Steve Waterman & Michael Anthony Whitehead

Our effort to discern the Top Numbers of Key Importance  within our little universe for The Big Board-little universe Project  began with this work by Steve Waterman and Prof. Michael Anthony (Tony) Whitehead (Chemistry, McGill University, Montreal).

Steve and Tony got together to publish an article, Self-Consistent Field Approach To The Physical Constants: All Physical Constants Are A Function Of Pi (Π) Though poorly received by the academic community, the article will be studied to attempt to understand their rather unique sense of numbers.

The generalized claim is that all the physical constants are derived from four math constants.  They did the work to explain this statement with 142 such constants.  Over time, every calculation and conclusion will be reviewed. Currently, our study look at their four math constants:

  1. Pi (Links go to Wikipedia when within the body of the subject)
  2. The isoperimetric quotient of a sphere
  3. Sphere packing  (CCP and NCCP)
  4. Number density

On its face, these four math constants do not seem to have enough substance to define all the better known physical constants. Yet, the Waterman-Whitehead team may have the kernel of an idea that truly opens the way to define the first 67 notations in ways that inform isotropy, homogeneity, the nature of infinity, the nature of the finite,  and the function of math constants to create the bridges to infinity and the physical constant to build bridges to physicality or space and time


What Did We Ever Do Without Our Universe View?

1957: The Beginnings of a somewhat Integrated Universe View

In 1957 Kees Boeke’s book, Cosmic Vision, The Universe in 40 Jumps, was published; it was the first integrated view of the known universe. He could have but did not engage the Planck base units. He could have, but did not consider any geometric calculations. Yet, he did get the attention of prominent scientists including Nobel-laureate, Arthur Compton. Thereafter, the Eames film, the Phylis and Philip Morrison book, Powers of Ten, the IMAX (Smithsonian) movie (guide), and the Huang’s scale of the universe opened this conceptual door for anyone who chose to walk through it.  Anyone could begin to have an integrated view using base-10 notation of the entire universe. It was a fundamental paradigm shift; all the attention given to it has been justified.

Most of the world’s people live within what we might call, their OwnView.  Even though subjective and often quite naïve, the elitists and the solipsistic and narcissistic among us, lift up that view as the best view, the only view, and/or the right view.

If and when we start to grow up, spread our wings and begin to explore beyond our horizons, we develop an objective view of the world.  As we integrate more and more facets of our subjective and objective views, it begins to qualify as a WorldView (in the spirit of the old Weltanschauung).

In light of Boeke’s work, the next step for all of us is to bring whatever WorldView we have, and see how it fits and works within a view of the entire universe. Kees Boeke’s work is historically the very first UniverseView. Although Boeke only had 40 jumps and used base-10 exponential notation, it is still the first systematic view of the entire Universe.

2011:  A Second Universe View Emerges From Another High School

A high school geometry class just up river from the French Quarter of New Orleans developed what appears to be the second systematic UniverseView.  It is quite a bit more granular than Boeke’s work and it originated from the students’ work with simple embedded and nested geometries. Using base-2 exponential notation this  group emerged with about 202+ doublings, layers, notations, or steps from the Planck Length  to the Observable Universe.  Eventually beside each length, the calculations from the Planck Time out to the Age of the Universe were added.

This fully-integrated UniverseView first emerged in December 2011 and was officially dubbed, “Big Board – little universe.” One of the initial boards was over eight feet high and the second and third generations were around 60 inches high.  The entire universe, mathematically-and-geometrically related within 200 or so notations, seemed to bring the universe down to a manageable size!

Now, what do we do with it?

The first thought was that this UniverseView with its 200+ notations could be a good container for Science-Technology-Engineering-Mathematics (STEM) education.  It puts everything in the known universe within a simple ordering system.  Then, in January 2012, in the process of trying to find scholarly references to understand the foundations of their work, the students and their teachers discovered Kees Boeke.  In so many ways, it was a vindication — “Somebody had been here before us.”  Yet, even with all the fanfare around Boeke’s work, not too much was done to extract meaning from that model.

The base-2 model is quite different. It has simple geometries and a more granular mathematics.  The students and teachers thought this ordering system might help to answer those historic queries by Immanuel Kant about (1) who we are, (2) why we are, (3) where we are going, and (4)  the meaning and value of life.

