Are the first 67 notations key missing links?

Could an integrated Universe View1 open new approaches to old academic questions? Because it redefines our understanding of space and time, it seems to open that possibility.

Sir Isaac Newton’s space and time became an absolute frame of reference2 and the basis of commonsense logic for well over 300 years. But, what if space and time are derivative, discrete and quantized? What if both have a specific beginning, the Planck Base Units, and today’s current time-length-mass-charge-temperature of the universe, or the Now, are always the endpoints? What if every notation is a highly-integrated part of the whole?

There are just over 200 base-2 exponential notations from the Planck Base Units to this moment, your “today.”  That range is best defined as the Age of the Universe.

Within our Universe View, also known as Universe Table and Big Board-little universe, the first 67 doublings cannot be measured at the singularity of the Planck base units. Planck time cannot even be measured at the 67th doubling. Though driven by a simple logic, simple mathematics, and very simple geometries, those first 67 doublings per se are not part of our observed universe and are currently not part of academic analyses.3

Given this situation, we advocate that the nature of space and time be reopened for scrutiny in light of these first 67 steps. Could these steps be the missing links that open real pathways to new insights into historic, impossibly-difficult and persistent, unanswered, academic questions that exist across all the disciplines? To open such a discussion, brief summaries and a few links will be provided within some of those large categories of human endeavor.  Eventually in-depth articles for each will be written and linked from here:

The longstanding goal of this article is to open the discussion to explore how these statements could be true and in what ways they are not true.

We invite you to join us to determine if and how these 67 notations just might be the infrastructure within which age-old questions can be addressed in new ways.


Endnotes, Footnotes, and References:
1. We have begun building a bridge to the 68th notation, the place where things begin to be measured in space and time. We will be using all the current research at research labs such as CERN, and speculative work such as the Langlands programs and amplituhedrons. Pi, all the dimensionless constants, the physical constants, pure geometries, and numbers and ratios will all be used in new ways.

This article is our appeal for help.

2. Sir Isaac Newton’s space and time became an absolute frame of reference

3. Within those first 67 notations are a different set of building blocks for the universe. If we bring to bear all of the geometries (projective, analytic, combinatorial..), including the simplest geometries (cellular automaton and the platonic solids), a little bifurcation theory and the 80-known binary operations, perhaps we can begin to intuit how forms become structure, then substance, then qualities, relations and systems, increasingly-and-exquisitely complex with literally trillions-upon-trillions of possible combinations and expressions, right on up to the fermions, bosons, skyrmions and all combinations of elementary particles that make up the Standard Model (SM) and the SMPP for Standard Model for Particle Physics.

Much more to come: Entropic forces and causal determinacy

On Developing A Rationale For A Working Model Of the Universe Based On A Quiet Expansion

Please note:  Reviewed,  January 14, 2016 at 1:01 AM GMT.

Abstract: Key embryonic insights from within high school geometry classes back in December 2011 are postulated as a most simple, logical, hypostatic structure of the universe, a base-2 progression from the Planck Length  to the Observable Universe. In September 2014 the naive question was asked, “Could the universe be based on a quiet expansion not a Big Bang?”  When the numbers from Planck Time to the Age of the Universe were added in December 2014, this concept of a quiet expansion seemed less speculative. In February 2015 the remaining Planck base units were added to the chart.  By August 2015 extensive questions were raised primary among them, “Can this chart be integrated with the Lambda-CDM model (the parameterized cosmological model)?” The immediate task of this paper is to explore those questions in light of the numbers at each doubling to help to discern reasons why this simple model warrants the attention of the academic community.

Simple facts, figures, and logic in search of a theory.  The chart with  five columns for each of the base Planck units and 204 notations down, 1020 boxes, provides real data to examine the logic flows both across and up-and-down.  The goal of this article is to examine no less than 10 boxes. Looking across would be 50 boxes; adding two or three boxes up and down would total over 150 boxes.  Ten percent should be enough to learn if a simple logic actually flows through the numbers.  If it does… well, perhaps we have a new science in the making.

201+ Notations Begin With The Planck Length.  The simple mathematical progression that rendered the 201+ base-2 exponential notations was the result of following embedded geometries going smaller and smaller until in the range of the Planck Length. Going in the other direction, larger and larger, was achieved by multiplying by 2 until in the range of the Observable Universe. The total, just over 201 doublings, could not be found within the writings of the academic community. The base-10 work done back in the 1957 by Kees Boeke and his high school classes in Holland was abundantly indexed; there were no references to a base-2 progression from the Planck Length to the Observable Universe, especially as a result of following embedded geometries within the tetrahedron and the primary octahedron within that tetrahedron.

Planck Time to the Age of the Universe is applied.  There is general scientific concurrence regarding the estimates of the age of the universe. That figure provides a better framework for the doublings of Planck Time,  from the beginning of time to this moment, right now, our current time, which always defines the endpoint.  Planck Time and Planck Length track together in informative ways. For example, the notation that defines one second is between 142nd notation (.6011 seconds) and the 143rd (1.2023 seconds).  The doublings of the Planck Length are 180,212.316 kilometers at Notation 142 and 360,424.632 kilometers at Notation 143. As one might have expected, the speed of light is confirmed in between the two at 299,792,458 meters for one second.  At this point in time the other three Planck base units have become quite large, larger than any common number within human experience.

This Quiet Expansion begins at the first doubling. Quite literally, there is no room for sound until out to the 108th doubling (the beginning of sound waves) and on out to the 119th (the full spectrum of sound ranges from Notation 108 to Notation 119).  There is something quite helpful within a visceral sense of the number and parameter.  Examining groups of numbers associated with a common human experience is more than helpful; it provides the infrastructure of logic.  Yet, there is no point where simple logic flows across all five Planck units. Yet, as demonstrated, it is quite informative when even two such numbers correspond.

For example, one of the very smallest notations with an experiential human equation is Notation 93 where observable light begins to manifest. Notation 101 is within the range of the thickness of human hair. This, of course, is where a large group within Planck Length and Planck Time correspond. This is the human scale universe. And, within that group there is one place where length and temperature correspond.

Planck Length and Planck Temperature. First, it was a leap of faith to hold to our working premise, “Everything starts simply” and to place the extremely hot Planck Temperature at the top of the chart.  That put a very common number between Notation 103 and 104 where the temperature has cooled to 98.6 degrees Fahrenheit.  Here we find among many other common things, the human egg cell. At Notation 105 the temperature has risen to 894 Kelvin or a very hot 1149.53° Fahrenheit and at Notation 102 it has dropped to a very cool –58° Fahrenehit.

Planck Mass. The very smallest notation with a common figure is the 31st doubling (Notation 31) where we find 103 pounds (46.74 kilograms). For many people, it is a key weight threshold signifying our coming of age, quickly approaching being  an adult. Within this doubling the other four figures are so small, it causes one to ponder. So much seems to be happening with each of these doubling, that 103 pounds encourages some speculation. How about this? Perhaps the 103 pounds is the sum total weight of this notation! At the top end of this column are the outrageously large numbers that come very close to estimates by some of the more speculative within the scientific community, especially if each number in this column is the sum total weight of that notation.  In some peculiar ways, this just may be a measurable concept.

The Human Scale Universe. Within the human and large scale universe, there are many familiar things within the Planck Length notations, yet the other Planck figures remain largely remote.

