The 13-sphere cluster that concluded Crystallizing Order was the result of a sphere placement process called rational accretion – a term meant to convey a reasoned approach to assembly, as opposed to the physical process of celestial body-building described by astrophysicists in their use of the word accretion.
If spheres had simply been nested around one sphere without forethought, by simply following unguided inclination, the inevitable result would have been some remaining space – slop that can not be filled by another (equal-sized ) sphere. Conversely, the cluster of 13 spheres set with a deliberative process was 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 that they make contact with, the completed underlying form is the cuboctahedron. Essentially, this form poses 6 perfect squares and 8 (equilateral) triangles, with the plane types sharing common edges, and the corners of 2 each converging in an alternating fashion at 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 placed behind the Platonic Solids (of which the cube is a member), because it exhibits more than one polygonal type. Accordingly, the cuboctahedron is categorized as a semi-regular polyhedron. More reflective of its historical context, the cuboctahedron is 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 form matched that of his icosahedral domes (that received much more attention), despite the cuboctahedron’s possession of squares, which he seemed to hold in contempt.
Because the cuboctahedron is the geometric basis of all applications of Geocentric Design Code, and because frequent reference to it in code discourse is cumbersome, I have shortened the term to cuboda. Its first syllable can be pronounced either like cube, and its last 2 syllables like that of the word pagoda. To relate or qualify another word with cuboda, the adjective form is cubodal as in cubodal this and thats.
Contemplation of the cuboda reveals it to have 3 fundamental expressions, each of which can be divided into an internal and external aspect. For example, the cuboda’s spherical expression poses a cluster of 13 spherical bubbles, and it can be seen as a constellation of the spheres’ center-points.
Joining those center-points constitutes the cuboda’s structural expression which is comprised of 36 lines: 12 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 match the 8 X 3 – sided triangles. Actually, these planes front a 4th expression that has only one aspect, but two polyhedral components: 8 triangular-faced tetrahedra and 6 square-faced semi-octahedra, which upon isolation appear as 4-sided pyramids. Can you see them in the below right depiction?
In considering the (external) faceted aspect of the cuboda’s planar expression, its portrayal almost invariably focuses on 4 prime perspectives that correspond to the cuboda’s prime geometric features brought to the fore: vertex; triangle; square; edge. Each of these exhibit at least some aspect of symmetry, and the discreet set of symmetry perspectives is 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. Such analogy might hold for any polyhedron, except that only the cuboda has a nucleus.
Although each of these 4 perspectives and the 3 cubodal expressions has individual worth, the interplay between expressions are especially interesting and useful. One most striking example of this virtue comes to light in the next search for how the ideal and most unique cubodal form is best applied to the real world.