Given this model has a starting point and an end point, the students and teachers opted to see the universe as finite.  Always encouraging students to go deeper in their understanding of mathematics, their teacher, Bruce Camber, commented To engage the Infinite it appears that we hold the objective and subjective in a creative balance and that balance is called geometry, calculus and algebra through which we can more fully discover relations.”

Boeke’s base-10 work has an important role in history.  It gave the human family a starting point to see an ordered universe.  The base-2 model takes the next step. Instead of just adding or subtracting zeroes, it adds 3.333 times more steps or doublings. It provides more data to explore the simplest continuities, relations and dynamics within and between each notation.  Base-2 is the heart and spirit of cellular division, chemical bonding, complexification (1 & 2), and bifurcation.

Perhaps it is here that the academic community might begin to create a truly relational, integrated and functional UniverseView. Surely it is here that we find the rough-and-tumble within science.

So, although base-2 UniverseView is the second UniverseView, it seems to hold some promise.  And though these are preliminary models,  just a crack in the doorway, what a sweet and simple opening it is.  Perhaps Kepler would be proud.

This high school group is now just starting to discover the work of  real-and-graciously-open scholars.  With the help of this larger academic community, our work just might  somehow capture the spirit of one of the world great physicists throughout history, John Wheeler, when he said, “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? How could we have been so stupid for so long?” 


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.

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.”

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,

Did A Quiet Expansion Precede The Big Bang?

A question about the question: It is difficult to know; however, a better question might be,
“Do the dynamics of a quiet expansion deflate the Big Bang?”

Last update: February 16, 2015 (also, small corrections since then)
Sequel: June 5, 2016, This Quiet Expansion Challenges the Big Bang

September 2014: If you think about it, most of the world’s people have never heard of the Big Bang theory (Reference 1 – the cosmological model, not the TV series). Of those who know something about it, a few of us are somewhat dubious, “How can the entire physical universe have originated from a single point about 13.8 billion years ago?” It seems incomplete, like there are major missing parts of the story.

To open a dialogue about this pivotal scientific theory is the reason for these reflections. And, if we are successful, all of us will have re-engaged our ninth grade geometry classes and we will begin to ask a series of “what if” questions about the origins of this universe.

Big Board – little Universe. Some of you are aware of our work within several high school geometry classes (Reference 2) to develop a model called the Big Board-little universe (Reference 3). Possibly you even know a little about the 201+ base-2 exponential notations from the Planck Length to the Observable Universe. It is a study that informally began on December 19, 2011, so most of us have only begun to explore the inner workings of each of the 201+ notations.

Because we believe all things start most simply, the first 60+ notations are potential keys for understanding a rather different model of our universe. These notations (also referred to as clusters, containers, domains, doublings, groups, layers, sets, and steps) have not yet been studied per se within our academic communities (Also, see reference 4). The best guess at this time is that the range of our elementary or fundamental particles begins somewhere between the 60th and the 67th notations.

The simple mathematics (Reference 5) and the simple geometries are a given; the interpretation is wide open.

This little article is an attempt to engage people who are open to new ideas to look at those first 60+ notations. What kinds of what-if questions could we ask? Can we speculate about how geometries could grow from a singularity to a bewildering complex infrastructure within and throughout those first 60+ domains, doublings, layers, notations, and/or steps? What if in these very first steps, there is an ultra-fine structure of our universe that begets the structure of physicality? What would a complexification of geometries give us? Might we call it a quiet expansion? Though we have always been open to suggestions, questions and criticisms, we are now also asking for your insight and help.
Updates of both models are being prepared whereby those first 60+ notations of the Big Board-little universe begin to get some projections to study and debate. Also, another version of the Universe Table (Reference 6) is in preparation to emphasize every notation from 1 to 65. Also, at the time this article was introduced, we initiated a chart of base-2 exponential notations of time from the Planck Time to the Age of the Observable Universe side-by-side with our chart for the Planck Length to the Observable Universe. And, to make this study a bit more robust, we also projected a time to add the other three basic Planck Units — mass, electric charge and temperature. (Note: The very-first rough draft of that work was completed in February 2015.)