Planck Time. Although we cannot meaningfully perceive much smaller than a tenth of a second (Notation 140), in 2010 machines at the Max Born Institute in Berlin measured down to 100 attoseconds (Notation 87).  Perhaps each notation with the Planck Time column describes a range in which relations are defined. Some elements of that statement may be measurable.

What Is Is? If looked to discern any special logic, one’s conclusion might be that each notation, with its vast array of vertices and multiples of the Planck base units, define the terms and conditions by which that notation-qua-notation is.  That is, these numbers define the “isness” of the notation.

So, let us look in depth at one second between Notation 143 and 144. The total mass ranges from 2.4268×1034 kilograms to 4.8537×1034 kg. It defines a range, “no greater than twice that amount, and not less than half that amount.” In a similar manner, the total energy has a range, 2.0913×1025 coulombs but not greater than twice this amount and no less than half that amount. The total of heat within the notation, a huge stretch of the imagination, is 2.4578×1014 K to 4.9156×1014 K. Though an unimaginable amount of heat to be spread out throughout this single Notation 142, it just may be a measurable concept.

Planck Charge.  Let’s look at which notations Planck Charge becomes a common number. For example, a lightning bolt is typically around 15 C, large bolts up to 350 C.  That is quite visceral, yet on the chart it is in the range of Notations 63 to 67, the run up to the transfer from the small-scale to the human scale universe.  If it represents the sum total charges within each notation, it certainly provides us with something to ponder.

These five Planck base units create very large continuity equations. Though imputed, remember that this schema is also based on the simplest geometries. Taking the entire chart and the weight of its simple logic, it suggests that the symmetries of these imputed geometries and these continuity functions are infinite, and that length (space), time, mass, charge and temperature are finite. These 201+ notations seem to define a finite universe and each notation defines a range in which particular subjects and objects are bounded by their Planck base units doubling, thereby each notation has a certain functional uniformity which provides a range within which particular groups or sets of things work.

Questions are asked, “Is this model the abiding, on-going, current structure of things as they are?  How?

201+ notations, divided by three, renders a small-scale, human scale, and large scale universe. The application of scaling laws and dimensional analysis to the first 60 notations resulted in learning about the power of base-8 expansion. By the 20th notation there are plenty of vertices with which to build structures; that is 1,152,921,504,606,846,976 or 1.152 quintillion vertices. By the 60th notation, add 36 more places (zeros). That is a robust infrastructure with 1152921504606846976000000000000000000000000000000000000 vertices (perhaps point-free vertices).

There is what would appear to be an infinite number of possible constructions. Add in the 131 better-known dimensionless constants and the fundamental physical constants, there should be enough variables to accommodate the Standard Model in physics as well as the science that has resulted from the standard model in cosmology. Please note that at the 60th notation, the size of the Planck Length doubling is not yet large enough to accommodate a fermion.  From the 1st doubling to at least the 60th doubling, all the “structure” may best be described as hypostatic, which means in this instance, the essence or underlying reality.

Humanity doesn’t physically appear within the Planck Time column until well into the 201st notation. There has been a dispersion of length (space) mass, charge and temperature throughout an ever-expanding universe.  Obviously there is a lot of science to learn between Notation 101 to Notation 202, and it will all be in relation to the deeper dynamics between Notation 1 and 101.

Reflections and Projections.  Our base-2 chart of the Planck Base Units was first published in February 2015. This is its first review. It is an introduction that requires many more years of work and analysis.  It frames a detective story whose final chapter could be written in many different ways.  To expand the grounds of the analysis will require going deeply inside the simple geometries within the first 60 notations to discern how these geometries extend undetected, but measurably present throughout the entire universe. The assumed universals — order and continuity,  multiple grids of relations with symmetries as well as asymmetries, and dynamics that seem to conjure up transformative instants of harmony, degrees of perfection and  the darkest forms of chaos within degrees of imperfection  —  will be studied in light of duality, finite and infinite sets, group theory,  and set theory.  That study will focus on the correlations with advanced combinatorics, matroids, amplituhedrons, and the Buckingham pi theorem.

All the questions raised within A Simple View of The Universe will now begin to be addressed.

Much more editing and perhaps a little more writing to come.

Working notes:  When this page is ready to be declared “a working first-draft,” I will post an index of related articles; and as a working first draft, this post will be the first in that list. -BEC

Editorial note:  Our world seems increasingly crazy. This model just might help to open new insights that might mitigate some forms of that craziness. So though still quite rough, it’s being brought into the light of the public rather early. Also, by working on it in public, perhaps others will have comments and suggestions to shape its potential.

This post is a continuation of a prior work, A Simple View of the Universe. There are more observations to make about the Planck Time progression and many more to make about the progressions of the other Planck base units.  So, to say the least, this document is very much in process and will be updated frequently throughout the day and throughout the month of September.

A More Simple View Of The Universe

Waterman“Behind it all is surely an idea so simple, so beautiful, that when we grasp it — in a decade, a century, or a millennium – we will all say to each other, how could it have been otherwise?” by John Archibald Wheeler, 1911-2008, physicist, How Come the Quantum? from New Techniques and Ideas in Quantum Measurement Theory, Annals of the New York Academy of Sciences, Vol. 480, Dec. 1986 (p. 304, 304–316), DOI: 10.1111/j.1749-6632.1986.tb12434.x

Is a simple mathematical and geometrical view of the Universe meaningful or useful?

Can we open a dialogue about the question?


Note: This article was initiated in July 2015 and it is now a Working Document. I fully acknowledge that the basic concept is idiosyncratic. There are profound challenges in many places, i.e. Planck Temperature and the order of dimensionless constants; and there are more questions than answers. Although I do not want to waste your time, the reason for working in public is to get your insights, suggestions and comments. Links, footnotes, and endnotes are rough. This article builds upon other work; two earlier articles and two sequels:

Thank you. – BEC


In December 2011 we began our work on a very simple mathematical and geometric model of the universe; it was playfully dubbed, Big Board-little universe.

We had started using the following parameters — base-2 exponential notation, the Planck base units, and the Platonic solids — in ways that created heretofore unobserved boundary conditions.

Our Three Initial Conditions

1. A basic chart. There are just 201+ base-2 exponential notations from the base Planck units of Length and Time to the Observable Universe and Age of the Universe respectively. In our chart these two base Planck units tracked together in informative ways and raised many questions. Here the operative function was multiplication by 2 while the two base Planck units were the known properties being multiplied. Notations took on a diversity of names depending on the functional qualities we were observing. A notation could be a cluster, domain, doubling, group, layer, set and/or step. The known universe was defined from about the 65th notation to the 201st-to-202nd doublings. A largely-undefined, very small-scale part of the universe was given a simple geometric and mathematical structure from the 1st to 65th doubling.

Is it significant? The mathematical progressions within the charts are simple, but raise questions. One key question is addressed within the Endnotes (below).

