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Foundations of the Big Board - little universe:
The Planck Length, Base-2 Exponential Notation, and Nesting Geometries

Author:  Bruce Camber
First posted: March 2012  Last update: February 26, 2015

Please note:  These pages extend the very first overview of the Big Board-little universe (January 2012), as well as a working-draft 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 base-2 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 point-free 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 on-going process for the foreseeable future.  It is a rather idiosyncratic access path to attempt to grasp the nature of reality by positing a small-scale universe that amounts to a pre-structure 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 base-2.  It creates a scale or orders of magnitude, a doubling (categories, clusters, groups, layers, notations, sets, steps and more). 

Powers-of-two and Exponentiation based on the Planck length.  Herein it is referred to as Base-2 Exponential Notation (B2). Can the universe, from the smallest to the largest, be seen in a more meaningful way using base-2 instead of base-ten 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 high-school 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 half-sized tetrahedrons and an octahedron.  Doing the same division within the octahedron,  we find six half-sized 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 base-ten notations. The second column is a Planck number based on the number of base-2 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





Primary vertices


Planck Length Multiples


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 3.23239994×10-35m 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 two-dimensional 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 6.46479988×10-35m 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 1.292959976×10-34m 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 close-packing of equal spheres in Wikipedia), tetrahedral close-packed 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 2.585919952×10-34m 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 5.171839904×10-34m 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 1.0343679808×10-33m 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 2.0687359616×10-33m 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 base-2 exponential notation and starting at the Planck length, necessary relations might be intuited.
3 8 2^8=256 4.1374719232×10-33m 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 8.2749438464×10-33m In that pentastar the 7.368 degree spread -- that is 1.54 steradians -- increases within the icosahedron.
4 10 1024 1.65498876928×10-32m _
4 11 2048 3.30997752836×10-32m _
4 12 4096 6.61995505672×10-32m _
5 13 8192 1.323991011344×10-31m _
5 14 16,384 2.647982022688×10-31m _
5 15 32,768 5.295964045376×10-31m _
6 16 65,536 1.0591928090752×10-30m _
6 17 131,072 2.1183856181504×10-30m _
6 18 262,144 4.2367712363008×10-30m _
6 19 524,288 8.4735424726016×10-30m _
7 20 1,048,576 1.69470849452032×10-29m _
7 21 2,097,152 3.38941698904064×10-29m more information
7 22 4,194,304 6.77883397808128×10-29m _
8 23 8,388,608 1.355766795616256×10-28m _
8 24 16,777,216 2.711533591232512×10-28m _
8 25 33,554,432 5.423067182465024×10-28m _
9 26 67,108,864 1.0846134364930048×10-27m _
9 27 134,217,728 2.1692268729860096×10-27m
9 28 268,435,456 4.3384537459720192×10-27m _
9 29 536,870,912 8.6769074919440384×10-27m _
10 30 1,073,741,824 1.73538149438880768×10-26m _
10 31 2,147,483,648 3.47076299879961536×10-26m _
10 32 4,294,967,296 6.94152599×10-26m _
11 33 8,589,934,592 1.3883052×10-25m _
11 34 1.7179869×1011 2.7766104×10-25m Actual number: 17,179,869,184 vertices
11 35 3.4359738×1011 5.5532208×10-25m 34,359,738,368
12 36 6.8719476×1011 1.11064416×10-24m 68,719,476,736
12 37 1.3743895×1012 2.22128832×10-24m 137,438,953,472
12 38 2.7487790×1012 4.44257664×10-24m 274,877,906,944
12 39 5.4975581×1011 8.88515328×10-24m 549,755,813,888
13 40 1.0995116×1012 1.77703066×10-23m 1,099,511,627,776
13 41 2.1990232×1012 3.55406132×10-23m 2,199,023,255,552
13 42 4.3980465×1012 7.10812264×10-23m 4,398,046,511,104




14 44 1.7592186×1013 2.84324906×10-22m 17,592,186,044,416
14 45 3.5184372×1013 5.68649812×10-22m 35,184,372,088,832
15 46 7.0368744×1013 1.13729962×10-21m 70,368,744,177,664
15 47 1.4073748×1014 2.27459924×10-21m 140,737,488,355,328
15 48 2.8147497×1014 4.54919848×10-21m 281,474,976,710,656
15 49 5.6294995×1014 9.09839696×10-21m 562,949,953,421,312
16 50 1.12589988×1015 1.81967939×10-20m 1,125,899,906,842,624
16 51 2.25179981×1015 3.63935878×10-20m 2,251,799,813,685,248
16 52 4.50359962×1015 7.27871756×10-20m 4,503,599,627,370,496
17 53 9.00719925×1015 1.45574351×10-19m 9,007,199,254,740,992
17 54 1.80143985×1016 2.91148702×10-19m 18,014,398,509,481,984
17 55 3.60287970×1016 5.82297404×10-19m 36,028,797,018,963,968
18 56 7.205759840×1016 1.16459481×10-18m 72,057,594,037,927,936
18 57 1.44115188×1017 2.32918962×10-18m 144,115,188,075,855,872
18 58 2.88230376×10 17 4.65837924×10-18m 288,230,376,151,711,744
18 59 5.76460752×1017 9.31675848×10-18m 576,460,752,303,423,488
19 60 1.15292150×1018 1.86335169×10-17m 1,152,921,504,606,846,976
19 61 2.30584300×1018 3.72670339×10-17m 2,305,843,009,213,693,952
19 62 4.61168601×1018 7.45340678×10-17m 4,611,686,018,427,387,904
20 63 9.22337203×1018 1.49068136×10-16m 9,223,372,036,854,775,808
20 64 1.84467440×1019 2.98136272×10-16m 18,446,744,073,709,551,616
20 65 3.68934881×1019 5.96272544×10-16m 36,893,488,147,419,103,232
21 66 7.37869762×1019 1.19254509×10-15m 73,786,976,294,838,206,464
21 67 1.47573952×1020 2.38509018×10-15m 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 Large-Scale Universe: Notations 134 to 205