Author: Bruce Camber
First posted: March 2012 Last update: February 26, 2015
Please note: These pages extend the very first overview of the Big Boardlittle universe (January 2012), as well as a workingdraft for an article for the academic community that was written for Wikipedia and within Wikipedia (March 2012). At that time, the working assumption was that a base2 chart from the Planck Units was out there in some academic journal, it just hadn't yet been indexed by Google. Wikipedia seemed to be a good place to have the academic community work together to write up something so simple. That was the starting point. The earliest draft of the article was first accepted and published early in April but then in May 2012 a group of specialists within Wikipedia rejected it as "original research" and it was deleted. Another article for the general public (and perpetual students) was also written in March 2012. We were attempting to find out what was wrong with our simple logic, mathematics and geometry.
What is simple? ... a point? Yet, by definition a point cannot have dimension. So then, is it a vertex? ...a node? Could this area be a pointfree geometry? What kind of singularity might these Planck Units be?
The very first doublings are a key to understand all the doublings. Yet, without question, this analysis will be an ongoing process for the foreseeable future. It is a rather idiosyncratic access path to attempt to grasp the nature of reality by positing a smallscale universe that amounts to a prestructure of all structure, one that gives rise to nodes or vertices, edges or lines, triangles and a multiplicity of 3D objects, and then to stuff of the human scale. So, it seems we must begin with most simple, the Planck Units, particularly the Planck Length, and begin to see what we can see in the process of multiplying by two. This type of exponential growth is called base2. It creates a scale or orders of magnitude, a doubling (categories, clusters, groups, layers, notations, sets, steps and more).
Powersoftwo and Exponentiation based on the Planck length. Herein it is referred to as Base2 Exponential Notation (B2). Can the universe, from the smallest to the largest, be seen in a more meaningful way using base2 instead of baseten scientific notation (B10) as proposed by Kees Boeke in 1957? B2 renders more granularity and a necessary relationality through nested geometries. The project originated with a series of five highschool geometry classes in December 2011. In looking at the five platonic solids, particularly the tetrahedron, the question was asked, "How far within could we go before hitting the walls of measurement or knowledge? Then, how far can we go before hitting the Planck Length?"
When we divide in half each of the six edges of a tetrahedron and connect those new vertices, we would find four halfsized tetrahedrons and an octahedron. Doing the same division within the octahedron, we find six halfsized octahedrons in each corner and eight tetrahedrons, one in each face. Within each object, we once assumed that we could divide those edges in half forever. Yet, unlike the limitless paradox introduced by Zeno (ca. 490 BC – ca. 430 BC), we had learned that the smallest conceptual measurement of a length within space and time was defined mathematically in and around 1899 by Max Planck. Though the Planck Constant, and Planck Length particularly, have not been universally accepted within the scientific community, it is a powerful concept based upon some of the most basic fundamentals of physics.
The Planck length is so small, it is written using exponential notation.
The number is 1.616199(97)x10^{35} meters. As a starting point we looked at many of the online references to the Planck length. In March 2012, there were just 276 Google links to the number, 1.616199, (virtually none). In our last review there were over 140,000. Over the next few years, we suspect those references will grow even more substantially. It has to be one of the more important numbers within space and time.
Professor Laurence Eaves of the University of Nottingham in England has a delightful YouTube video that explains this length that is used to define a point.
In this simple exercise, take the Planck length and multiply it by 2, until we reach something that is measurable today (the diameter of a proton) and then to objects within the human scale, and finally to the edges of the observable universe. Mathematically, it will require somewhere over 201 notations or doublings. We arrived at several different numbers, one by a senior NASA scientist, now retired, and another by a French astrophysicist who gave us the figure of 205.1 notations and explains the difference (See footnote #5).
In five columns, the first column is the baseten notations. The second column is a Planck number based on the number of base2 notations from the Planck length. The third column is the number of vertices, the powers of two. The fourth column is for the incremental increase in size (length). And, the fifth column will continue to be used for simple reflections.
Notations: Domains, Doublings, Groups, Layers, Sets Human Scale: Notations 67 to 137 Large Scale: Notations 134 to 205 Updates

1

0

2^0=1

Planck Length Multiples
1.616199(97)×10^{35}m

Discussions, Examples, Information, Speculations:
At the Planck Length, though it is a truly just a concept, let us take it as a given and that it is a special kind of vertex that is pointlike and a special kind of singularity.

