The Teikemeier/Drasny Sphere

 

Key Background Assumptions

There are two important assumptions in my analysis that need to be made explicit at the outset. The first we might consider a basic axiom of the geometry of space, and the second concerns the underlying structure of the hexagrams.

The Geometry of Space

I take it to be an uncontroversial property of space that two distinct n-dimensional entities cannot occupy the same location in n-dimensional space at the same time. This is true whether the dimensions are spacial or conceptual. Any violation of this property takes us up to (at least) n+1 dimensional space.

Consider the case for a two dimensional plane and two dimensional objects. Imagine two filled squares (two dimensional shapes) on a piece of paper (a two dimensional space). The two squares may touch without violating their containing spacial structure, but they cannot both be in the same location. In such a case, in 2 dimensions, we would talk about one being "on top" of the other, but such talk clearly takes us to a 3 dimensional space. The argument readily generalizes to n dimensions.

In organizing the gua into a spherical structure, it is reasonable to assume that this is taken to be an essentially three dimensional arrangement.

The Underlying Geometry of the Hexagrams

It is a basic assumption of my own work that the underlying geometry of the hexagrams, their natural structure, is the structure defined by the Boolean lattice. This assumption is implicitly shared by a number of other authors. My paper Flowers and Steps shows how the apparently distinct constructions of three different authors are unified through an analysis in terms of the lattice. Further, although they seek a simpler spatial arrangement, the basic rules of construction for both the Chorand Spheres and the Teikemeier/Drasny Sphere follow the basic rules of the Boolean lattice.

A two dimensional projection of the Boolean lattice is given here. However, the structure itself is naturally understood as a 6 dimensional hypercube.

An Assumption in the Construction

Finally, I wish to note an identical assumption that is made by Chorand, Teikemeier and Drasny in their different constructions. At the outset, they assume the existence of a spherical surface onto which they project their structure (in Teikemeier's case, he also allows symbols within the volume of the sphere). This is done because of the "perfect" nature of a sphere and its role as a metaphysical ideal. However, it seems to me a better approach would be to make the construction without any such assumption and then infer the actual geometry of the structure that results.