Chorand Spheres

General Chorand Spheres

It is possible to generalize the notion of Chorand Spheres using Boolean algebra to define a construction which can be applied to any hexagram.  At first, this might seem an unnecessary exercise. Afterall, the first Chorand Sphere is oriented on Receptive/Creative, the most pure hexagrams, and the second on the two poles of resonance / , the most mingled hexagrams. However, consider the following analogy.  

The north/south pole of the earth is typically defined with respect to the axis of the planet's spin; this is so-called "true north".  But there is also "magnetic north". Magnetic north deviates a few degrees from true north, but it nonetheless provides a useful frame of reference. In generalizing the Chorand construction, we acknowledge that / is the true axis for the hexagrams, the fundamental polarity, but allow for the possibility that we might also need to accommodate alternative orientations.

Let us call the set of hexagrams defined by ieb < 3 the "Chorand Set" (or CS for short).  Remember, this is set of hexagrams that appear in the Receptive/Creative sphere. We can now define a function as follows:

This says that the Chorand function of a hexagram H is the set of hexagrams composed of G x H for each G in the Chorand Set CS. If we take G as Receptive, then we get the original Chorand Sphere as previously defined. If we take G as , then we get the second Chorand Sphere. However, by taking G to be some other value, we can generate different spheres.  

As an example, consider taking the hexagram as the reference point for the sphere. The Boolean structure that results from this is shown below. If you compare this to the Boolean strucuture for the original sphere, it is clear that it is composed of a slightly different set of hexagrams, representing the slight shift in axis. However, as expected, the two structures are topologically equivalent. The red lines show the axis of the structure when considered as a sphere.  The actual construction of the sphere is left as an exercise for the reader.

As noted above, by taking for G, we get the second Chorand Sphere.  This leads us directly to a general result:

This says that for any Chorand Sphere defined on hexagram H, the corresponding hypersphere composed from all the hexagrams not in the first sphere is simply the Chorand Sphere defined on H x . We know these two spheres are disjoint, they share no hexagrams. Together, they encompass the totality.

Another general result of interest is

This says that the Chorand Sphere defined on a hexagram H is the same as the Chorand Sphere defined on its Boolean opposite ~H. Exploring the extent of algebraic theorems that can be derived from these definitions remains an open topic.