Chorand Spheres

The Principle (or External) Sphere

Receptive Chorand HemisphereThe animation on the right shows the Chorand Hemisphere constructed by starting with Receptive in the centre and applying Henri's algorithm.

The six gua added immediately around Receptive are those that are obtained by changing each line of Receptive in turn.  This first part of the construction is the same as Hacker's "hexagram flower" method.

The next layer of the construction creates links between pairs of the first 6 petals by combining their changes into a single hexagram. At this point a clear star shape is visible.

The final layer of the construction creates links between pairs of the point hexagrams of the star to create the large hexagon of the complete hemisphere. Hexagrams in this final layer are 3 lines different from the centre hexagram.

The diagrams below show the Receptive hemisphere alongside the corresponding construction beginning with Creative (note that I have modified the ordering of the hexagrams to make the correspondence between the middle layers clearer).

The first thing to notice is that the hexagrams in the outermost ring of the Creative hemisphere (shown in red) are the same as those in the Receptive hemisphere.  This is the basis for fitting the two hemispheres together into a single sphere.

In this joined arrangement, Creative is the top pole and Receptive is the bottom pole. Between them, the other hexagrams form a rhomboidal grid over the surface of a sphere.  The six hexagrams picked out in red, common to both hemispheres, form the equatorial ring.

If we compare the hexagrams shown in the two hemispheres above with those from the ieb < 3 structure, shown again on the left, it is clear that the hexagrams in both constructions are the same.

Notice that the equatorial hexagrams from the Chorand Sphere appear as the middle layer of the lattice structure, whilst Creative and Receptive, the poles of the sphere, form the top and bottom elements of the lattice.

Compare the other layers of the lattice with the Chorand Hemispheres and note how they correspond. The two structures are topologically equivalent.

I would like to suggest that this construction provides an interesting answer to the question of how the six-dimensional Boolean lattice could project into three-dimensional space. The Chorand Sphere generated on the Receptive/Creative axis can be seen as the visible surface of the lattice in 3-space.  We will return to this topic later in the thread.

We shall now consider the next aspect of the Chorand construction, the sphere generated from the / axis.