2. Geometries. We imputed a pervasive, simple geometry throughout the universe. This project started within our high school geometry classes by going inside the simple tetrahedron by dividing the edges by 2 and by connecting those six new vertices. We could see four half-sized tetrahedrons in each corner and an octahedron perfectly in the middle. We then went inside the octahedron; there we found six half-sized octahedrons in each corner and a tetrahedron within each face. Our geometry classes were exploring the question, “How far within could we go by continuously dividing by 2 each tetrahedral-octahedral layer?” Then we asked, “How far out can we go by continuously doubling what we had?” With just these two Platonic solids, we could tile and tessellate each layer and between layers or doublings throughout the entire model. We learned about the limits in both directions and we have begun learning about this progression called base-2 exponential notation.

Our initial structures were all three-dimensional. When we found many two-dimensional plates across all the notations, coherence throughout the universe seemed possible.

The cross-notational plates were quickly recognized within nature. The one with just hexagons was an easy analogue of graphene. Within manifold geometries, the analogue would be to fullerenes.

Although there is no evidence that these analogical constructions exist within every layer, we imputed, hypostatized, or hypothesized that in some manner of speaking, such analogues do exist, especially within the first 60 doublings. We could then ask the question, “Given this ubiquitous, four-dimensional web (continuum, matrix, grid), why does the universe work in the manner that it does?” In looking for answers, we have begun to see a means to attract, relate, bind, break or repel constructions within each, and between each, of the 201+ layers.

3. Logic. Our current chart redefines the continuity function to start with the infinitesimally small measurements, the base Planck units, and go out to their largest possible measurements using the Observable Universe and the Age of the Universe as the primary outer limits. Though imputed, this continuity function became our first principle for order in the universe yet it took a period of contemplation of the Big Board-little universe charts and images to begin to see the universe as a natural container for space and time.

As a container with a definitive beginning and current limits, the weight of logic seems to favor the conclusion that the universe is finite. That quickly raises questions about the infinite, such as, “If it is not defined by space and time, how is it defined?”

Within the tilings and tessellations of our pervasive-but-simple geometries and with our base-2 expansion from the base Planck units, we began finding an extraordinary diversity of possible symmetries and potential relations. We asked, “Could symmetry-making and symmetry-breaking through time be the basis for all dynamics? Could the illusive harmony be a perfection of those symmetries within a moment in time?” Unto itself, this logic seemed to become its own system of value and for valuations.” Perhaps the very nature of space and time is derivative; and order, relations, and dynamics and their three functional qualities — continuity, symmetry and harmony — somehow constitute the infinite and are infinite.”

This simple logic became an important building block to postulate our first principles. Our charts had become a model of the known and a largely-unknown, infinitesimally-small universe.

Who? What? Why? When? Where? How?

4. History. This highly-integrated view of the universe must now be tested within the history of logic, mathematics, philosophy and physics. If this embryonic model is to have a place within the work of scholars, it must be critically analyzed. And, we know it has a long way to go before it earns such a place within scholarship. It must address very basic related questions about duality, finite and infinite sets, group theory, set theory, then advanced mathematical concepts that seem to be necessarily related like advanced combinatorics, matroids, amplituhedrons, and the Buckingham pi theorem. Like breadcrumbs, these topics will be followed up in the near future.

We are still within a very young and naive stage in our development and there are many very-very basic questions to explore:

  • Who are the players — the scientists and mathematicians — who are experts within this small-scale domain?
  • What are the “somethings” that are doubling within each notation?
  • Why have these first 65-or-so notations been declared irrelevant by academics? Why haven’t the philosophers and brain-mind scholars explored the possibility that this continuum is the domain of the mind and values?
  • When does simple logic and simplicity itself override experimental data?
  • Where are the indicators that there is a domain that gives rise to gluons, hadrons, and the rest of the particle zoo?
  • How do the doublings of space and time work to become the container within which those “somethings” begin to expand? Could those somethings best be defined by causal set theory, pi, the dimensionless constants, symmetry making, and perfected states?
  • Does the Michaelson-Morley experiment provide insights from their historic quest to define the aether?
  • Does this small-scale domain have anything to do with the continuum (Cyclic Conformal Cosmology) that was proposed by Roger Penrose of Oxford?
  • Is it the matrix or grid that Frank Wilczek (MIT) delineates? Why? How?
  • Could this small-scale universe be all of the above?
  • Thinking about CERN and their current research from quarks to gluons, how does this small-scale universe work in such a manner to give rise to the impeccable successes of the Standard Model (including confirmation of tetraquarks and pentaquarks) as well as the standard model in cosmology (Lamda CDM)?
  • Might this small-scale domain be the basis for homogeneity and isotropy in the universe? How do dimensional analysis and dimensional homogeneity apply?
  • Might this domain be the basis of fundamental interactions giving rise to dark matter and dark energy?

These are some of the subjects (or objects) that occupy our attention and focus our time. “Let’s go over the details just one more time to attempt to learn how this model provides new footings and foundations that could give rise to some of our current perceptions and accepted models and theories.”


5. Starting point or domain or … The key question is, “What is being measured by the doubling of each Planck base unit?” Something is being doubled within each notation of those five columns and 201+ notations. First, we assume that Planck’s base units are the singularity (the Void), yet, we now ask, “What happens when each is doubled? What is manifest that doubles?” …only natural units? These are always based solely on universal dimensionless physical constants. But, all of them? Some of them? If so, which come into play and when do they come into play and why do they come into play? There are many books and articles about these constants, however, our primary reference is the 2006 article by Tegmark, Aquirre, Rees, Wilczek (TARW), “Dimensionless constants, cosmology and other dark matters” where they identify 31 dimensionless physical constants (PDF). The Planck Length (space) and Planck Time are two of their 31.

Once we have begun to understand the TARW conceptual frame of reference, we will attempt to take on the other 104 dimensionless constants defined within Wikipedia.

Our short-term work is to begin to understand the published works of an expert with each of these constants. Perhaps we will begin to see how our two base units create a nondimensionalized plenum and vinculum so an “archetype” of mass(kg) and electric charge (q) begin to manifest and we begin to discern how the parameterizing functions of the Planck constant (h) including the speed of light in vacuum (c), the gravitational constant (G), the electric constant (ε0) and the elementary charge (e) as each comes into play. We assume somewhere along our progression of doublings, the fine-structure constant (α) will present itself as will all the other dimensionless constants.

“What is manifest?” First, we have the actual calculations by Max Planck for length, time, mass and electric charge. Questions abound. “How do these manifest? Though infinitesimal, is there a manifestation of something?”

Our first assumption is that the “somethings” could be either simple vertices or what are known as point-free vertices. Part of our on-going study, we are told by Freeman Dyson that we should be using dimensional analysis and scaling laws to count the vertices within base-2 exponential notation; thus, we should be multiplying the number of vertices by 8 with each doubling. If so, there could be eight vertices within the first or second doubling.

With the second doubling we have the simple calculations — multiplying by 2 — of base Planck units of length, time, mass and electric charge. Then we have the scaling number or 64 vertices. To observe this progression, we will eventually make a chart for our base units to the 65th notation.

The first twenty doublings open our analysis. The first eight vertices constitute the first chapter of the story. Theoretically or conceptually, here is the first abiding step to construct and sustain our little universe. Here we will start our analysis with the tools of causal set theory, cubic close packing, Pi, the dimensionless constants, and a perfected state with continuity, symmetry, and an infinitesimally short moment of harmony.

Then the story becomes increasingly complex with each doubling.

