Chorand Spheres

Projected in Three-Space

How does all this look to us in three-space? The Chorand Sphere takes half of the hexagrams and links them together in a 2-dimensional grid across the surface of a sphere. Imagine looking up at the Receptive pole in the first Sphere.

The curvature of the surface means that you can actually see less than half of the hexagrams the structure contains; the equatorial hexagrams will be lying on the horizon, nearly perpendicular to the line of sight. Assuming that the surface of the sphere is opaque, only around 13 hexagrams would be clearly visible from any particular view point.

Remember that the second sphere is invisible, because it does not occupy the same 3-dimensional space. However, some of the missing structure will be inside the visible sphere, so if the surface were not opaque, it would be possible to see further substructures within, involving some of the other hexagrams. This is a topic that I hope to return to later.

I want to suggest that the first Chorand Sphere makes an excellent candidate for a canonical presentation of the Boolean lattice in 3-space. The seasonal sequence is a simple, natural example of a looped string; a one-dimensional sequence, closed in the second dimension. As such, it has always seemed like a candidate to be a visible ring of hexagrams in 3-space. This looped string is an order of magnitude simpler than a Chorand Sphere, and the fact that it, and the coherent wave sequences, are part of the first Chorand Sphere forms an elegant dimensional progression. Under such a view the Internal Chorand Sphere could be seen as the result of tunring the whole Boolean lattice "inside out" through three dimensional space.

There is one caveat that I need to add to this description. In his construction, Chorand assumes the surface of a sphere and then projects his structure on to it. Although I have gone with this assumption, it seems a better solution would be to look at the lattice dynamics of the structure and infer the resulting geometry. Considering the diagram given here, it seems to me that a more accurate description of the projection would be a hexagonal cross-section prism (in the geometric sense) with faceted faces, with corresponing hexagonal pyramid ends. Of course, from a topological perspective, this is equivalent to a sphere. However, most people think geometrically, not topologically and, from a geometric perspective, this is a prism is not a sphere.

There is another projection that needs to be considered in this thread. The two topological primitives of 3-space are the sphere and the torus, and we must also consider the latter.  This is a topic that I will return to later.