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The Teikemeier/Drasny Sphere

  I was first made aware of an arrangement of the 64 hexagrams into a single sphere by Lothar Teikemeier on the old Hexagram-8 discussion group. Now a similar arrangement has surfaced in József Drasny's Yi-Globe construction. Although motivated by different initial concerns, the resulting structure is very similar. In the following analysis I clarify the relationship of these structures to the Boolean Lattice and to Chorand's Spheres and also suggest a slightly different alternative construction that I believe is structurally more consistent. In recognition of its origins I refer to this alternative version as the Teikemeier/Drasny Prism.

Key Background Assumptions

There are two important assumptions in my analysis that need to be made explicit at the outset...

An Analysis of the Construction

I begin by describing some aspects of the construction of the sphere in detail...

Dimensionality, Chorand's Spheres and the Boolean Lattice

The most natural way of describing the structure of the hexagrams of the Yijing is as a six dimensional hypercube...

A Reconstruction of the Sphere

In the light of the background assumptions and the structure of the Boolean lattice, I suggest a slight modification to the Teikemeier/Drasny sphere...

The Layers

This shows the six dimensional Boolean lattice flattened into a three dimensional prism...

The Extra-Dimensional Residual

To conclude this analysis, I investigate exactly which symbols cannot be flattened into 3 dimensional space by the Teikemeier/Drasny construction...

Conclusions

The Teikemeier/Drasny construction is a valuable and interesting addition to the contemporary canon of geometric/structural analyses of the Symbols of Change.

If the six dimensional hypercube formed by the Boolean lattice is taken as the base structure, as the natural organization of the hexagrams, then the External Chorand Sphere can be taken as the normally visible surface of the lattice in three dimensional space - it contains 32 symbols. The Internal Chorand Sphere then describes the additional structure of the lattice that cannot normally be apprehended in three dimensional space - it contains the other 32 symbols.

Following on from this, the Teikemeier/Drasny prism shows what happens if you try and flatten more of the six dimensional lattice into three dimensional space. The Internal Chorand Sphere is inverted, and some of its symbols are flattened into an extended three dimensional structure: 44 of the hexagrams are then unproblematically to be found within the three dimensional structure, either on the surface, or within the volume, of the resulting prism. However, this still leaves 20 symbols which remain folded up in extra dimensions within certain points in the structure. These are symbols which cannot be flattened out from the higher dimensional space because the structure does not have enough positions to accommodate all of the symbols. The algebraic relationships between these higher dimension, co-located symbols tells us about the limitations imposed on the symbols by the three dimensional structure.

Thus, the Boolean Lattice, the Teikemeier/Drasny Prism, and the Chorand Spheres, taken as a family of structures, show a useful progression from the high, natural dimensionality of the symbols, through to ever flatter, more apprehensible structures.

 

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