By Bruce Camber
Initiated: January 8, 2016 Small Updates: Monday, May 12, 2017 Related: Top Ten
“I have learned that many of the Greeks believe Pythagoras said all things are generated from number. The very assertion poses a difficulty: How can things which do not exist even be conceived to generate? But he did not say that all things come to be from number; rather, in accordance with number – on the grounds that order in the primary sense is in number and it is by participation in order that a first and a second and the rest sequentially are assigned to things which are counted.”
– Theano, On Piety (as reported by Thesleff, Stobaeus, and Heeren)
Using the model of the universe generated through the Big Board – little universe Project where there are 202 doublings from a so-called singularity * of the Planck base units (particularly from Planck Time) to the Age of the Universe, the question to be addressed is, “Which numbers come first and why?” Mathematical logic calls out the most simple-yet-powerful numbers that can be used to build and sustain a highly-integrated universe. Our other assumptions are here. Each of these key numbers and number groups are introduced; each will then become the focus of additional study, further analysis, and the basis for a more-in-depth report about each number. Our initial numbers are:
(1) 3.1415926535897932384626433+ or π or Pi
(2) 74.04804896930610411693134983% or the Kepler Conjecture
(3) 0, 1 where the numbers are: zero and one
(4) 7.356103172453456846229996699812+ degrees, a geometric gap (see old Chrysler logo)
(5) 1:1.618033988749894848204 or the Phi ratio
(6) 4.6692016091029906718532 which is a ratio called the Feigenbaum constant
(7) 110 of Stephen Wolfram’s rules
(8) 6.6260709×10−34 J·s or Planck constant plus all related numbers
(9) Groups of dimensionless constants, all known mathematical and physical constants
(10) 13.82±0.021 billion years, the Age of the Universe
* Please note: The first priority of all links inside the body of the article is to other articles within this website. The second priority is to specific ArXiv articles. And the third priority is to Wikipedia pages. For those occasional links that do not open new tabs or windows, please use the back-arrow key to return to the referring page. All links within the Endnotes will eventually go to the original source materials if posted on the web.
Page 1 of 10 https://81018.com/number/
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Most of us know the universe is infused with numbers. It seems nobody really knows how all these numbers are organized to make things and hold it all together.
In our work with high school students there is a constant demand that our numbers be intellectually accessible. Simplicity is required. So, it is rather surprising that we ended up engaging the Planck Length (and the other Planck base units) very early in our study of the platonic solids. We also started to learn about base-2 notation and combinatorics. We had to do it. We had divided our little tetrahedron and octahedron in half so many times, we knew we were in the range of that limit of a length, and we wanted to find a place to stop. Eventually, to get more accurate, we started with the Planck Length, alongside the base-2 exponential notation of the numbers and a base-8 expansion of vertices, and we multiplied our way out to the Observable Universe.1 It took just over 202 doublings. What?
That fact is as unknown as it is incredible (even as of January 2016 when this article was posted).
In December 2011 we could find no references to the 202 notations in books or on the web. We eventually found Kees Boeke’s 1957 work with 40 (of the 64 jumps) using base-10 notation. It was a step in the right direction, but it had no lower and upper boundary, no Planck numbers, and no geometry. It had just 40 steps amounting to adding zeroes.
We were looking for anything that could justify our “little” continuum. We didn’t know it at the time, and we later learned that we were looking for those deep relations and systems that give us homogeneity and isotropy, a cosmological constant, and an equation of state. Though it seemed that this chart places everything, everywhere throughout all time in an ordered relation, we had no theoria, just the praxis of numbers. We tried to set a course to go in the direction of a theory that might bind it all together.
The first 64+ doublings constitute a range that scholars have been inclined to dismiss over the years as being too small; some say, “…meaninglessly small.” Yet, being naive, it seemed to us that the very simple and very small should be embraced, so we started thinking about the character of the first ten (10) doublings. Trying to understand how to “Keep It Simply Simple,” we were pleasantly surprised to discover that there was so much work actively being pursued by many, many others throughout academia and within many different disciplines to develop the logic of the most simple and the most small.
