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Last Update: Saturday September 25, 2021

"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?"
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,
Volume 480, December 1986 (p. 304 of 304–316)

DOI: 10.1111/j.1749-6632.1986.tb12434.x

A Simple View of the Universe

Key question: Is a simple mathematical and geometrical view of the Universe meaningful or useful?
The goal of this article: Try to open a dialogue about the question


Note: This article was initiated in July 2015, updated throughout August 2015, and it is now a "working first draft." 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 prior work:

Thank you.   - BEC

Navigation notes: Cursor over any link and it may pop up a very small summary window. You will discover that clicking on links that are in brackets [ ] go to the bottom of this page. Unbracketed links open a new window. Unacknowledged links are for us (our bread crumbs).


Back in December 2011 we began our work on a very simple mathematical model of the universe that was playfully dubbed,
Big Board-little universe.1 We had started using the following parameters -- base-2 exponential-and-scientific notation,2 the Planck base units,3 and the Platonic solids4 -- in ways that created heretofore unobserved boundary conditions.

Our Three Initial Conditions

1. A basic chart.[5] 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. The 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 mathematical structure from the 1st to 65th doubling.

Is it significant? Science knows experimentally that light travels 299,792.458 km/second (a light second). Between notations 142 and 143 it is confirmed using the Planck Time and base-2 exponential notation. To our knowledge, this is the first time it has been confirmed in such a manner.

Also, a Planck Time multiple, either 1.202 seconds or .6011 seconds, could be used as a standard unit of time that is based on a theoretical constant. We will further explore our simple calculations for a day, week, month and year, particularly in light of recent work to define the theoretical chronon.

2. Geometries.[6] 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 new vertices.6a We could see four half-sized tetrahedrons in each corner and an octahedron perfectly in the middle. We then went inside the octahedron and found six half-sized octahedrons in each corner and a tetrahedron within each face (8). With just these two Platonic solids, we could tile and tessellate each layer and between layers or doublings throughout the entire model.6b Our geometry classes were exploring the question, "How far could we go deep within the simple tetrahedral-octahedral structure?" Then we asked, "How far out can we go by continuously doubling what we had?" It was here we began to learn that this progression is called base-2 exponential notation. Academically,  the tetrahedral-octahedral structure has been largely limited to chemistry and the analysis of silicates, cobalt, antiferromagnetism, and network structure and framework bonds.

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.7  Within manifold geometries, the analogue would be to fullerenes.8

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, bond, bind, break or repel constructions within each, and between each, of the 201+ notations.

3. Logic.[9] 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 limits9a. Though imputed, this continuity function became our first principle for order in the universe9b yet it took 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, "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."9c Perhaps the very nature of space and time is derivative; and, order-relations-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, infinitesimal universe.

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

4. History. [10] This integrated universe model must now be tested within the history of logic, mathematics, philosophy and physics. Any model, if it is to have a place within the work of scholars, must be critically analyzed. And, we know this model has a long way to go. It must address very basic related questions about duality,11 finite and infinite sets,12 group theory,13 set theory,14 then advanced mathematical concepts that seem to be necessarily related like advanced combinatorics,15 matroids16 amplituhedrons,17 and the Buckingham pi theorem.18 Like breadcrumbs, these topics will be followed 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 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, dimensionless constants, and perfected states?
  • Does the Michaelson-Morley experiment19 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 Penrose20 of Oxford?
  • Is it the matrix or grid that Frank Wilczek21 (MIT) delineates? Why? How?
  • Could our small-scale universe be all of the above?
  • Thinking about CERN and their current research from quarks to gluons,22 how does this small-scale universe work in such a manner to give rise to the impeccable successes of the Standard Model23 and CERN's more recent confirmation of tetraquarks24 and pentaquarks?25
  • Might this small-scale domain be the basis for homogeneity26 and isotropy27 in the universe? How does dimensional analysis and dimensional homogeneity28 apply?
  • If so, then what does it infer about the most distant objects from the Hubble Space Telescope?

These are some of the subjects (or objects)29 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.

Calculations, Measurements and Observations

5. Starting point or domain or ...[30] The key question is, "What is being measured? Something is being doubled within each notation that is defined by the Planck base units. First, we assume the singularity28 (the Void) of these base units, yet, we now ask, "What happens within that first doubling? What gets doubled?" ...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, we cite just one as our primary reference -- Wilczek et al -- 31 dimensionless physical constants identified (PDF).31 The Planck Length (space) and Planck Time are two of their  31.