Notations: Doubling: Scaling Vertices* (units)(zeroes):
0 0 0
1 2 8
2 4 64
3 8 512
4 16 4096 (thousand) (3)
5 32 32,768
6 64 262,144
7 138 2,097,152 (million) (6)
8 256 16,777,216
9 512 134,217,728
10 1024 1,073,741,824 (billion) (9)
11 2048 8,589,934,592
12 4096 68,719,476,736
13 8192 549,755,813,888
14 16,384 4,398,046,511,104 (trillion) (12)
15 32,768 35,184,372,088,832
16 65,536 281,474,976,710,656
17 131,072 2,251,799,813,685,248 (quadrillion) (15)
18 262,144 18,014,398,509,481,984
19 524,288 144,115,188,075,855,872
20 1,048,576 1,152,921,504,606,846,976 (quintillion) (18)
*Vertices or point-free vertices

With every one of the TARW 31 dimensionless constants, a guess will be made to see what happens to the number within each doubling. We will watch the simple logic of each doubling, especially between the 65th and the 70th doublings. When does that number punch out and become something that is reduced to practice? Or, in what notation does a dimensionless constant combine with anything that is manifest? When is there an apparent effect?

By the 20th notation, our vertex figure using dimensional analysis is up to an exabyte, the same number as 2-to-the-65th or 1.1529 quintillion vertices. We can see therefore that count continues out to 54 places (18 x 3) by the 60th notation. These numbers are so far beyond “large numbers” that it may seem meaningless. Certainly we all need to begin getting accustomed to very large and very small numbers! It seems that we could conclude that with so many vertices there is enough potential structure to contain every part of the Standard Model known to date.

Anything and everything seems possible.

6. Identity: Humanity at the center of this model of the universe. In December 2014, when we tracked the Planck Time next to the Planck Length, we found 201.264+ notations. Our very first chart in December of 2011 had 209 notations. We did not know where to stop. A NASA scientist helped us; he calculated 202.34 notations. Then a prominent French astrophysicist who did a calculation of 205 notations (See footnote 5).

From the 100th to 103rd notations we find sperm, hair, the thickness of today’s paper from a book or magazine, and the human egg, clearly a few of the basics that evolve to become humanity. And, of course, we recognize that there are many other objects within these four notations. Yet, within its simplicity, there was a quiet affirmation, “Perhaps we, the swarming sea of humanity, are not irrelevant. This model places us squarely in the middle of it all.”

7. The small-scale, human-scale, and large-scale Universe. In our chart of the Big Board – little universe there are 201.264+ notations. When divided by three, each scale would ideally have just over 67 notations. Following a longstanding convention within scholarship, we call these groups, the small-scale universe, the human-scale universe and the large-scale universe.

The small-scale universe ranges from the singularity of the Planck base units to notations 67 and 68. Within the 66th and 67th notations, protons, fermions and neutrons are indexed. Leptons, quarks may well be within the 64th and 65th domain. Some posit them at much smaller sizes. But, the measuring tape is mathematics and it is oblique mathematics to be sure. Common elements of the aluminum and helium atoms show up in the 68th notation.

This human-scale universe ranges from the 68th notation to the 135th notation. There have been times when we have been boldly speculative, perhaps just imaginative, thinking about the transition from the human scale to the large scale.

The large-scale universe ranges from the 135th notation to just over the 201st notation. Not just the domain for governments anymore, here the truly imaginative, speculative, and bold have gone where others would fear and tremble.

These three scales provide the second most-simple division of the universe and by studying the transitions between each, we will engage combinatorial mathematics, group theory and set theory in fundamentally new ways. The continuity conditions are redefined. Symmetry functions are expanded. And, there is a possibility of understanding something new about the harmony of the universe (see history of the Greats who used such terms, i.e. Pythagoras, Plato, Aristotle, Kepler, Newton and Leibniz).

We have begun to analyze other progressions or scales based on fourths, fifths sixths, and so on. In time, we may find something of interest.

8. Numbers and Operands (from Sequential Real Numbers, to Base-2 to Dimensional Analysis). We have observed how the simple mathematics of both base-2 exponential notation and dimensional analysis become unwieldy rather quickly by the 60th and 21st notations respectively. Virtually every day we say, “We need to go over this one more time. It seems that we are missing something.”

First, the notations (doublings or steps) are sequentially ordered, 1 to just over 201. What is that sequence? Is there any possibility that it could be related to the Fibonacci sequence? What is the very nature of addition?

Next, there is multiplication, division, and ratios. A former NIST scientist and mathematics professor at Brown, Philip Davis, cautioned that the circle and sphere are more simple than the tetrahedron. Of course, he is right. We are now learning more about cubic-close packing (ccp) and the world of pi. Within the first notation with its eight vertices, we now know that we have to understand ccp and anticipate that the entire small-scale universe is driven by ccp.

That will be an article in the near future.

At the top of this article is a quote from John Archibald Wheeler who was thinking about the standards for measurement within quantum mechanics. If Pi drives this small scale universe, we know Pi is an irrational number and transcendental number that never ends and never repeats. It gives each construction those qualities and those qualities reflect an essence of quantum mechanics; we know there is a lot to chase down here.

Also, one of the most simple ccp configurations will be the pentastar with seven vertices in the form of five tetrahedrons. There is a 7.38° (7° 21′) gap that we have called squishy geometry as well as quantum geometry; here are degrees of freedom that continue within the icosahedron (20 tetrahedral structure) and the pentagonal dodecahedron (60 tetrahedral structure). What is it all about? We are not sure, but we do know it is worth more study.

There are many notations as those Planck base units are being multiplied by 2, that raise questions. We say, “There are doctoral dissertations in there!” It is within our scope of work. Then it came time to ask, “What has over a quintillion units of something?” Today, we have answered, “Vertices or point-free vertices.” Are there any other possibilities?

What are the key operands? It seems that a vertex is a reasonable answer. It is a special kind of point defined by axioms, and these have no “…length, area, volume, or any other dimensional attributes.” Yet, within our logic these points give functional capabilities to continuity, symmetry and harmony. And, these points have within them the conditions for order, relations and dynamics.

We take the universe as a whole, just as it is given; however, we assume that it is all complete, integrated, where the historic is the current, the here and now.

Thank you.



  • At some notation, the geometries, logic, and all the somethings of the universe, must begin sharing a common space and time and as we approach the first doubling, everything shares it. We assume this shared space begins somewhere between the 60th and 67th notation. We call this domain, hypostatic, because it provides a working foundation for everything everywhere for all time. We have also referred to it as a substrate.
  • Of course, these observations, guesses, and working conclusions will be revisited often.
  • The model also works as a simple Science, Technology, Engineering, Mathematics (STEM) tool; it organizes data in a robust way and it opens many new doors for exploration. That seems to be a worthwhile use of our time.
  • Part of this project began in 1979 at MIT.


1. Four key charts and key question:

2. The Platonic Solids: The simple geometries still hold new insights

3. A Simple Logic: Continuity, symmetry and harmony

4. History within Logic, Mathematics, Philosophy, and Physics:

5. Starting Points:

6. Identity: Humanity at the center of this model of the universe.

7. Three Scales of the Universe: Small Medium and Large (more to come)

8. Numbers and Operands (more to come)


Pi equals 3.1415926535897932384626433832795028…


An arc of a circle with the same length as the radius of that circle corresponds to an angle of 1 radian.