Within the studies of combinatorics, cellular automaton, cubic close packing, bifurcation theory (with Mitchell Feigenbaum’s constants), the Langlands program, loop quantum gravity, mereotopology, point-free geometry (A.N. Whitehead, Harvard, 1929), the 80-known binary operations, scalar field theory, and string theory (with LQG), we found people working on theories and the construction of the simple. Yet, here the concepts were anything but simple.2
It is from within this struggle to understand how all these numbers relate, we began our rank ordering of all possible numbers. This exercise helps to focus our attention.
Planck Length and Planck Time. One might assume that we would put the Planck base units among the most important numbers to construct the universe. As important as each is, it appears at this time that none of them will be among the Top 5. Although very special, the Planck numbers are determined by even more basic and more important concepts and numbers. At the very least, all those numbers will come first.
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First Principles. The work to find the Top Numbers was preceded by an end-of-year report after four years of studying and using the Big Board-little universe charts. That report titled, Top Ten Reasons to give up little worldviews for a much bigger and more inclusive UniverseView 3, was done with comedian David Letterman in mind. He often had a Top Ten on his show.
“#10” for us it is, “Continuity contains everything, everywhere, for all time, then goes beyond.” One of the key qualities to select our most important numbers is the condition of continuity and discontinuity starting with the simplest logic and simplest parts.
A Quick Review of the Top Ten Numbers in the Universe.
Because many scholars have addressed the question, we did a little survey.
- Pi was somewhere on most everybody’s list among our sample.
- 0 and 1 are obvious keys for most scholars.
- Numbers like e and Phi (and the Fibonacci sequence) are on most of these lists.
- There is this one equation by Euler — ei*Pi + 1 = 0 — where five key numbers are used in what Richard Feynman considered to be the crown jewel of mathematical equations about identity.
- Though infinity is not a number but a concept, it was on many lists.
- The numbers for the speed of light, c, and the gravitational constant, G, are also on many lists. These numbers are keys used by Max Planck for his calculations of the Planck base units back in 1899.
- The Planck Constant is quite rightly on many lists.
- The Feigenbaum constants, Buckingham Pi theorem, the fine-structure constant, and other dimensionless quantities and physical constants were cited less often.
- We added two numbers not cited at all: mathematician Thomas Hales‘ number from his proof of the Kepler Conjecture and what we call the Pentastar 7.38 degree gap.
- Astronomer, Martin Rees, author, Just Six Numbers: The Deep Forces That Shape The Universe (See Section 3, “Well-known subsets,” and click on 3.2, “Martin Rees’ Six Numbers)” or go to our summary page for Lord Martin.
- Cosmologist, physicist, mathematician, John D. Barrow, author, The Constants of Nature: The Numbers that Encode the Deepest Secrets of the Universe. 2003.
- Philosopher/scientist, Roger Penrose, Oxford, The Penrose Number
- Physicist-mathematician, James D. Stein, author, Cosmic Numbers (begin on page vii) and a summary, Popular Mechanics (September 2011).
- Clifford Pickover, Wonders of Numbers summary
- Journalist David Pegg has a Top 25 (May 9, 2013).
- Business analysts Ben Duronio and Walter Hickey have a top 10 (July 8, 2012)
- Geometer, Steve Waterman, and chemistry professor, Michael Anthony Whitehead have just four math constants that they used in the derivation of 142 physical constants. Because of their conclusions, their numbers are being scrutinized.
Doublings, the powers of 2, and base-2 notation. Yes, our work with base-2 notation originated from within a high school. We have no published scholarly articles and there has been no critical review of our emerging model. Nevertheless, we forge ahead with our analysis of numbers and systems.
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Goals. Our singular goal is to try to construct our universe using mathematical logic. We begin with the magic of the sphere as the beginnings of a nexus of transformation between the finite and infinite.
Our #1 number is Pi (π).
