Rethinking the Map

The mapping I have been using to date has a problem. My theory suggests that a mapping exists between the open and closed octahedral graph because their dimensional scaling factors are identical. The open graph was projected by taking the limit of the ratio of densities between generations. The closed graph was projected into 3D space and the dimensional scaling factor calculated using Euclidean geometry. Is there a dimensional difference in these target projections?

The open graph one may consider is a single dimensional projection as it relates to taking a limit between adjacent generations, or is it. Does it introduce a degree of freedom or remove one, that is, it has taken knowledge of a solution and generalised it into a comparative metric. In the closed case we go from zero to three, that introduces three degrees of freedom. How should this be represented in the mapping.

There is one more thing to consider, this is a mapping of solutions, and the TPPT does not represent all the solutions … only the primitive solutions. Typically the ternary tree has a scaling factor of 1/3 and a dimensionality of 3. From any one solution you can move to any one of three child solutions (or freedoms), or along a continuum to another parallel solution set. That’s four degrees of freedom.

There are also four paths from any one solution to any one of four connected solutions in the closed graph.

 

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