A full circle corresponds to an angle of 2π radians.


  1. Pi is a constant.
  2. Pi is an irrational number.
  3. Pi is a transcendental number.
  4. Pi is a non-repeating number – no pattern has been identified using computer analysis within over twelve trillion places.
  5. Pi ( π ) is the exact ratio of the circumference of a circle to its diameter.   It is that simple.

Thank you, Wikipedia, for the graphics (above) that demonstrate this simple definition.  There are over 45 Wikipedia articles about pi.

So, what do you make of it?  What is going on?

Perhaps a few more questions and comments would help.

  1. What is it about a circle and sphere that pi is always-always- always true?
  2. How does a number become a constant, irrational and transcendental all at the same time?
  3. Let us compare pi to other unique numbers that have a special role among all numbers.  These are e, 0, 1, and I. They are all magical, but π stands out. So, let’s ask, “What are the shared qualities of these numbers?” Let’s study them to see if we can find any necessary relations.
  4. We have the ratio between a circle and a line. Perhaps this is the fundamental transformation between the finite and infinite? Are circles and spheres always implicating or imputing the infinite?

That is a big question and enough to ponder for awhile.

Notwithstanding, there are many more questions to ask.

Some speculations: Pi may be the key to unlock the small-scale universe within the big Board-little universe
1.   To get to the application of pi  within the Planck Units, we’ll need to emerge from the singularity of the Planck Units.  Is the radian a key to understanding this process?  First, a radius is extended from the singularity.  A radius extends into the preconditions for space and time, a now emergent small-scale universe. It makes that first arc equal to its own length.  It does it again and again and again and again and again (six radians) and then makes that last leap, 2 pi, to complete the circle. Is this a reasonable scenario? Why? Why not?

2. We need to run through dozens of scenarios, often, and slowly and carefully.  What scenarios are perfect and obvious?

3. We are at the singularity of the Planck Units.  We are establishing the foundations for the physical world.  If all things start simply, this must be the place to start.  It doesn’t get more simple and more mysterious. Nothing is a mistake, everything comes from a perfection to a space-time moment, so what could possibly happen?

What happens within the first six doublings?    (to be continued)

For further discussion:
1.  Is the Small-scale Universe the basis for the homogeneity and isotropy of space and time?
2.  Does everything in the universe share some part of the Small-Scale universe?
3.  How is Planck Temperature calculated?  Does it begin with the other Planck Units and expand from that figure at the first notation?

Note:  All of human history has occurred in the last doubling.  Yet, all doublings remain active and current and dynamic.  Continuity trumps time. Symmetries trump space.

What does sleep have to do with anything?  If all time is current, within the moment, we particularize by the day and uniquely within a given waking day.  Sleep seems to bring us into the infinite.  Dreams seem to be the helter-skelter bridge between the finite and infinite.  It seems that these naïve thoughts are worth exploring further.

This Shifting Paradigm Changes Our Perception Of Everything

Editor’s note:  This page was first posted within Small Business School, a television series that aired for over 50 seasons on PBS-TV stations (1994-2012).  It is the author’s business website, so many of the links go to that Small Business School website.    Eventually all links will be redirected to pages within The Big Board – little universe Project.


Background: Our study of the Planck Length to the Observable Universe began formally on December 19, 2011. Though we thought about the matrix from the Planck Time to the Age of the Universe, it took until December 8, 2014 to add it the Planck Length chart. Logically, but non-intuitively, the two tracked well together. Based on that work, we started looking at our own foundations for understanding first principles, universals and constants.

First, our television series began in 1994 based on first principles (linked from here). These were a direct reflection of our faith and our belief that faith and science must cohere or one of them is wrong.

Second, we used those first principles in all that we have done. That’s how one knows the first principles work. Yet, eventually, those first principles inform in new ways. It is not automatic. It takes time. But, there is always a next step. We can always improve on the initial conditions.

Third, we all need to extend our principles globally, then extend them throughout the universe. That drove our work on the Big Board-little universe back which started in December 2011.And oddly enough, we can now see how such principles just might become the core of a new small business revolution.

Here is a paradigm shift that just might change our perceptions of everything.

1. The Universe appears to be finite. That’s huge. It has measurable smallest units for space and time. It has measurable units for the largest dimensions of space and time, the Observable Universe and the Age of the Universe, respectively.More

2. The Universe has an ethical bias. Yes, hard to believe, but it seems to be true. If so, the theological among us have some very real work to do because theology will be informed by science and science will actually be informed by theology. And, those within radical Islam will learn that they still have much to learn from their Allah and our science!

3. The Universe is smaller and more ordered than we think.In 202+ steps, you go from the smallest measurement to the very largest.Initially it sounded ridiculous and it seemed inconceivable, yet over time, it sinks in.

4. The Universe is more connected than we think. In fact, everything is related to everything, all within 202+ steps! Seems impossible; it’s not.

5. The Universe gets structure from space-and-time, but not its essence. The structures go back to basic geometries that have become exquisitely complex (Also, see reference #4). One might conclude that the essence of that structure comes from the Infinite through our constants and universals which appear to be best engaged through the Planck Units.

Now, with all these references, we now say, “Let’s get focused; there are great things to do to get us all on track for a brilliant future.”

Part II: The Finite and the Infinite

There is “The Observable Universe” (and everything within it) and then there is Beyond All Space and Time  (link goes to Part I: First Principles)   Author: Bruce Camber.

1.  Introduction

After three years of reflection on our Universe View,  I wrote up yet another summary listing of some of the steps we had taken since December 19, 2011. On that December day, as those five classes were happening, it seemed like we had gone through a door that had not been opened. I wondered, “Why can’t we find any discussions about this simple structure of our universe?”  Beyond a simple ordering system based on the nested geometries of octahedrons and tetrahedrons, it seemed like it could have other useful applications. Nevertheless, my precautionary instincts kicked in. We would go slowly. Our work would be incrementalism at best. Plus, it has been difficult to get solid feedback.  In these days there seems to be a bit of fear of being wrong.

We asked, “How can it be wrong especially if it’s based on such simple logic, simple math, and simple geometries?” Of course, our model became a teaching tool.  It involved science, technology, engineering and mathematics, four of the cornerstones of invention and innovation. We imagined that the worst thing that could happen is being faulted for being overly simplistic.

So what?

The stakes are obviously very high. Our world is coming apart at its seams simply because there is no compelling integrative system of understanding of the sciences, the world’s theologies, and the diversities within the human family and her cultures.

Yet, we all share great commonalities that start from conception and birth.

What happens to us?

As a very little baby, each of us quickly learns there is a “You” and a “Me.”  There is an object out there and there is a subject in here. The emphasis is usually all about the “Me” so much so it has become a common expression in the culture, “It’s all about me.”  Narcissism is all about Me. Barack Obama is all about Me. Vladimir Putin is all about Me. Throughout history leaders are often clearly narcissistic and it is usually quite obvious these models ultimately do not work very well.

2. History Lessons: The Subject-Object problem is as old as history.

Which is more fundamental, the Subject or the Object?  The question has been debated in some form for millennium. It is only in this century and in this time that we can finally break through this historic problem.  We have to.  It seems that human survival is dependent on it.