1 
1 
2^1=2 
3.23239994×10^{35}m 
At the first notation or doubling, there are two vertices or nodes, perhaps the shortest possible line or edge. Nobody knows where current string theory comes into play This is a domain for speculative work, but we suspect even superstring theory as it is currently understood comes much later. Perhaps we can only say that a twodimensional object, a simple circle and a possible sphere emerge here and with every subsequent doubling. One might say that this notation is a necessary condition or initial condition for every subsequent doubling. Perhaps this might be called source code. 
1 
2 
2^2=4 
6.46479988×10^{35}m 
At the second notation there are four vertices or nodes. One might imagine that there are several logical possibilities yet within this speculative system, the simplest seems most logical. Three vertices form a triangle that define a plane and the fourth vertex forms a tetrahedron that defines the first three dimensions of space. Within the confines of the sphere, Pi or TT unfolding within the tension of its equation, it seems that a perfect tetrahedron must dynamically emerge inside the sphere. 
2 
3 
2^3=8 
1.292959976×10^{34}m 
At the third doubling there are eight vertices. Again, one might imagine that the activity is still within the sphere. Also, at some point the logical possibilities expand to include placing the vertices either inside the tetrahedron, on the edges of the tetrahedron or outside the tetrahedron. Again, it would seem that an octahedron and four tetrahedrons begin to emerge. If added within (see the closepacking of equal spheres in Wikipedia), tetrahedral closepacked structures emerge. If added externally, with just three additional vertices, a tetrahedral pentagon is created of five tetrahedrons (picture to be added With all eight additional vertices added externally,a cube or hexahedron could be created. 
2 
4 
2^4=16 
2.585919952×10^{34}m 
At the fourth doubling there are sixteen vertices. If any one of the vertices were to become a center point, and 10 vertices are extended from it, a tetrahedral icosahedron chain begins to emerge (picture to be added). With twenty vertices a simple dodecahedron is possible. And with the icosahedron, all five platonic solids are accounted. Among the many possibilities, in another configuration, a cluster of four polytetrahedral clusters (a total of 20 tetrahedrons) begin to emerge and completes with twenty vertices (picture to be added). With the tetrahedron these vertices could also divide the edges of the internal four tetrahedrons and one octahedron. If the focus was entirely within the octahedron, the first shared center point of the octahedron would begin to be defined and by the 18th vertex of the fourteen internal parts, eight tetrahedrons (one in each face) and the six octahedrons (one in each corner) would be defined
(picture to be added). 
2 
5 
2^5=32 
5.171839904×10^{34}m 
At the fifth notation, there are 32 vertices. Here there is a possibility for a cluster of eight tetrahedral pentagons to emerge and complete with 34 vertices. Simple logic and the research within the work on cellular automaton suggest that the most simple possible structures emerge first 
3 
6 
2^6=64 
1.0343679808×10^{33}m 
At the sixth notation, there are 64 vertices. With just 43 of these, a hexacontagon could be created. It has 12 polytetrahedral clusters with an icosahedron in the middle. 
3 
7 
2^7=128 
2.0687359616×10^{33}m 
By the seventh doubling, the possibilities become more textured. The results are not. Simple exponential notation based on the power of two is well documented. Of course, by using base2 exponential notation and starting at the Planck length, necessary relations might be intuited. 
3 
8 
2^8=256 
4.1374719232×10^{33}m 
Geometric complexification will be discussed. The nature of the perfect fittings, octahedrons and tetrahedrons, and the imperfect fitting, tetrahedrons making a pentastar or icosahedron, need review. 
3 
9 
2^9=512 
8.2749438464×10^{33}m 
In that pentastar the 7.368 degree spread  that is 1.54 steradians  increases within the icosahedron.