#1 = π
Numerical constant, transcendental and irrational all rolled into one
Pi (π) seems to be a good starting point. Dimensionless, non-ending or continuous, it is also non-repeating which is discontinuous (uniquely infinite). This most simple construction in the universe requires just two vertices to make the sphere. How does it work? It appears to give form and structure to everything. Using dimensional analysis and scaling laws, this progression of the first 20 notations shows the depth of possibilities for constructions when multiplying by 8. Our open question: In what ways do the Feigenbaum constants within (bifurcation theory) apply?4
|Domains||B-2 Vertices||Scaling Vertices||Bifurcation*||Ratio*|
|14||16,384||4,398,046,511,104 (trillion:12)||columns come|
|20||1,048,576||1,152,921,504,606,846,976 (quintillion: 18)||constant.|
Discussion. Pi still holds many mysteries waiting to be unlocked. Among all numbers, it is the most used, the most common, and the most simple but complex. We assume, that along with the other mathematical constants, pi (π) is a bridge or gateway to infinity. We assume it is never-repeating and never-ending. It is “diverse continuity.” There are enough scaling vertices within ten doublings to construct virtually anything. So, to analyze a possible logical flow, any and all tools that have something to do with pi (π) will be engaged. Again, among these tools are combinatorics, cellular automaton, cubic close packing, bifurcation theory (with Mitchell Feigenbaum’s constants), the Langlands program, mereotopology and point-free geometry (A.N. Whitehead, Harvard, 1929), the 80-known binary operations, and scalar field theory. Perhaps we may discover additional ways to see how pi gives definition — mathematical and geometric structure — to our first 60-to-67 notations. What are the most-simple initial conditions?
More Questions. What can we learn from a sphere? …by adding one more sphere? When does a tetrahedral-octahedral couplet emerge? When do the tessellations emerge? At the third notation with a potential 512 scaling vertices, surely dodecahedral and icosahedral forms could emerge. Within the first ten notations with over one billion potential vertices, could our focus shift to dynamical systems within the ring of the symmetric functions?
#2 = Kepler’s Conjecture
Not a very popular topic, one might ask, “How could it possibly be your second choice?” Even among the many histories of Kepler‘s voluminous work, his conjecture is not prominent. To solve a practical problem — stack the most cannon balls on the deck of a ship — he calculated that the greatest percentage of the packing density to be about 74.04%. In 1998 Professor Thomas Hales (Carnegie Mellon) proved that conjecture to be true. By stacking cannon balls, all the scholarship that surrounds cubic close packing (ccp) enters the equation. The conjecture (and Hales 1998 proof) opens a huge body of current academic work.5 There we found an animated illustration on the right within Wikipedia that demonstrates how the sphere becomes lines (lattice), triangles, and then a tetrahedron, and then three tetrahedrons surrounding a center triangle which becomes the base of an octahedron (pictures below).
Within that second layer of green spheres the very first tetrahedral-octahedral couplet emerges. This discussion continues: , , , , , , , , , , , and . Link to this paragraph: https://81018.com/number/#Kepler
This image file (right) is licensed under the Creative Commons Share-Alike 2.5 Generic license. The two images just below on the right replicate it with three tetrahedrons around a triangle and then an octahedron laid in on top of that triangle.
Revisions. As we find experts to guide us within those disciplines where pi has a fundamental role, undoubtedly sections of the article will be substantially re-written and expanded. Our goal has been to find the most logical path by which all of space and time becomes tiled and tessellated. Perhaps there is a new science of the extremely small and the interstitial that will begin to emerge. These just might be the foundations of foundations, the hypostatic, the exquisitely small, and the ideal.6 We plan to use all the research from Kepler to today, particularly the current ccp (hcp and fcp) research from within our universities, in hopes that we truly begin to understand the evolution of the most-simple structures.
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#3 = 0, 1
The numbers, zero (0) and one (1) begin the mapping of pi to Cartesian coordinates. Beginning with a circle, each sphere is mapped to two-or-three dimensional Cartesian coordinates. It is the beginning of translating pi to sequences and values. The first iterative mapping is a line, then a triangle, then a tetrahedron, then an octahedron. When we focus solely on this subject, with experts to guide us, perhaps we can engage the study of manifolds that are homeomorphic to the Euclidean space.6
#4 = A Geometric Gap =
The little known 7.356103+ degree gap is our fourth most important number, the possible basis for diversity, creativity, openness, quantum indeterminacy, uniqueness, chaos and, yes, even quantum fluctuations.7 That Aristotle had it wrong gives the number some initial notoriety; however, it is easily observed with five regular tetrahedrons (pictured) there is a gap of about 7.35610° and there are eight (not seven vertices). The number for that gap in degrees appears to be transcendental, non-repeating, and never ending. The potential for the indeterminate resides deep within the system. We believe that place is right here. By the way, the tetrahedron (four vertices) AND the octahedron (six vertices) taken together are whole, ordered, rational, and perfectly tessellate and tile the entire universe.
