The Teikemeier/Drasny Sphere


An Analysis of the Construction

I need to begin by describing some aspects of the construction of the sphere in detail. Although the Teikemeier sphere and the Drasny globe are very similar structures, the description of the construction given by Drasny is well presented and a model of clarity. Therefore, I shall generally work from Drasny's presentation.

Layer 1Readers wishing to start with Drasny's own description of the construction should read this page.

Let's start at the base, with the Receptive Receptive and call this Level 0. Arranged above this are the six symbols with a single yang line. They are arranged symmetrically so that each is the same distance from the Receptive and they are equally spaced from each other. These six symbols form Level 1. The two dimensional projection of this is shown in the diagram on the right. At this stage the construction is identical the External Chorand Sphere.

Note that as a result of this, hexagrams that are opposite each other in any given layer will be exchanged trigram opposites. For example, the hexagram at the "bottom" of the ring is 100000 and opposite it in the layer is the hexagram 000100 - these two hexagrams are related as 100000 = e(000100).

The next stage is to introduce the symbols with two yang lines, connecting them to the symbols already in the construction, following the basic principle of symmetry.

Level 2For the Chorand construction, each pair of adjacent symbols in Level 1 are linked to give a symbol in Level 2 of the structure. The result of this is shown on the left. Now this clearly does not include all of the gua with two yang lines. In the External Chorand Sphere there are six gua in Level 2, but there are 15 gua with two yang lines. For Chorand, this is not a problem, because there is a second Sphere, the Internal Sphere, where the other symbols are to be located. However, the Teikemeier/Drasny construction attempts to get all of the symbols into a single structure.

Layer 2bAs a first step, further symbols can be added to Level 2 by allowing links between pairs of non-adjacent symbols in Level 1. The result of this additional construction is shown on the right with the new elements, and their connections, shown in red. Now an issue immediately becomes apparent. Up to this point it has been reasonable to assume that the symbols were being arranged on the surface of the sphere, but that assumption now fails.

Consider the red connecting lines on the diagram - these cross not only the connecting lines of the original symbols, but also the red connecting lines of other intermediate symbols. Now, the lines cannot be passing through each other, as that would violate the dimensionality of the surface. So, instead, they must be either above or below the other lines in the structure. Of course, as soon as we introduce relations such as above and below, the lines, and therefore the symbols they connect, are no longer strictly on the surface of the sphere, but instead are above it or below it. This is not a serious problem: instead of describing the surface of the sphere, we must allow that these additional symbols are actually inside the sphere.

Both Teikemeier and Drasny allow symbols inside the volume of their construction: both place symbols on the central axis; and Teikemeier, but not Drasny, place some other symbols inside the sphere. However, both authors place all of the symbols at Level 2 on the surface. I suggest that this is not correct, and that some of the symbols must be inside the structure. My own reconstruction, presented later, clarifies this matter.

So far, this suggests that the External Chorand Sphere correctly represents what is on the surface of the sphere, and that these additional symbols create additional structure within the interior volume of the construction.

However, this is still not the end of the process for Level 2. We now have 12 symbols in Level 2, but there are actually 15 hexagrams with two yang lines. We will see that integrating these three remaining symbols into the construction will force us out of three dimensional space.

Doubled TrigramsFollowing the principle that the symbols are always placed equidistant from their root symbols, we need to consider their placement. The remaining symbols are the three hexagrams composed of doubled yang trigrams, that is: 100100, 010010 and 001001. Each of these symbols has a pair of root symbols that are on opposite sides of the circle of Level 1. Therefore, each of these three symbols must be in the centre of Level 2. This means that these three symbols all occupy exactly the same location in the Teikemeier/Drasny sphere. This is consistent with the idea that trigram exchange opposite are always opposite each other in a layer, because these trigrams are all self-inverse under exchange: that is, 100100 = e(100100) and similarly for the other two symbols.

It puzzles me that this is introduced by Drasny without further comment; and yet three symbols in the same spatial location is not logically possible under normal circumstances. Drasny does have a note referring to the "secondary" positions of these symbols, but there is no detail on his web page.

There are two possibilities that I can imagine. Firstly, that the node in the structure containing these co-located symbols is somehow in a simultaneous superposition of all three symbols, as the animated diagram here suggests. However, I am not sure how this would work with respect to the rest of the structure: the decoherence of quantum phenomena when they encounter classical structures is well known. Instead, I wish to suggest that this problem in fact means that the sphere does not actually exist in three dimensions, but must have extra dimensions folded within it to accommodate the symbols in this way. That is, although the symbols appear to be in the same location from a three dimensional perspective, they are actually at distinct locations in a higher dimension.

There are a number of other places in the construction where similar co-locations occur, in Level 3 and Level 4. I will not detail them at this point, and the reader is referred to Drasny's own description for the full details.

Now this extra dimensionality is not a problem in itself, after all the Boolean lattice of hexagrams is a six dimensional structure. What surprises me is that Drasny does not seem to understand the implications of this feature of the construction for the dimensionality of the structure, and continues to discus it as if it is purely three dimensional, which it cannot be.