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A Study of Notation #4  (still rough notes)

The Numbers: Fourth doubling of the Planck base units.

The fourth doubling
of Planck Time 
The fourth doubling 
of Planck Length
The fourth doubling
of Planck Mass
The fourth doubling
of Planck Charge
8.62585×10−43 (s)
512 or 4096

Sphere to tetrahedron-octahedron couplet

The fourth notation is a simple doubling of the third notation which is a simple doubling of the second which is a simple doubling of the first. We will be going over these doublings many times, just to see if we are missing anything. There may  be another kind of higher-order doubling of Notation #2 to Notation #4 and not just to Notation #3. There may be complex systems just starting to evolve, the mathematics of which is not within our current understanding.
Projective and Euclidean geometries are striking out on their own yet there are as few as 512, and possibly many more than 4096 vertices to share. There are many many possibilities for combinations of numbers, geometries and functions.

What do we call these configurations? There is a very special language among the professional geometers and we will turn to them for advice.
The fifth notation, another prime, will open yet another possibility for a mathematical system to emerge. There will be at least 4096 and possibly many more than 32,768 vertices with which to develop an integrated mathematical system. Within the sixth notation the vertex count jumps again, no less than 32,768 and possibly many more than 262,144 construction vertices to create an perfect or close-to-ideal infrastructure.  Within the seventh — it’s a prime — a new system is postulated to emerge and there will be  from 262,144 to many more than 2,097,152 vertices with which to work.

We now need to begin working with professional mathematicians with a broad scope to help guide our thinking. If you have some suggestions, you know what to do!
I will continue my research.   Together we can get this done.