Chorand Spheres

The Internal Sphere

The sphere discussed on the previous page can be seen as the primary sphere generated on the / axis. In addition to this, the Chorand construction generates a second sphere on a perpendicular axis. In this case, the axis is /. The construction is applied in exactly the same way as before, starting with the new axis, and the resulting hemispheres are shown below.

As before, the two hemispheres can be fitted together to form a complete sphere. Clearly, the method ensures that this is topologically identical with the previous construction, except that we now have a different set of hexagrams in the nodes. You can check that none of the hexagrams in the previous sphere appear in this sphere, and vice verse.  The two spheres split the set of hexagrams into two sets of equal size.

Given the topological equivalence of the two Chorand Spheres, it is interesting to consider the Boolean structure generated by the hexagrams in this new configuration. This is repeated below.

This is, superficially, a completely different structure to that shown previously for the Receptive/Creative sphere.  However, topology is concerned with the fundamental spacial properties of a structure, and in that case, the two Boolean structures are equivalent. There is a well known mathematicians' joke about topologists: a topologist can't tell the difference between the mug of coffee and the donut! This is because the mug of coffee and the donut are both, topologically, a torus.

Relating these structures back to the Boolean lattice: the first Chorand Sphere has 30 facets, so between them, both spheres have 60 facets.  There are 240 facets in the complete Boolean lattice. So, 180 facets are hidden.These "missing" facets connect the two spheres together.