From 1973 through 1980 I worked with a professor who uniquely focused on the Subject-Object problem.   His focus was on the hyphen between the Subject-and-Object.  He would say things like, “The relation is the primary real and space and time are derivative.”

But again, how so? So what?

If we add the words, “The relation is the primary real between the Finite and the Infinite and space and time are derivative,” we begin getting closer to being able to explore the question, “How does all that work?”

First, we could observe that our relatively new Universe View with its 205+ notations, now called our Universe Table, has taken all of space and time and put the two into a finite container. It necessarily brings the Infinite into the equation yet also appears to puts the Infinite out of reach. That could be controversial, however, it is not out of reach.

3. Constants, Universals, and Reality

The universals and constants seem to provide a bridge between the two.  The universals and constants seem to exist independent  of all space and any time yet also seem to be necessarily dependent  on all space and all time.

Also, along our path we discerned that the 205+ exponential notations imposed a simple ordering scheme.  The notations impose a certain continuity within the universe. The simple geometries within this scale impose an inherent structure that has both symmetries and asymmetries. As the two create relations, the ;door opens to an actual   time or applied time (historic time) and there are dynamics that have a certain harmony and an abundance of dynamics that are clearly dissonant.

Using just this schema alone, we then discerned that these categories imposed an inherent value chain within the very being of science, theology, business and culture.  If order / continuity, relations / symmetry and dynamics / harmony were taken as our first-phase definition of Infinity, it seemed as though we were able to duck under the most specialized language of science – theology – business – culture yet use language that is applicable to all four.

We believe that these three groups are the most simple perfections of form / function.

So what do we do with it?

4.  Perhaps the beginning of a breakthrough:  Could all of life be a ratio?

In December 2013 I sent a note out to an online group called the Polyhedrons.  Mostly mathematicians, and most geometers within that group, they are quite sophisticated and often I barely understand what they are discussing.  Yet, I wanted some feedback on our little project and now we had a student who had entered his work on the Universe Table into the National Science Fair.

Of the few responses, one came from Steve Waterman, a geometer-mathematician who in the 1990s defined an entirely new class of Polyhedron.   Yet, within his voluminous website, he especially wanted us to focus on his work with the constants.  One of the leading global arbiters of scientific constants is the US National Institute for Science & Technology (NIST).  In March 2014, after a few lengthy conversations about how NIST defined these constants (over 300) and how the same constants could be generated through ratios of any number of combinations of constants, I finally began to grasp the extraordinary thing that Steve Waterman has done.

His work is so profound it took awhile to sink into my thick skulls.  I had to have some confirmation that I wasn’t racing ahead to erroneous conclusions.  I contacted a Brown University professor of mathematics, a former NIST scientist, and the author of several basic books about the foundations of mathematics.  He brushed it aside, ” 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”

Of course, he is right.  And de facto, we fall into the group that he has criticized.  Yet, with our simple starting points, we have discovered an exceeding complex universe within relatively simple domains.

There is something more going on here.

If we add the three ratios together, 1/3 + 1/3 + 1/3 we get 1.  If we calculate the ratio and add them together we get .999999+.  Something is lost.  In a dynamic tension, we get wholeness.  When we look at the parts as an object, .33333+ we lose something and the result is slightly off.

NIST lists 335 constants ; all have been defined as a ratio in much the same way Planck calculated his constants.  Reducing them to a number, an actual size that corresponds with the NIST measurements, gives us a few clues as to how things are ordered, key components of the relations, and a door to explore the functionalities in the transformations from one notation to the next.

There is a lot of work to do here and as of this writing, all 300+ NIST constants are now in the pipeline for scrutiny and analysis.

What do you think?

Tiling the Universe In Just Over 201 Exponential Notations: A Great Chain Of Being

Initiated: December 1, 2014  Most recent update: Monday, February 15, 2016

Tetrahedral-Octahedral-Tetrahedral (TOT) couplets tile and tessellate the  universe.1   In earlier writings, we have observed how the Known Universe could be tiled in less than 202 exponential notations or steps, layers, doublings, or domains. ≡

Please note:  Many links will open a new tab or window.

TOT_3.jpgThe TOT Structure2 appears to be the “simplest, strongest, most perfect, interlocking three-dimensional tiling” within the Observable Universe. The TOT can be used to make ball-like structures, clusters, lines, domains or layers.  Here we can find, perfectly-nesting within every possible layer, a great chain of being seemingly suggesting that everything is related to everything throughout the universe.

December 2011: The Start of Our Research Using Base-2 Exponential Notation, Planck Length, And Plato’s Geometries.3  We used very simple math and got simple results yet also found hidden complexities. After doing a fair amount of analysis of our initial results, we continue to make new observations, conjectures and speculations about the forms and the functions within this universe. From all our data and study, it seems logically to follow that this tiling is the first extension of geometry and number (the sequence of notations) in a ratio.

The most simple engaging the most simple: Here may be the beginning of value structure.4 If so, it necessarily resides deep within the fabric of the universe, the very being of being.  Could these very first doublings be the essential tension of creation?

Here simplicity is based on a very simple logic, “Everything starts simply.” 5

NOTE: The TOT as a tiling would be the largest-but simplest possible system that spatially connects everything in the universe.  Yet, even with just octahedrons and tetrahedrons, it is also exquisitely complex; we’ll see the beginnings of that complexity with the many variations of R2 tilings (two dimensional) within this initial R3 tiling (three dimensional).6  Thus, the TOT would also be expanding every moment of every day opening new lines instantaneously. One might say that the TOT line is the deepest infrastructure of form and function. Perhaps some might think it is a bit of a miracle that something so simple might give such order to our universe.

The purpose of this article is to begin to introduce why we believe that this could be so.

Notwithstanding, we acknowledge at the outset that our work is incomplete. By definition tilings are perfect and the TOT tiling is the most simple. In our application these tilings logically extend from the within the first doubling to the second doubling to all 201+ doublings necessarily connecting all the vertices within the universe.

In earlier articles we observed how rapidly the vertices expand7  Yet, that expansion may be much greater once we understand the mathematics of doublings suggested by Prof. Dr. Freeman Dyson,6 Professor Emeritus, Mathematical Physics and Astrophysics of the Institute for Advanced Studies in Princeton, New Jersey. We are still working on that understanding.

We are taking baby steps. It is relatively easy to get a bit confused as to how each vertex doubles. The first ten doublings will begin to tell that story.

And, of course, we are just guessing though basing our conclusions on simple logic.

THE MOST SIMPLE TILING. Using very simple math — multiplying by 2 — the first tetrahedron could be created in the second doubling (4 vertices). Then, an octahedron might be created in the third doubling. That would require six of the 8 vertices. The first group of a tetrahedral-octahedral-tetrahedral chain requires all eight. Today we are insisting on doubling the Planck Length with each notation and to discern the optimal configurations. By the fourth doubling, there could be 16 vertices or six tetrahedrons and three octahedrons. At the fifth doubling (32 vertices), we speculate that the TOT extends in all directions at the same time such that each doubling results in the doubling of the Planck Length respective to each exponential notation.

