The Teikemeier/Drasny Sphere

 

Dimensionality, Chorand's Spheres and the Boolean Lattice

As already suggested, the most natural way of describing the structure of the relationships between the hexagrams of the Yijing is as a six dimensional hypercube. This comes about by considering the dimensionality of the lattice structure induced by the Boolean algebra of the symbols. Of course, it is extremely difficult, if not impossible, for the human mind to grasp a six dimensional structure except as a mathematical abstraction. Hence the need for various simplifications of this base structure.

Chorand's spheres, the Teikemeier/Drasny sphere, and my own construction of Unfolding are, essentially, all ways to project this complex six dimensional structure into an apprehensible form in three dimensional space.

With Unfoldings, the full six dimensional lattice structure is systematically unfolded into a sequence of lower dimensional sublattices with clear relationships between each substructure and the whole. The dimensionality of the sublattices is well-defined and uniform for each unfolding. I have suggested that this provides an algebraic model of Bohm's notion of the unfolding of the explicate order from the implicate order. Technical details of Unfoldings can be found in my paper The Yijing as a Symbolic Language for Abstraction.

With Chorand's spheres, the six dimensional hypercube is split into two spheres, each sphere is unproblematically a two dimensional surface curved in the third dimension. One sphere, the External Sphere, starts with the Receptive 000000 and Creative 111111 as the poles of the principle axis and builds the structure according to similar principles to Teikemeier/Drasny, but restricts the construction to retain the appropriate dimensionality. This sphere is composed of 32 hexagrams. Chorand's Internal Sphere starts with Before Completion 010101 and After Completion 010101 as the poles of the principle axis and applies the same construction principles to build a second sphere composed of the remaining 32 symbols. Thus, Chorand partitions the six dimensional hypercube into two independent three dimensional spheres. The full description of the Chorand construction can be found here.

In contrast, the Teikemeier/Drasny construction attempts to place all 64 hexagrams into a single, consistent three dimensional structure. If this is its aim, then it clearly fails: the structure cannot be three dimensional, it must have extra dimensions folded within it to accommodate the symbols which "overlap" or occupy the same spatial locations. However, I believe that it is this very failure which gives the structure its true value: the structure very usefully demonstrates just how much of the hypercube can be consistently flattened into three dimensional space.