The Teikemeier/Drasny Sphere

 

An Reconstruction of the Sphere

In the light of the original background assumptions and the structure of the Boolean lattice, I wish to suggest a modification to the construction of Teikemeier/Drasny sphere. The result is closer to Teikemeier's version than Drasny's, but with changes to the positions of some of the symbols. Nothing significant in my subsequent analysis hinges on these modifications, and the reader who is content with the existing construction, as offered by either Teikemeier or Drasny, may safely skip the material on this page and the following, and move directly to my analysis of the dimensionally residual symbols, which are identical for both my reconstructed version and Teikemeier's original version.

Motivation

As already stated, my primary reason for this reconstruction is that I wish to proceed without the assumption that the geometry of the resulting projection is spherical. Instead, we will infer the geometry of the end result from the construction. The reconstruction also gives me an opportunity to reposition some of the symbols from their locations in the Teikemeier/Drasny structure, resulting in (what I believe to be) a more consistent analysis of the geometry and structure.

Constructing in Layers

I will follow the basic construction method that we have already discussed, building up the structure of each "hemisphere" in parallel, in layers, and then fitting the two halves together.

Clearly, the lower half of the sphere starts with the Receptive 0000000, Level 0; and the upper starts with the Creative 111111, Level 6. The next two layers are the polar rings, Level 1 and Level 5. We have already seen the construction of Level 1, and that proceeds exactly as before. There is a question about the choice of circumferential order for the gua, but I shall leave that unaddressed for the moment, and go with the "standard" ordering. Remember that this ordering means that hexagrams on opposite sides of the same layer are the trigram exchange of each other. The construction of Level 5 follows the same principles, with the additional rule that hexagrams that are Boolean opposites should be located at diametrically opposite positions on the structure (i.e. the line connecting them must pass through the centre of the structure). This gives the following two arrangements for Level 0/1 and Level 6/5.

Level 0 and 1      Levels 6 and 5

The next step introduces Level 2 (and Level 4) - now differences start to emerge between this construction and the Teikemeier/Drasny approach.

Levels 1 and 2In the diagram shown on the left we see the construction of Level 2. Hexagrams circled in cyan are symbols on Level 1, the level above, included for context. Around the edge of the structure we see that each pair of adjacent symbols in Level 1 gives rise to a new symbol in Level 2. Also, we see that each pair of alternate symbols in Level 1 gives rise to a new symbol in Level 2. Positioned optimally relative to their parent symbols in the level above, this is the middle ring of symbols. Finally, although the connecting lines are not shown, each pair of diametrically opposite symbols in Level 1 give rise to a new symbol in Level 2. Because there are three such pairs in Level 1 there are three distinct symbols (the doubled yang trigrams) that all appear in the centre position, labelled as A2 in the diagram. Note that in every case, the new symbol in Level 2 is equidistant from its two parent symbols in Level 1.

This gives three distinct rings of symbols in Level 2, the outer ring, which will appear on the surface of the structure; the middle ring which will be located within the volume of the structure; and finally the three symbols co-located on the central axis. Both Teikemeier and Drasny conflate the outer and middle rings, putting all the symbols on the surface.

Levels 4 and 5The same construction can be applied to Level 5 to give the symbols and their relative positions in Level 4. This diagram is shown on the right, using the same conventions are the previous example.

Again, we end up with three distinct sets of symbols: one on the surface of the structure; one set of co-located symbols (the doubled yin trigrams) on the central axis, marked as A4; and one set of symbols between the surface and the central axis.

This only leaves Level 3 to be constructed. This is the equatorial layer that connects the two halves of the structure built so far. As a first pass, I will proceed from Level 2, using the positions of the symbols in that layer to create the layer above.

Level 2 and 3 (Part 1)I shall make the construction in stages. The first thing to note is that whilst symbols in Level 1 are derived from 1 symbol in Level 0, and each symbol in Level 2 is the composition of two symbols in Level 1, the symbols in Level 3 are the composition of 3 symbols in Level 2. The first stage of the construction is shown on the left. Here, all of the symbols in Level 2 are shown ringed in cyan to provide the full context as we build Level 3. We can see that each symbol around the edge of Level 3 is composed from 3 symbols in Level 2 and that, as before, each resulting symbol is equidistant from all of its parent symbols. This gives all of the symbols on the surface of the equatorial ring.

The second set of symbols in Level 3 are slightly more problematical and, as Teikemeier shows, give rise to more co-located hexagrams - in this case pairs.

The pair of diagrams below show the construction for one such pair. Starting with the diagram on the left, we see that the symbol in the red ring in Level 3 011010 comes from the three indicated symbols in Level 2, including one of the symbols co-located in the A2 group. However, the diagram on the right shows that the symbol 101100 in Level 3 is derived from different Level 2 hexagrams, including a different symbol from the A2 group, but the result is positioned at the same location. The same reasoning holds for the other positions in the middle ring of Level 3, resulting in 6 pairs of co-located hexagrams forming the middle ring of Level 3.

Level 2 and 3 (Part 2a)      Level 2 and 3 (Part 2b)

The final pair of co-located symbols in Level 3 occur in the very centre of the structure, again arising from 3 symbols in Level 2. The following pair of diagrams show this.

Level 3 (Part 3a)      Level 3 (Part 3b)

Level 3So, the final structure for Level 3 has only 6 uniquely located individual hexagrams, located on the surface. All of the remaining positions contain co-located pairs.

Of course, to be sure of the construction we would also need to work from Level 4 back to Level 3 to make sure the two different ways of constructing Level 3 agree on all the placements. This is easy to check and is left as an exercise for the keen reader.

We now have all of the levels constructed, and it is clear that many of the hexagrams are positioned within the volume of the structure rather than on the surface. Further, it does not look like its geometry is spherical - it looks like a hexagonal cross-section prism, as I have previously suggested for the Chorand Sphere.

The next page presents the individual layers and offers some initial comparisons.