We Can Only Speculate. We can only intuit the form-functions of this tiling as it expands. And, yes, within the first 60 or so notations, it seems that it would extend equally in all directions. With no less than two million-trillion vertices (quintillion), using our simple math of multiplying by 2, we will see how that looks and begin to re-examine our logic. Again, this tiling is the most simple perfection. And although we assume the universe is isotropic and homogeneous, there is, nevertheless, a center of this TOT ball, Notations 1, 2 and 3.8

That center even when surrounded by no less than 60 layers of notations is still smaller than a fermion or proton.  This model uniquely opens up a very small-scale universe which for so many historic reasons has been ignored, considered much too small to matter.

Nevertheless, it seems to follow logically that this TOT tiling is in fact the reason the universe is isotropic and homogeneous.9

Key Evocative Question from the History of Knowledge and Philosophy: Could this also be the Eidos, the Forms, about which Plato had been speculating? Could this be the domain for cellular automata that John von Neumann, Alan Turing, and others like Steve Wolfram have posited? Here we have an ordering system that touches everything and may well be shared by everything. Within it, there can be TOT lines that readily slide through larger TOTs. There could be any number of cascading and layering TOTs within TOTs.10 (A new image is under development with at least ten layers.  A link will be inserted as soon as we have it.)

A SECOND GROUP OF TILINGS. Within the octahedron are four hexagonal plates, each at a 60 degree angle to another. Each of these plates creates an R2 tiling within the TOTs that is carried across and throughout the entire TOT structure.

These same four plates (R2 tilings) can also be seen as triangle.  There ares six plates of squares. One might assume that all these plates begin to extend from within the first ten notations from the Planck Length, and then, in theory, extend throughout our expanding universe.

Only by looking at our clear plastic models could we actually see these different R2 tilings.

We have just started this study and we are getting help from other school teachers.

Jo Edkins, a teacher in Cambridge, England made our study even more dramatic by adding color in consistent patterns throughout the plate  We can begin to intuit that there could be functional analysis based on such emphases.11

We were challenged by Edkins work to see if we could find her plates within our octahedral-tetrahedral models. We believe we can find most of her tilings within the models.

Within the Wikipedia article on Tessellation (link opens a new window), there is an image of the semi-regular tessellation.  We stopped to see if we could find it within our R3 TOT configuration.  It took just a few minutes, yet we readily found it!  One of our next pieces of work will be to highlight each of these plates within photographs of our largest possible aggregation of nesting tetrahedrons and octahedrons.

Here the square base of the octahedrons couple within the R3 plate to create the first manifestation of the cube or hexahedron.  We will also begin looking at the very nature of set theory, category theory, exponential objects, topos theory, Lie theory, complexification and more.12

Obviously there are several R2 tilings within our R3 tiling. How do these interact? What kinds of relations are created and what is the functional nature of each? We do not know, but we will be exploring for answers.

A THIRD TILING BY THE EXPERTS. Turning to today’s scholars who work on such formulations as mathematical jigsaw puzzles, I found the work of an old acquaintance, John Conway. In 2011 with Professors Yang Jiao and Salvatore Torquato (all of Princeton University), they defined a new family of three-dimensional tilings using just the tetrahedrons and octahedrons.13

Hexagon JoWe are studying the Conway-Jiao-Tarquato (CJT) tiling. It is not simple. Notwithstanding, conceptually it provides a second R3 tiling of the universe, another way of looking at octahedrons and tetrahedrons. Here are professional geometers and we are still attempting to discern if and how their work fits into the 201+ base-2 notations.  And, we are still not clear how the CJT  work intersects with all of the R2 tilings, especially the four hexagonal plates within each octahedron.


It takes on a new meaning within this domain of the very-very-very small. Fine structures and hyperfine structures? Finite and infinite? Delimited infinitesimals? There are many facets — analogies and metaphors — from the edge of research in physics, chemistry, biology and astrophysics that can be applied to these mathematical and geometric models.

From where do these expressions of order derive?  “From the smallest scale universe…” seems like a truism.

Perhaps this entire domain of science-mathematics-and-philosophy should be known as hypostatic science (rather loosely interpreted as “that which stands under the foundations of the foundations”).


Notes & Work Areas:

Endnotes, Footnotes, and other References

1.  This article is linked from many places throughout all the articles and documents.   It is a working document and still subject to updates.

2. In 2006 I wrote to Dr. Francis Collins, once director of the National Genome Research Institute and now the National Institutes of Health. His publisher sent me a review copy of his book, The Language of God, and we spent several hours discussing it with her. The genome, the double helix and RNA/DNA have structure in common and it all looks a lot like a TOT line. Collins, a gracious and polite man, did not know what to say about the more basic construction.

Also, on a somewhat personal note, although we call it a TOT line it is hardly a line by the common definitions in mathematics; it’s more like Boston’s MBTA Orange Line. Now here is a real diversion.  Thinking about Charlie on the MTA  in the Boston Transit (a small scale of the London Underground or NYC Transit), this line actually goes places and has wonderful dimensionality, yet in this song, it is a metaphorical black hole. Now, the MBTA Orange Line is relatively short. It goes from Oak Grove in Malden, Massachusetts to Forest Hills in Jamaica Plain, a part of Boston where I was born.

3. Classroom discussion on December 19, 2011 in metro New Orleans high school where we went inside the tetrahedron by dividing in half each of the edges and connecting those new vertices. We did the same with the octahedron discovered inside that tetrahedron and did the same process with it. We divided the edges of these two objects in half about 110 times before we finally came into the range of the Planck Length. We then multiplied each edge by 2 and connected those vertices. In about 100 notations we were somewhat out to the edges of the Observable Universe. We are still learning things from this basic construction.

4. Where is the Good in Science, Business and Religion is located in several places on the web, however, it was first published on September 2, 2014 within a LinkedIn blog area. The chart was first used in another blog, “Is There Order In The Universe” which was published on June 5, 2014.

5. The Concept of the Expanding Universe is part of the concept of the Cosmological Principle (metric expansion of space) that resides deep within the concept of the Observable Universe.

6. As of this writing, there does not appear to be any references anywhere within academia or on the web regarding the concept of counting the number of vertices over all 201+ notations.  Using the simplest math, multiplying by 2 (base-2), there is a rapid expansion of vertices. Yet, it can also be argued that vertices could also expand using base-4, base-6, and base-8. That possible dynamic is very much part of our current discussions and analysis. It is all quite speculative and possibly just an overactive imagination.

7. We have made reference to Prof. Dr. Freeman Dyson’s comments in several articles. If he is correct, his assumption adds so many more than a quintillion vertices, it gives us some confidence that everything in the universe could be readily included as a whole. Within this link to fifteen key points, the Dyson reference is point #11.

8. If the Planck Length is a vertex from which all vertices originate, and all vertices of the Universe in some manner extend from it, the dynamics of the notations leading up to particle physics (aka Particle Zoo) become exquisitely important. Questions are abundant: How many vertices in the known universe? What is the count at each notation? Do these vertices extend beyond particle physics to the Observable Universe? In what ways are the structures of the elementary particles analogous? In what ways are the periodic table of elements analogous? What is the relation between particle physics and these first 60 or so notations? Obviously, we will be returning to each of these questions often.

9. Isotropic and homogeneous are working assumptions about the deep nature of the universe. Homogeneous means it has a uniform structure throughout and isotropic means that there is no directional bias. This work about tilings provides a foundation for both assumptions.