Within this infinitesimal space may well be the potential for creativity, free will, the unpredictable, and the chaotic. Here may well be the basis for broken symmetries. Of course, for many readers, this will be quite a stretch. That’s okay. For more, we’ll study chaotic maps and the classification of discontinuities.
#5 = phi = φ = The Golden Ratio
= φ = 1:1.618033988749894848204586
Of all the many articles and websites about the golden ratio and sacred geometry, our focus is on its emergence within pi and within the platonic solids. Phi is a perfection. It is a mathematical constant, a bridge to infinity. We are still looking to see if and how phi could unfold within the tetrahedral-octahedral simplex. Could that answer be within Petrie polygons? The magic of the golden ratio does unfold with the dodecahedron, the icosahedron, and the regular pentagon. Within this listing, phi has bounced back and forth with the Pentastar gap. Which manifests first? Is it manifest if it is inherent?
Starting with this article, we have begun an active study of Phi and its relations to pi and the Platonic solids. Although there are many, many papers about phi, none are from our special perspective of 201+ notations.
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#6 = Feigenbaum constants
δ = 4.669 201 609 102 990 671 853 203 821 578
We are the first to admit that we are way beyond our comfort zone, yet to analyze and interpret the processes involved within each of the doublings, each an exponential notation, requires tools. This Feigenbaum constant gives us a limiting ratio from each bifurcation interval to the next…. between every period doubling, of a one-parameter map. We are not yet sure how to apply it, but that is part of our challenge.
It gives us a number. It tells us something about how the universe is ordered. And, given its pi connection, we need to grasp its full dimensions as profoundly as we can. We have a long way to go.
#7 = Rule 110 cellular automaton
There are 255 rules within the study of elementary cellular automaton. Rule 110 was selected because it seems to define a boundary condition between stability and chaos. All 255 rules will be studied in light of the first ten notations to see in what ways each could be applied. Any of these rules could break out and move up or down within this ranking. Steve Wolfram’s legacy work, New Kind of Science (NKS) is online and here he lays the foundations for our continued studies of these most basic processes within our universe.8
#8 = Max Planck numbers
We have been working on our little model since December 2011. Over the years we have engaged a few of the world’s finest scientists and mathematicians to help us discern the deeper meaning of the Planck Base Units, including the Planck Constant. We have studied constants from which the Planck numbers were derived, i.e. the gravitational constant (G), the reduced Planck constant (ħ), the Speed of light in a vacuum (c), the Coulomb constant, (4πε0)−1 (sometimes ke or k) and the Boltzmann constant (kB sometimes k). This engagement continues. We have made a very special study of the Planck Base Units, particularly how these numbers work alongside base-2 exponential notation and the doubling of the Platonic solids. We had started with the Planck Length, then engaged Planck Time. Finally in February 2015, we did the extension of Planck Mass, Charge, and, with a major adjustment to accommodate simple logic, Temperature. We have a long, long way to go within this exploration. Essentially we have just started.9
Mead & Wilczek. Notwithstanding, there is a substantial amount of work that has been done within the academic and scientific communities with all the Planck numbers and those base numbers that were used to create the five Planck base units. Perhaps chemistry professor, C. Alden Mead of the University of Minnesota began the process in 1959 when he first tried publishing a paper using the Planck units with serious scientific intent. Physics professor Frank Wilczek of MIT was the first to write popular articles about the Planck units in 2001 in Physics Today (312, 321, 328). From that year, the number of articles began to increase dramatically and experimental work that make use of these numbers has increased as a result. https://81018.com/number/#7
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We will also bring in Lord Martin Rees “Six Numbers” as well as the current work within the Langlands programs, 80 categories of binary operations, scalar field theory, and more (such as the third law of thermodynamics and zero degrees Kelvin).
In studying the functionality of these many numbers, especially those among the dimensionless constants, we believe this list will evolve and its ordering will change often. In searching the web for more information about about dimensionless constants, we came upon the curious work of Steve Waterman and an emeritus chemistry professor at McGill University in Montreal, Michael Anthony (Tony) Whitehead. I showed their work to a former NIST specialist and now emeritus mathematics professor at Brown University, Philip Davis. He said, “There are always people who wish to sum up or create the world using a few principles. But it turns out that the world is more complicated. At least that’s my opinion. P.J.Davis” Of course, he is right; Einstein did a good job with e=mc2. Because claiming to find all the physical constants derived by using pi, the isoperimetric quotient, close cubic packing and number density is not trivial10, we’ll be taking a second look. Perhaps they are onto something! We have brought their work out in the open to be re-examined and in so doing we will re-examine over 140 physical and mathematical constants. This work is also ongoing.
