The Teikemeier/Drasny Sphere

 

The Extra-Dimensional Residual

To conclude this analysis, I wish to investigate exactly which symbols cannot be flattened into three dimensional space. These are simply those symbols where more than one symbol occupies the same location in the structure. The reconstruction of the structure that I have offered in the preceding pages agrees with Teikemeier's structure on which symbols remain in higher dimensions, so this part of the analysis applies regardless.

Identifying the Symbols

Firstly, there are the eight symbols that overlap on the central axis of the sphere. These are found in one pair in Level 3, at the centre of the structure, and two groups of three, in Level 2 and Level 4. As these groups are on the axis, I refer to them as A2, A3 and A4.

A4: {110110, 101101, 011011}

A3: {101010, 010101}

A2: {100100, 010010, 001001}

In addition, there are a number of further pairs in the middle structure in Level 3 (in Teikemeier's construction these pairs are on the surface of Level 3).

E1: {110010, 101001}, E2: {110100, 011001}, E3: {101100, 011010}

E4: {010110, 001101}, E5: {100110, 001011}, E6: {100101, 010011}

What can we say about these groups of symbols? Taking them as a whole, one of the first things to observe is that they are all from the Internal Chorand Sphere. Now if, as I suggest, the External Chorand Sphere is taken to be the natural surface projection of the hypercube into three dimensional space, what we see in the Teikemeier/Drasny structure is an attempt to pull more of the hypercube structure out into three dimensional space.

Some of the Internal Chorand Sphere can be successfully flattened into three dimensions. However, the hexagrams detailed above represent aspects of the six dimensional structure that cannot. In the attempt, it seems as if the Internal Chorand Sphere is turned somewhat inside out. In particular, the hexagrams in A3, which are at the very centre of the Teikemeier/Drasny structure are at the poles of the Internal Chorand Sphere. Similarly, the hexagrams in A2 and A4 are equatorial on the Internal Chorand Sphere, but have migrated inwards to the central axis in the Teikemeier/Drasny structure.

Analysed As Opposition Patterns

In my detailed formal analysis of the Symbols of Change, one of the techniques is to consider relationships of opposition. There are two main techniques: firstly relating symbols in groups described by equations, and secondly clustering together all the symbols related through opposition. So, now let's consider the specific sets of symbols from these two perspectives.

Firstly note that the equatorial pairs, E1...E6, are all examples of Taiji pairs (see the description here). That is, for the two symbols, X and Y in each pair, it is the case that X = e(~Y). Take the set E2 {110100, 011001} as a concrete example:

X  = e(~Y)
110100  = e(~011001)
   = e(100110)
   = 110100

The same derivation can be done for the two symbols in each of the equatorial pairs. Therefore, we have a strong algebraic connection between the paired extra-dimensional symbols. In fact, this reflects the inadequacy of a three dimension structure in capturing the relationships between the symbols that are set up in that structure. Specifically, we have seen that symbols opposite each other in the same layer are exchanged opposites and, also, that symbols diametrically opposite each other across the whole structure are Boolean opposites. Now, for symbols in the equatorial plane, being opposite in the layer is also being opposite across the whole structure. This means that the structurally opposite symbol must be both the exchange opposite and the Boolean opposite, and those symbols in Level 3 that are not co-located are symbols for which ~X = e(X). The other pair of symbols in the Yijing for which this equation holds are the pair of symbols in A3. The remaining symbols in the layer are therefore forced into co-location to maintain the consistency of the structure.

As an aside, when we look at the position of these co-located symbols in the Internal Chorand Sphere, it turns out that the two members of each pair are at the same latitude (one of the tropics), but on opposite sides of the sphere.

Further, the equatorial symbols can be grouped as contrast clusters. Specifically:

So, all of the extra-dimensional symbols in the middle layer (the equatorial plane) of the structure can be naturally grouped into contrast clusters.

What of the two remaining sets of symbols, A2 and A4? Of course the union of these two sets is a well-defined set:

But they can also be analysed in terms of contrast clusters, if not in such a natural way as the equatorial pairs. Specifically, the overturned symmetric symbols in A2 ∪ A4 form one contrast cluster, and the remaining symbols form another.

  • A2 ∪ A4 = CxCy

However, the grouping is not as natural as the previous sets. In this case the reader will observe that one symbol form each of A2 and A4 form one contrast cluster, whilst the remaining two symbols from each set form the other. That is, the contrast clusters cross-categorize the sets A2 and A4; whilst for the other sets, A3 and E1...E6, the contrast clusters are formed from simple set unions.

Summary

So, what does this tell us? Firstly, it is important to realize that a well-defined lattice structure has a definite number of places where symbols can be placed, if we have more symbols than we have places in the structure, then we have a problem. Secondly, the rules used to build the Boolean lattice implicitly assume a six dimensional structure and this gives enough places for all the symbols. However, when we build the Teikemeier/Drasny prism we are applying those same rules that presume six dimensions, but restricting the construction to three dimensions. It should not be a surprise that things don't fit perfectly. The algebraic analysis of which symbols are co-located in the resulting structure simply show where the inter-symbol relationships force us out of three dimensions.