10. The two small images in the right column are of a very simple four-layer tetrahedron.  The Planck Length is the vertex in the center.  The first doubling creates a dynamic line that can also be seen as a circle and sphere. The next doubling creates the first tetrahedron and the third doubling, and octahedron and another tetrahedron, the first octahedral-tetrahedral cluster also known as an octet. The fourth doubling may be sixteen vertices; it may be many more.  When we are able to understand and engage the Freeman Dyson logic, the number of vertices may expand much more rapidly.  Again considering the two images of a tetrahedron  in the right column and its four layers, today we would believe that it amounts to three doublings of the Planck Length.  When we begin to grasp a more firm logic for this early expansion, we will introducing an image with ten layers to see what can be discerned.

11. I went searching on the web for images of tetrahedrons and tessellations or tilings of hexagons. Among the thousands of possibilities were these very clean images from Jo Edkins for teachers.  Jo is from the original Cambridge in England and loves geometry.  She has encouraged us in our work and, of course, we thank her and her family’s wonderful creativity and generosity of spirit.

12. Virtually every mathematical formula that appeared to be an abstraction without application may well now be found within this Universe Table, especially within the very small-scale universe.  We will begin our analysis of  set theory, category theory, exponential objects, topos theory, and Lie theory to show how this may well be so.

13. “New family of tilings of three-dimensional Euclidean space by tetrahedra and octahedral” Article URL:
Authors: John H. Conway, Yang Jiao, and Salvatore Torquato

14 Our example of a TOT line was introduced on the web in 2006. In July 2014, this configuration was issued a patent (USPTO) (new window). That model is affectionately known within our studies as a TOT Line.

This patent is for embedding a TOT line within a TOT line. There are two triangular chambers through the center; and for the construction industries, we are proposing four sizes to compete with rebar, 2x4s-to-2x12s, and possibly steel beams.

The Patent Number: US 8.769.907 B2, July 8, 2014 is fully disclosed at the WordPress website,


Center for Perfection StudiesThe Big Board–Little Universe Project 

What is finite? And, what is truly infinite?

“Finite or Infinite? Is That The Question?”    (link goes to Part II)

Some of our high school students think our scientific community makes the study of Science, Technology, Engineering and Mathematics (STEM) all too difficult to understand and overly complex by defying a certain commonsense logic. (Reference #1)

We have been studying simple math and simple geometries from the smallest possible measurement of a length to the largest (Reference #2). It appeared to some of the students, based on this work, that the universe is obviously finite. They have been told that intellectually and historically, it is an open question. For them, “Make a choice and see where it takes you.”

The students with strong faith statements said, “Only God is Infinite. All things within space and time are finite.” (Reference #3) When asked about all the universals-and-constants and space-and-time, the concurrence is that these are the access paths, interconnections and transformations between the Finite and the Infinite.

For the best of these students, asking the question, “What is the Infinite?” is like asking the question, “Who is God?” And, they have answers.

Of course, as a result of a little coaching, they say, “First, God is Perfect.” When asked, “What is perfection?” they echo their coach: “Perfection is order-continuity, relations-symmetry and dynamics-harmony, all rolled into one.” (Reference #4) That amounts to an understanding of the Infinite without importing all the related history and revelation from the various faith statements within our very short history throughout our little world. The Finite is another story. We turn to many people from Euclid to Einstein for inspiration to provide the academic and religious communities with our simple observations and assumptions.

Hardly postulates and axioms, our statements are a praxis in-search-of theoria:

If these statements are taken as a given, then what kind of universe and what kind of science do we have? Should we re-examine the use of infinity throughout the ages going back to the ancient Greeks? Should we reconsider the theory of indivisibles? And, perhaps we should even reconsider the very nature of the Big Bang and its theory.

Of course, that is our agenda (Reference #8),   our current focus for the immediate future.


1. One of two key general overview and working article,  Order in the Universe

2. One of the earliest reflections on all our efforts and work: Is it true that everything starts most simply?

3. Initially written in November 2012, just what is the nature of belief?

4. In light of those constants, universals and the finite-infinite relation, the nature of perfection seems to follow:

5. Examining basic structure in basic ways: Simple View of the Universe

6. Our first look at the progression of doublings.  This listing was written to accompany an article for Wikipedia: Written in March 2012 to support an article for Wikipedia

7. There are somewhere over 201 base-2 exponential notations (doublings, domains, layers or steps) within the Known Universe.

8. Another analysis of key points: There are 15 in this article.

Even between atheists and believers

Perhaps all it comes down to is an answer to the question, “Whose metaphor is more meaningful?” You will not find many atheists who deny science. They do not deny the constants and universals that are always in the back of the science textbooks.

There are three constants within the sciences that remain clear, in spite of quantum mechanics. The first is that there is order and continuity in the world. It is the basis of knowing. In every discipline there are multiple parameter sets where this is true. Beginning in mathematics, a rather pure form of thought, abstraction and representation, we then move into physics. It has multiple parameter sets as well. There is one for Newtonian mechanics, another for General Relativity and Special Relativity and yet another for quantum mechanics. Then chemistry and biology have their own parameter sets. All these parameters simply establish the boundary conditions of what is being measured within them.

Each has a formalized language. And, each has a metaphorical language that pushes into the edges of the unknown.

The sciences all embrace varying definitions of relations yet all of these definitions are understood by a symmetry function.

Specialized disciplines with each of the sciences hypothesize about the nature of the unknown, just beyond their limits of knowledge, and all these hypotheses are a study of the deepest dynamics of their discipline. The experience of insight, the “ah-ha” of the creative surge, is experienced as a concrescence of symmetries or harmony.

The atheists mostly object to the use of specialized language. They understand rules, mores, and societal law and order even though many are nihilistic, others narcissistic, and many both.

Yet, change will come. Some of these folks will begin to realize that time is not a fundamental frame of reference and that there are qualities of life that permeate everything in every way, and that these qualities empower order, relations, and dynamics, and that these three scientific functions with the faces of continuity, symmetry and harmony just might also be understood with very personal language. When and if they do, they are on their way to create a personal bridge to religion and some of the brave among them just may cross it.

Notations 83 down to 66 (out of 1-to-202)

Please note: Chart the five Planck base units, Planck Length & Time and Planck Length
The first number, the Notation (out of total of 202), is also the number of times the Planck Length has been doubled.
83:  .156309264 nanometers
or 1.56309264×10-10meters
82:  7.81546348×10-11m
81:  3.90773174×10-11m
80:  1.95386587×10-11m
79:  9.76932936×10-12m
78:  4.88466468×10-12m
77: 2.44233234×10-12 m
76: 1.22116×10-12m
75:  6.10583084×10-13m
74:  3.05291542×10-13m
73:  1.52645771×10-13m
72:  7.63228856×10-14m
71:  3.81614428×10-14m
70:  1.9080×10-14 meters
69:  9.54036072×10-15m
68:  4.77018036×10-15m
67:  2.38509018×10-15m
66:  1.19254509×10-15 m

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This image below is from a PPT chart, a presentation of physicist, Prof. Dr. Emily L. Nurse.  She first gave it as a Masterclass in Manchester University back in 2005Dr. Prof. Nurse is affiliated with University College London, Fermilab in Chicago, and CERN Laboratories, Atlas Project, in Geneva.Atoms


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