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A Study of Notation #2
The Numbers: Second doubling of the Planck base units.
The second doubling

The second doubling

The second doubling

The second doubling

Scaling

2.14546×10^{−43} (s) 
6.464916×10^{35}.(m) 
5.176.470(51)×10^{8 }(kg) 
7.75109×10^{18} (C) 
8 or 64 
December 2017: This is our first draft of a working document (on the sixth anniversary of the early beginnings of our project).
Imagine such a dynamic from the beginning of time. Here every sphere is pushing out the next. Then, this image of Kepler’s stacked cannonballs demonstrates one possibility for the emergence of simple tetrahedrons from an alignment of the centerpoints of each sphere. This simple fact of geometry tells us much more than we can imagine.
At one time I was quite focused on the number of vertices. Then with Freeman Dyson’s suggestion, we began focusing on the scaling vertices. Now, we are considering the possibility that there are many forces that are generating vertices. It begs the question, “Which numbers and constructions are the primary building blocks or the seed structures?”
Because everything starts simply and Euclidean geometry appears to have the deepest intellectual foundation, I initially thought it would be a good place to start. I have a special affection for those five basic solids. Then I read that Euclidean geometry is a subset of projective geometry. That can’t be ignored.
Projective Geometry. Until convinced otherwise, let us start with projective geometries! From this second notation let it extend into every one of the 200 notations going forward. These planckspheres may well be the dark matter and dark energy that cannot be measured for obvious reasons!
Now, that, too, is a special image that cannot be minimized.
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Along that path I ran into the creativeandadventuresome energies of those focused on defining a theory of everything. Often these people use simple geometries that have become complex. Here, for example, is an image of a Quasicrystalline Spin Network (QSN) from the Quantum Gravity Network scientists who consider to be the Planck scale substructure of spacetime.
They may be right, however, my guess is that such an object could not manifest any earlier than the fourth notation where we’ve gone from spheres (Notation 1) to projective geometry (Notation 2) and then to Notation 3, Euclidean geometries.
By following the scaling vertices another story is told.
At one time, I thought there would only be eight scaling vertices for Notation #1. Now I can see possibilities for another eight on another track being carried by neverending, never repeating numbers. Logically and most simply there would be 64 for Notation #2, 512 within Notation #3, 4096 for Notation #4, and 32,768 for Notation #5.
Then why not extend that thought even further? What if there were several points within that first notation where dimensionless constants are scaled such that there are multiple tracks that open within the first notation. It could make such complex constructions as the QSN (just above) readily possible by that fourth doubling (86451232,768). What was initially thought to be only 32,768 vertices within the fifth notation could in fact be many different 32,768 dimensionless vertices by the fourth!.
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Certainly, where our first notation is primarily spheres orienting themselves to each other and the second notation is the beginning of projective geometry, I believe most of the third notation will be that tetrahedral generation pictured within that dynamic image of the stacked cannonballs. Within the tetrahedron is an octahedron. Within the octahedron are four hexagonal plates around the centerpoint, a halfsized octahedron in each of the six corners and a tetrahedron in each of the eight faces.
I believe a deep complexity could begin here with this basic, basic structure.
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With all the other forces in play, there are many possibilities for bonding, cinching and grappling things together.
So, to build out this second layer of the fabric of the universe, we will begin earnestly studying projective geometries and then we will be checking on them throughout all the notations.
We know the full complexity of all geometries is about to emerge. We have studied a range of intellectual positions — Langlands, Barrows, Penrose, Rees, Wilczek (and so many others). It’s not easy for somebody over 70 years old (me), but given that I require, especially within the first group of notations, simplicity, perhaps I can muddle through the first ten until somebody rescues me!
In a series of four articles, I tried rather unsuccessfully to get into the Langlands programs to answer the question, “How would this group envision a logical construction path and its modalities?” I am quite sure these Langlands programs has a lot to say rather early in the notational sequence.
There is much more to come. This is a working first draft.