrahedron) from lines joining the center-points of 4 spheres. Under one special set of circumstances characterized by simple equality, the idea of the right angle was abstracted from the 4-sphere cluster, and that concept was made real by the reasoned placement of sphere 5.

Another sphere similarly nested formed a circuit of 4 right angles, and a perfect square. An analysis of the 6-sphere cluster revealed that only one sphere contacted all the remaining 5, spheres that comprised the triangle and the square clusters with 2 spheres being common to both underlying planes that thus share a common edge between them.

Equipped with these abstractions, the incompleteness of the 6-sphere cluster is addressed by sphere 7′s placement in contact with the common sphere, to in essence presume it to also be a *central sphere*. Nesting sphere 7 between the center sphere and any 2 other spheres seems to be the only reasonable course available, with the 5 choices either being nesting into (2) triangular sphere-pairs, or (3) square sphere-pairs.

Nesting sphere 7 into any of the square’s 3 pairs forms another triangle and as such follows the observed square/triangle alternation. If it is nested into the pair adjacent to the pre-existing sphere-pair common to both triangle and square, a new right angle is formed as well. With sphere 8 placed to complete a second square, an underlying alternation of squares and triangles completely surround one sphere’s center-point, such that the 4 corners converge in a kind of full-point crystallization.

The more established and explicit alternating pattern is easier to build on with spheres 9 and 10, which are nested between sphere-pairs from the original and new square clusters. As these spheres sustain the pattern of planar alternation, another kind of circuit is completed with formation of a hexagonal layer of 6 alternating triangles.

Observing the 10-sphere cluster on its flip side, the 7-sphere hexagonal layer poses 6 vacancies for nesting 3 spheres with 2 possible placement choices. Again, preserving plane pattern alternation rules, and so placed to follow the precedent the alternation is seen all the way around the completed form.

The 13-sphere is complete with the one central sphere surrounded by 12, each indistinguishable in the sense that neighboring spheres are arranged identically around them. Connecting the spheres’ center-points delineates the underlying form termed the *cuboctahedron*. Because this form is involved in all code applications, it will be explored exhaustively in the next post.