Big Bang Up. Most people start time with the Big Bang. Is there a possibility that there are events between Planck Time and the bang (or whatever sounds there were when things became physical somewhere between notations #66 to 67)?

In their 2014 book, Time in Powers of Ten, Natural Phenomena and Their Timescales, Gerard ‘tHooft and Stefan Vandoren of Utrecht University (Reference 7), use base-10 notation and assume there is nothing in the gap between the known time intervals of within theoretical physics and Planck Time.

We are doing a little fact check to see if the authors give those notations from Planck Time any causal qualities. It appears that they were not concerned about those base-10 notations until we pointed them out to them.

The first time period of interest to us is the first 20± base-10 notations which would be the first 67 base-2 notations. What happens between the Planck Units and the emergence of the elementary particles? These are real durations in time. A lot can happen.

We will be exploring this small-scale universe in much greater detail. By the 60th doubling there are quintillions-upon-quintillions of vertices with which to create many possible models. Also, in light of the work to justify the Big Bang theory, there is an abundance of information from all the years of research since the concept was first proposed in 1927 by Georges Lemaître.

Steven Weinberg, the author of The First Three Minutes (Reference 8), begins his journey through the origin of the universe at 1/100th of a second. Our hypothesis is that we can mathematically go back to a much, much smaller duration. We believe that we should start at the Planck Time and multiply it by 2. And, just as the fermion within notation 66 would be the size of a small galaxy compared to the Planck Length, 1/100 of a second between notations 137 and 138 represents an even greater gap of the ignored and unknown. We suspect starting one’s analysis so late misses key critical interactions and correlations (Reference 8b).

We’ve just started to see what the numbers can tell us.

A lot of pre-structuring of the universe could be quietly happening within such a duration (1/100th of a second). Using our most metaphorical, speculative thinking, one could imagine that the actual event within those first sixty notations was a gentle, symphonic unfolding, fully homogeneous and isotropic.* Although we should embrace all the key elements of today’s big bang theory, we should also be constantly asking, “What kinds of geometries would be required within each of the first 60 notations to render these effects?”

Perhaps the universe and our future belong to the geometers.

So, this article is to empower all of us to find the best geometers around the world to engage the Big Board-little universe model within what we call “the really-real small scale universe.” Of course, some of the work has already been done within the study of spheres, tilings, and combinatorial geometries.

If you would like to comment politely, please drop me a quick note (

Thank you.

Bruce Camber

* homogeneous Having the same property in one region as in every other region
isotropic Having the same property in all directions.


Endnotes and References:

1 A Wikipedia summary of the basic Big Bang theory. As you will see within this Wikipedia article, the basic theory has been highly formulated with a fair amount of scientific evidence. If our rather-naïve, quaint-little challenge to that model is ever to catch some traction, it will have to account for the results of every accepted scientific measurement about the Big Bang theory that has been thoroughly replicated.

2 Is There Order In The Universe? There are nine references within this article and each opens to a page that has been written since the first class on December 19, 2011.

3 This image of the Big Board-little universe is Version 2.0001.

4 This article is our very first attempt to provide a somewhat academic analysis of the work done to date. It was rejected by several academic journals so it was first released within WordPress, then the LinkedIn blog pages, and finally re-released right here.

5 The debate within Wikipedia about the importance of base-2 exponential notation resulted in their rejection of the original article. It was judged to be “original research.” We thought that judgment was just a little silly. The concepts were all out there; these articles were just to organize that data.

6 A WordPress blog page for our emerging UniverseView.

7 This article about the book, Time in Powers of Ten by Gerard t’Hooft and Stefan Vandoren, is the most comprehensive that I could find at this time. If you happened to find a better review, please advise us.

8 An online version of the entire book, The First Three Minutes by Steven Weinberg. There are many reviews, yet this one provides a little counterweight. Weinberg also wrote the forward to Time in Powers of Ten. Gerard t’Hooft (1997) and Steven Weinberg (1979) are Nobel laureates.

A chart showing the correlations between Planck Time and Planck Length at the 136th and 137th notations is here.

9 A WordPress article about very small and very big numbers. There is our initial discussion about the first 65 notations.

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