4 
10 
1024 
1.65498876928×10^{32}m 
_ 
4 
11 
2048 
3.30997752836×10^{32}m 
_ 
4 
12 
4096 
6.61995505672×10^{32}m 
_ 
5 
13 
8192 
1.323991011344×10^{31}m 
_ 
5 
14 
16,384 
2.647982022688×10^{31}m 
_ 
5 
15 
32,768 
5.295964045376×10^{31}m 
_ 
6 
16 
65,536 
1.0591928090752×10^{30}m 
_ 
6 
17 
131,072 
2.1183856181504×10^{30}m 
_ 
6 
18 
262,144 
4.2367712363008×10^{30}m 
_ 
6 
19 
524,288 
8.4735424726016×10^{30}m 
_ 
7 
20 
1,048,576 
1.69470849452032×10^{29}m 
_ 
7 
21 
2,097,152 
3.38941698904064×10^{29}m 
more information 
7 
22 
4,194,304 
6.77883397808128×10^{29}m 
_ 
8 
23 
8,388,608 
1.355766795616256×10^{28}m 
_ 
8 
24 
16,777,216 
2.711533591232512×10^{28}m 
_ 
8 
25 
33,554,432 
5.423067182465024×10^{28}m 
_ 
9 
26 
67,108,864 
1.0846134364930048×10^{27}m 
_ 
9 
27 
134,217,728 
2.1692268729860096×10^{27}m 
9 
28 
268,435,456 
4.3384537459720192×10^{27}m 
_ 
9 
29 
536,870,912 
8.6769074919440384×10^{27}m 
_ 
10 
30 
1,073,741,824 
1.73538149438880768×10^{26}m 
_ 
10 
31 
2,147,483,648 
3.47076299879961536×10^{26}m 
_ 
10 
32 
4,294,967,296 
6.94152599×10^{26}m 
_ 
11 
33 
8,589,934,592 
1.3883052×10^{25}m 
_ 
11 
34 
1.7179869×10^{11} 
2.7766104×10^{25}m 
Actual number: 17,179,869,184 vertices 
11 
35 
3.4359738×10^{11} 
5.5532208×10^{25}m 
34,359,738,368 
12 
36 
6.8719476×10^{11} 
1.11064416×10^{24}m 
68,719,476,736 
12 
37 
1.3743895×10^{12} 
2.22128832×10^{24}m 
137,438,953,472 
12 
38 
2.7487790×10^{12} 
4.44257664×10^{24}m 
274,877,906,944 
12 
39 
5.4975581×10^{11} 
8.88515328×10^{24}m 
549,755,813,888 
13 
40 
1.0995116×10^{12} 
1.77703066×10^{23}m 
1,099,511,627,776 
13 
41 
2.1990232×10^{12} 
3.55406132×10^{23}m 
2,199,023,255,552 
13 
42 
4.3980465×10^{12} 
7.10812264×10^{23}m 
4,398,046,511,104 
14 
43

8.7960930×10^{12}

1.42162453×10^{22}m

8,796,093,022,208 
14 
44 
1.7592186×10^{13} 
2.84324906×10^{22}m 
17,592,186,044,416 
14 
45 
3.5184372×10^{13} 
5.68649812×10^{22}m 
35,184,372,088,832 
15 
46 
7.0368744×10^{13} 
1.13729962×10^{21}m 
70,368,744,177,664 
15 
47 
1.4073748×10^{14} 
2.27459924×10^{21}m 
140,737,488,355,328 
15 
48 
2.8147497×10^{14} 
4.54919848×10^{21}m 
281,474,976,710,656 
15 
49 
5.6294995×10^{14} 
9.09839696×10^{21}m 
562,949,953,421,312 
16 
50 
1.12589988×10^{15} 
1.81967939×10^{20}m 
1,125,899,906,842,624 
16 
51 
2.25179981×10^{15} 
3.63935878×10^{20}m 
2,251,799,813,685,248 
16 
52 
4.50359962×10^{15} 
7.27871756×10^{20}m 
4,503,599,627,370,496 
17 
53 
9.00719925×10^{15} 
1.45574351×10^{19}m 
9,007,199,254,740,992 
17 
54 
1.80143985×10^{16} 
2.91148702×10^{19}m 
18,014,398,509,481,984 
17 
55 
3.60287970×10^{16} 
5.82297404×10^{19}m 
36,028,797,018,963,968 
18 
56 
7.205759840×10^{16} 
1.16459481×10^{18}m 
72,057,594,037,927,936 
18 
57 
1.44115188×10^{17} 
2.32918962×10^{18}m 
144,115,188,075,855,872 
18 
58 
2.88230376×10 ^{17} 
4.65837924×10^{18}m 
288,230,376,151,711,744 
18 
59 
5.76460752×10^{17} 
9.31675848×10^{18}m 
576,460,752,303,423,488 
19 
60 
1.15292150×10^{18} 
1.86335169×10^{17}m 
1,152,921,504,606,846,976 
19 
61 
2.30584300×10^{18} 
3.72670339×10^{17}m 
2,305,843,009,213,693,952 
19 
62 
4.61168601×10^{18} 
7.45340678×10^{17}m 
4,611,686,018,427,387,904 
20 
63 
9.22337203×10^{18} 
1.49068136×10^{16}m 
9,223,372,036,854,775,808 
20 
64 
1.84467440×10^{19} 
2.98136272×10^{16}m 
18,446,744,073,709,551,616 
20 
65 
3.68934881×10^{19} 
5.96272544×10^{16}m 
36,893,488,147,419,103,232 
21 
66 
7.37869762×10^{19} 
1.19254509×10^{15}m 
73,786,976,294,838,206,464 
21 
67 
1.47573952×10^{20} 
2.38509018×10^{15}m 
147,573,952,589,676,412,928 
This page is Notations 0 to 67: The Small Scale Universe Go to notations 67 to 134: The Human Scale Go to the LargeScale Universe: Notations 134 to 205
