Exponentiation, Expansion, Inflation, and Acceleration

Tetra200Most recent updates: Throughout August 2016.
WORK-IN PROGRESS  (Not a first draft)
The enclosed chart will definitely go from 112th notation (about 3.3″ inches in length) down to the 67th notation (2.38509018×10-15 meters), the place of particle physics.

Introduction. There are four processes within the progression of possible geometries within the our chart of the base-2 numbers from the Planck scale to the Age of the Universe. There are many more processes that we can imagine, however, these are the most simple. Once we have analyzed these four progressions of geometries, we will attempt to define how these geometries interact to create more complexity. With this posting, we open more questions than we answer. Many more calculations need to be done, listed, and analyzed. Two of earliest progressions have also been introduced within the large horizontally-scrolled chart on lines 8 and 9; one is called Base-2 Vertices and other, Scaling Vertices.

With this posting the concern is to define the whole/parts relations.

The first process/progression. Up arrow, also a caret (^), hat and/or tentOur initial sample starts with a 3.3-inch tetrahedron; we divide the edges in half and connect the new vertices.  The result is a new number of necessarily related tetrahedrons (4) and an octahedron. This combination is sometimes called an octet. Typically by using division, it is assumed at each step, things are getting smaller and smaller.  In the very first exercise in December 2011, that was the process. But, we were not concerned about the resulting numbers of objects,  we were just following one of the smaller objects until we were at about the size of the Planck Length. At notation 67 (on our way, going further inside, to notation 1), the question would eventually be asked, “What is getting smaller now? Do the number of objects continue to increase or in some special way does everything begin to share the same structures and do the numbers actually begin to decrease?”  We didn’t know then and we do not know today, however, thoughts about homogeneity and isotropy began crossing through our minds and continue to do so.

Here the total number of objects is being considered for the first time from a common size of about two inches increasingly smaller, down to to the 67th notation.  Perhaps when we get to that point, somebody will have some advice for us. Do the number of objects continue to increase or is there a consolidation or is it both?

The second process. The V, also symbol for Von Neumann universe, a class of hereditary well-founded sets. This collection is formalized by Zermelo–Fraenkel set theory (ZFC).

The numbers are related to just the objects as they are. Multiplied by 2, the number of same-sized objects would be getting larger, all expanding in an identical fashion. Analogues with cellular division will be studied closely. This progression of numbers has been taken out to the 20th notation here: https://bblu.org/2016/01/08/number/#4  It was taken out to the 201 notation as a simple count of vertices here:  https://bbludata.wordpress.com/vertices

Now the counting would begin as close to the Planck scale as possible, possibly notation 2 with its eight scaling vertices or notation 3 with its 64 scaling vertices or perhaps even later. That logic is being examined.

It all needs to be understood in light of the first, third and fourth processes for all 201 notations.

The third process. The line, bottom-to-top and top-to-bottom.  Here is the most direct path between 1 and 201.  The numbers are related to the edges being multiplied by 2.  It could be either a single dimensional line, or a two-dimensional plate, and have shared edges with the three-dimensional processes as iterated  above and below.  Here we are describing the two-dimensional plates that tile and tessellate the universe.

The fourth process: Up/down, V and ^ from any point, any notation.  Possibly the first up/down process began at the 67th notation.

All four processes can in some manner be applied to each notation, to each group of 67 notations, and to all 200+ notations.

Calculations of the Numbers of Tetrahedrons & Octahedrons

This is a discovery process; we are learning as we go along this path. Your suggestions are most welcomed.

0 Tetrahedron(s) Octahedron(s)
1 a= 4
b=1
Note: There are 4 tetrahedrons (a) and one octahedron (b) within every tetrahedron.
2 16 + 8 = 24 c+d=#2a where (4)(a)=c and (8)(b)=d (4 + 6) = 10
Note: There are 6 octahedrons and 8 tetrahedrons within every octahedron. Each column provides the running total of each object going smaller. It is a notation-by-notation  count of identical objects.
In each step, there are always numbers from each column to be added together (within the parenthesis) to get a total.  The first number in each parentheses is tetrahedral-related and the ratio is always 4 tetrahedrons to one octahedron. The second number in each parentheses is octahedral-related and the ratio is always 8 tetrahedrons to 6 octahedrons.
3 (96+ 8o) = 176 (24+ 60) = 84
4 (704 + 672) = 1376 (176 + 504) = 680
5 (5504 + 5440) = 10944 (1376 + 4044) = 5420
6 (43776 + 43,360) = 87136 (10944 + 32,520) = 43,416
7 (347776 + 347328) = 695,104 (87136 + 260,496) = 347,632
8 (2780416 + 2781056) = 5,561,472 (695,104 + 2085792) = 2,780,896
What questions could we ask about the whole/parts relations? What might be imputed by watching the ratios between the two columns?
9 (22245888 + 22247168) = 44,493,056 (5,561,472 + 16,678,464) = 22,238,400
10 (177972224 + 177907200) = 355,879,424 (44,493,056 + 133430400) = 177,923,456
11 (711888896+ 1423387648) = 2,135,276,544 (355,879,424 + 1067540736) = 1,423,420,160
12 (8,541,106,176 + 11,387,361,280) = 19,928,467,456 (2,135,276,544 + 8,540,520,960) = 10,675,797,504
13 (79,713,869,824 + 85,406,380,032) = 165,120,249,856 (19,928,467,456 + 64,054,785,024) = 83,983,252,480
176/84=2.09523809524 1376/680=2.02352941176 10944/5420=2.01918819188 87136/43416=2.0070020269 695104/347632=1.99953974318 19,928,467,456/10,675,797,504=1.86669590244
14 (to be continued) (to be continued)
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