#10 = The Age of the Universe
13.799±0.021 billion years
This number is important because it creates a boundary condition that is generally recognized for its accuracy throughout the scientific and academic communities. Though it may seem like an impossibly large number of years, it becomes quite approachable alongside base-2 exponential notation. Without it, there is no necessary order of the notations.
Although there are many different measurements of the age of the universe, for our discussions we will use 13.799±0.021 billion years. The highest estimate based on current research is around 13.82±0.021 years. Also, within this study there are some simple logic problems. In 2013, astrophysicists estimated the age of the oldest known star to be 14.46±0.8 billion years. That is peculiar given most estimates of the age of the universe are between 13.81 to 14.1 billion years.
Notwithstanding, all these measurements come within the 201st notation. At the 143rd notation, time is just over one second. Within the next 57 doublings, we are out to the Age of the Universe. So, with the Planck Time as a starting point and the Age of the Universe (and our current time) as the upper boundary, we have a container within which to look for every possible kind of doubling, branching and bifurcation. We can study hierarchies of every kind, every set, group or system. Eventually we can engage holomorphic functions within our larger, ordered context, i.e. the seen-and-unseen universe.11 https://81018.com/number/#8
Page 9 of 10 EARLY DRAFT.
1 Our Initial exploration of the types of continuity and discontinuity: Continuous-discrete, continuous-quantized, continuous-discontinuous, continuous-derivative… there are many faces of the relations between (1) that which has a simple perfection defined in the most general terms as continuity yet may best understood as the basis of order and (2) that which is discrete, quantized, imperfect, chaotic, disordered or other than continuous. These are the key relations that open the gateways between the finite and infinite.
Questions: What is a continuum? What is a discrete continuum?
2 We are simple, often naive, mathematicians. We have backed into a rather unique model of the universe. To proceed further we will need to understand much more deeply a diverse array of relatively new concepts to us; we are up for the challenge. We have introduced just a few of those many concepts that attempts to define the very-very small and/or the transformations between the determinant and the indeterminant. There will be more!
3 Of the Top Ten Reasons, the first three given are our first principles. We know it is an unusual view of life and our universe. The sixth reason advocates for a Quiet Expansion of our universe whereby all notations are as active right now as they were in the very earliest moments of the universe. When space and time become derivative, our focus radically changes. It opens a possible place for the Mind down within the small-scale universe. Our current guess is between the 50th and 60th notations. The archetypes of the constituents of our beingness are between notations 67 (fermions) to notation 101 (hair) to notation 116 (the size of a normal adult). Then, we live and have our sensibility within notation 201, the current time, today, the Now. So, this unusual view of the universe has each of us actively involved within all three sections of the universe: small scale, human scale, and large scale. To say that it challenges the imagination is a bit of understatement.
4 Open Questions. There are many open questions throughout this document. It is in process and will surely be for the remainder of my life. All documents associated with this project may be updated at anytime. There should always be the initial date the document was made public and the most recent date it was significantly updated. Although the Feigenbaum constants are our seventh number selected (and there are more links and a little analysis there), we will attempt to find experts who can guide us in the best possible use of these two constants within our studies. Bifurcation, it seems, has an analogous construct to cellular division, to chemical-and-particle bonding, to cellular automaton (especially Rule 110,) and to the 80 categories of binary operations.
5 Wikipedia, ccp, and genius. Jimmy Wales is the founder and CEO of Wikipedia. His goal is to make the world’s knowledge accessible to the world’s people. He has a noble vision within precarious times. In order to be published within Wikipedia, the material has to have its primary sources of information from peer-reviewed publications. As a result, Wikipedia is not where “breakthrough” ideas will first be presented. Blogging areas like WordPress are a more natural spot and Google quickly indexes all those blogging areas. It took only a day before they found this article. So with a little ingenuity one can quickly find many new references to new ideas and then go to Wikipedia to find the experts on that subject. Prior to this research, we had barely scratched the surface of ccp. We did not know about the Feigenbaum constants or Kepler’s conjecture. For sure, we had never seen the cannonball stacking illustration that helped us to visualize the process by which a sphere becomes a lattice, becomes a triangle, and then becomes a tetrahedron. We are quite confident that our first four numbers are the right selections possibly even within the right order. If you believe otherwise, of course, we would love to hear from you.
