A prime code virtue is that every architectural and engineering application derives from the same conceptual model – an applied geometry essentially built from spheres. Although the construction is an abstract exercise, the use of spheres as “building blocks” receives its inspiration from nature. Spheres are the shape of the universe at large, its innumerable celestial bodies and their smallest particles. No wonder the word “ball” is of the first learned by an infant child.
A Rational Accretion of Spheres – 1
To construct the model from spheres, being mindful of their integral center-points is key. As the first few spheres are assembled in an instinctive kind of way, joining their center-points forms the 3 dimensions of lines, planes (equilateal triangles), and the solid tetrahedron.
To build meaningfully beyond this elemental assemblage first requires abstracting a right angle from the tetrahedron, then realizing the idea of it in the next sphere placements to form a square. In assessing the cluster at this point, a line is shared by both the square and triangle, and only one sphere is in contact with all the others. If subsequent sphere placements were to follow these cues, a fully enveloped cluster results.
The Bode – 2
The 13-sphere cluster’s underlying form was identified at least as far back as the time of Archimedes. More recently it has been known by various names such as “middle crystal”; “dymaxion” or “vector equilibrium” (Buckminster Fuller); and the “triangular gyro-bicupola” (Johnson Solid).
Still, the form’s most enduring term is cuboctahedron; but because such a name becomes wearisome with repeated reference, it is shortened to cuboda – or further still to bode. The bode manifests in 3 basic ways: as a spherical cluster; a crystalline alternation of faceted planes; or a structural space frame of radial and connecting lines. Each of these manifestations also possess distinct internal and external expressions.
Furthermore, the bode may be oriented in 4 different ways such that a particular geometric element – point (vertex), line (edge), triangle, or square – is dominant. These various orientations and manifestations – as well as the interplay between them – sets the stage for pairing the abstract bode to the real world.
The Geocentric Cuboda – 3
The first and foremost bode application pairs the form’s central sphere with earth for this simple reason: It is the one sphere common to all people at all times. So matched, the outer cluster of spheres enveloping earth is viewed as a kind of anti-entropic thunderhead that imparts guiding bolts of spatial order. Happily the cluster can be relatively oriented so that a pair of radial lines joining the center-points of opposing outer spheres and transfixing the earth-sphere coincide with its natural axis of rotation. Aligned thus, the radial lines joining the opposing equatorial spheres’ center-points serves as a complementary axis for the geocentric cuboda.
Together, the natural and imaginary axes function to spin any bode element to any longitude and latitude – and are regarded to do so instantaneously. Furthermore, spinning elements about axes joining other opposing equatorial elements other than the vertices affords some pattern orientation alternatives.
Pattern Attributes – 4
With the bode pattern, what you see is very indicative of what you get. Each triangle heralds the first underlying form of assembled spheres – the tetrahedron. So there are 8 of them in the full bode. For each square-based form that fills space between the tetrahedra is mirrored outward, a full octahedron results. In doing so, it is evident that octahedra are intrinsic to tetrahedra and visa versa.
The implication of this is that the bode pattern may be generated outward without bound – or divided inwardly to any infinitesimal degree. In other words, the bode pattern poses an infinitely customizable one characterized by triangles representing both octahedra and tetrahedra; and by every line being the juncture of one square and two triangular planes, with any intersection of lines constituting a potential center point of a sphere of any size.
Indefinite Accretion of Spheres – 5
All of this is to say that the bode pattern may be extended with additional spheres. However, special caution must be taken to avoid a glob of random disorder. Although the procedure is straightforward in theory, nesting into triangular clusters is an iffy proposition.
If the right clusters are chosen, however, the bode pattern may be generated indefinitely with this unique attribute: every sphere potentially becomes a center sphere surrounded by 12 in a manner identical to the original 13-sphere bode cluster.
For serious study and reference, the above is detailed in the Orientation PDF.