The 13-sphere cluster concluding the post on Crystallizing Order was the result of a sphere placement process termed *rational accretion – *which specified a reasoned construction as opposed to the purely natural process of celestial body-building that astrophysicists describe with their use of the word accretion.

If spheres had been nested around one sphere without forethought and simply followed unguided inclinations, the inevitable result would have been odd remaining spaces – slop that can not be filled by more (equal-sized ) spheres. Conversely, the cluster of 13 spheres is characterized by an attribute unique among all forms – the possession of a natural center which is exactly filled by a sphere equal in diameter to those surrounding it.

When the center-points of the 12 outer spheres are each connected to those of their 4 immediate neighbors, the underlying form is called a *cuboctahedron*. On its surface, this form features 6 perfect squares and 8 (equilateral) triangles, with these plane types sharing common edges, and the corners of 2 each converging in an alternating fashion at each of the 12 identical vertices.

Among the polyhedra – geometric solids comprised of polygonal faces of equal sides and angles – the cuboctahedron is kind of a second class citizen. It is not among the Platonic Solids (of which the cube is a member), because it exhibits more than one polygonal type. Thus the cuboctahedron is categorized as a semi-regular polyhedron as well as being counted among the Archimedean Solids – a grouping named after the great ancient scientist, philosopher, inventor, mathematician to whom the form is usually ascribed.

Other names for the cuboctahedron have come down through the centuries: Middle Crystal; Triangular Gyro-Bicupola; Dymaxion; and the Vector Equilibrium. The latter two terms were coined by Buckminster Fuller whose interest in the cuboctahedron matched that of the icosahedron (from which his domes had their geometric basis), despite the cuboctahedron’s possession of squares, which he seemed to hold in contempt.

Because the cuboctahedron is the geometric basis of all Geocentric Design Code applications, and because frequent reference to it in code discourse is cumbersome, it is shortened to “cuboda*.” *Its first syllable is pronounced like cube, and its last 2 syllables like that of the word pagoda. Cuboda’s adjective form is “cubodal” as in cubodal spheres.

Examination of the cuboda shows it to have 3 fundamental expressions, each having an internal and external aspect. For example, the cuboda’s *spherical expression* can be seen as a cluster of 13 spherical bubbles, or as a constellation of the spheres’ center-points.

Joining those center-points constitutes the cuboda’s *structural expression* which is comprised of a total of 36 lines: 12 inner radial lines joining the center-point of the central sphere to those of the outer 12 spheres; and the 24 lines joining the outer sphere center-points, each to its 4 neighbors to form an external skeleton.

The planar expression of the cuboda is characterized by 4 interlocking hexagons on the inside, and the previously observed gemlike alternation of 14 facets on the outside. The 6 X 4-sided squares equals the 8 X 3 – sided triangles. Actually, these planes signify a 4th expression: 8 triangular-faced tetrahedra and 6 square-faced semi-octahedra, or square pyramids evident in the below right depiction.

Portrayal of the cuboda’s planar expression is usually on 4 prime perspectives that correspond to the cuboda’s prime geometric features brought to the fore: vertex; triangle; square; an edge. Each of these exhibit some aspect of symmetry, and they are the most important because they are the only ones that have any practical use. Such a notion has parallel in the natural world, where only certain quantized energy states have physical meaning in the atom where nothing exists between whole number of electron wavelengths surrounding the nucleus. That analogy might hold for any polyhedron, except that only the cuboda has a natural nucleus.

Each of the 4 perspectives and the 3 cubodal expressions has individual worth, with the interplay between expressions being especially interesting and useful. One most striking example of this attribute comes to light in the search for how the ideal and most unique cubodal form is best applied to the real world.

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