Page 10 of 10 IN PROCESS
6 A hypostatic science. Our small-scale universe, defined as the first 1/3 of the total notations, ranges from notation 1 to just over 67. It is established only through simple logic and simple mathematics. Because it cannot be measured with standard measuring tools or processes, validating its reality requires a different approach. Our first indication that it may be a reality is found between notations 143 and 144 at exactly one second where the speed of light “can be made” to correspond with the experimental measurement of the distance light travels in a second. Currently it appears to be one notation off which could be as brief as just one Planck Time unit.
One of our next tasks is to carry that out to a maximum number of decimal places for Planck Time and Planck Length, and then to study the correspondence to a Planck second, a Planck hour, a Planck Day-Week-Month, a Planck Light Year, and finally to the Age of the Universe and the Observable Universe.
Our goal is to determine if this is the foundational domain for the human scale and large-scale universe. We are calling this study a hypostatic science because it is a study of the foundations of foundations.
7 From SUSY to Symmetry Breaking and Everything In Between. One of the great hopes of the Standard Model and many of the CERN physicists is that supersymmetries will be affirmed and multiverses will wait. Within the Big Board-little universe model, their wish comes true. Plus, they gain a reason for quantum indeterminacy and embark on a challenge to apply all their hard-earned data acquired to embrace the Standard Model to the most-simple, base-2 model.
• The Lagarias-Zong article (a PDF), “Mysteries in Packing Regular Tetrahedra (PDF)” is where I learned about Aristotle’s mistake.
• Frank, F. C.; Kasper, J. S. (1958), “Complex alloy structures regarded as sphere packings (PDF). I. Definitions and basic principles” (PDF), Acta Crystall. 11. and Frank, F. C.; Kasper, J. S. (1959), and “Complex alloy structures regarded as sphere packings. II. Analysis and classification of representative structures”, Acta Crystall. 12.
• “A model metal potential exhibiting polytetrahedral clusters” by Jonathan P. K. Doye, University Chemical Laboratory, Lensfield Road, Cambridge CB2 1EW, United Kingdom, J. Chem. Phys. 119, 1136 (2003) Compete article, ArXiv.org as a PDF: http://arxiv.org/pdf/cond-mat/0301374
• “Polyclusters” by the India Institute of Science in Bangalore, illustrations and explanations of crystal structure.
My recommendation is to start with “Mysteries in Packing Regular Tetrahedra” by Jeffrey C. Lagarias and Chuanming Zong.
Also, let’s begin to grasp the systems-networks-topology of the Barabási–Albert model.
8 Cellular Automaton. Although the discipline is intimately part of computer science, its logic and functions are entirely analogous to mathematical logic, functions, and binary operations. We have just started our studies here with great expectations that some of this work uniquely applies to the first ten notations.
9 The Planck Platform. All the numbers associated with the generation of the Planck Constant and the five Planck base units, plus the Planck units unto themselves are grouped together until we can begin to discern reasons to separate any one number to a notation other than notation 1.
10 The Magic Numbers. Mathematical constants, dimensionless constants and physical constants are studied in relation to the isoperimetric quotient, close-cubic packing, number density and to bifurcation theory and to the 80 categories of binary operations. We will working with the processes developed by geometer, Steve Waterman, and chemistry professor, Michael Anthony Whitehead and the generation of the 142 physical constants.
11 The first 67 notations. Given the work of CERN and our orbiting telescopes, we can see and define most everything within notations 67 to just over 201. The truly unseen-unseen universe, defined only by mathematics and simple logic, are: (1) the dimensionless constants, (2) that which we define as infinite, and (3) the first 60-to-67 notations. It is here we believe isotropy and homogeneity are defined and have their being. It is here we find the explanation for the most basic cosmological constant. It is here the Human Mind takes its place on this grid which claims to include “everything-everywhere-for-all-time.”
Please note: Work on this article. A running commentary is being developed within the LinkedIn blogging area for Bruce Camber. Besides editing the overall document, we’ll are still working on the end notes using some of these reference materials.
fHere are four of our key