Once we have begun to understand that paper, 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 experts with each of these constants.  Perhaps we will begin to see how our two base units create a nondimensionalized plenum32 and vinculum33 so an "archetype" of mass(kg)34 and electric charge (q)35  begin to manifest and we begin to discern how the parameterizing functions of the Planck constant (h),36 including the speed of light in vacuum (c),37 the gravitational constant (G),38, the electric constant (ε0)39 and the elementary charge (e)40 as each comes in to play.  We assume somewhere along our progression of doublings, the fine-structure constant (α)41 will present itself as will all the other dimensionless constants.

So, to focus on the very first doubling, we ask, "What is manifest?" First, we have the actual calculations by Max Planck for length, time, mass and electric charge. "How are these manifest?" Though infinitesimal, there is a manifestation of something.

Our first assumption is that the "somethings" could be either simple vertices or what are known as point-free vertices. More study is required.  We are told by Freeman Dyson42 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. If so, there would be eight vertices within the first doubling.

When we more fully understand these scaling laws, dimensional analysis and point-free vertices that figure could actually increase.  It will not get smaller!

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 this story.  Theoretically or conceptually, here is the first abiding step to construct and sustain our little universe. It is its own perfection, yet it must also provide degrees of freedom.  Here we will start our analysis with the tools of causal set theory, cubic close packing, Pi, the dimensionless constants and perfected states with continuity, symmetry, and an infinitesimally short moment of harmony. 

The story becomes increasingly complex with each doubling.

Notations:   Doublings:  Scalings (of vertices or point-free vertices):
0 0 0
1 2 8
2 4 64
3 8 512
4 16 4096 (thousands) (3)
5 32 32,768
6 64 262,144
7 138 2,097,152 (millions) (6)
8 256   16,777,216
9 512 134,217,728
10 1024 1,073,741,824 (billions) (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 (trillions) (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 (quadrillions) (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 (quintillions)  (18)

With every one of the 31 dimensionless constants, we will engage in a series of thought experiments made to see what happens to each number within each doubling. We will watch the simple logic of doubling and scaling, especially between the 60th and the 70th doublings. When and why does a number punch out and become something that is reduced to practice?   Or, in what notation and in which combinations does a dimensionless constant become manifest?

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 undergird every part of the Standard Model.

Anything and everything seems possible.

6. Identity: Humanity at the center of this model of the universe.[43] 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 scientist44 helped us; he calculated 202.34 notations. Then a prominent French astrophysicist45 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.[46] We then divided our chart of the Big Board - little universe by three so each scale would ideally have just over 67 notations. Following a longstanding convention within scholarship, we call these 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 the history of the Greats who used such terms, i.e. Pythagoras,47 Plato,48 Aristotle,49 Kepler,50 Newton51 and Leibniz52).

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.[53] From Sequential Real Numbers, to Base-2 exponential notation, and then to Dimensional Analysis. We have observed how the simple mathematics of both base-2 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-or-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,54 area,55 volume,56 or any other dimensional attributes57 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.

Bruce Camber


  •  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,58 because it provides a working foundation for everything everywhere for all time.
  • These observations and possible conclusions will be revisited often.
  • The model works as a simple Science, Technology, Engineering, Mathematics (STEM)59 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.60

Endnotes (To return to the article, click on the number between the two bars [ ].)

[5] Four key charts:

[6] The Platonic Solids: The simple geometries still hold new insights

[9] A Simple Logic: Continuity, symmetry and harmony

[10] History within Logic, Mathematics, Philosophy, and Physics:

[30] Starting Points:

[43] Identity: Humanity at the center of this model of the universe.

[46] Three Scales of the Universe: Small Medium and Large

  • More to come...  "This parameter problem can be viewed as the logical continuation of the age-old reductionist quest for simplicity."

[53] Numbers and Operands

  • Numbers and Operands. (to be continued)
  • More to come... 


Note on simplicity:

Definitions (partial listing):
1. Not hard to understand or do.
2. Having few parts
3. Not complex or fancy.  Plain, basic, not special or unusual.
4. The fundamental and straightforward nature of something.
5. A group with no proper subgroup.
6. Composed of a single element, not a compound.

Cultural Flavoring:
KISS:  "Keep It Simply-Simple."
Camber Flavoring:
Everything starts simply.

"Either science is irrational [in the way it judges theories and predictions probable] or the principle of simplicity is a fundamental synthetic a priori truth." -Swinburne , Oxford1997. 

An example of complexity

""The Schwarzschild radius appearing in the spheristatic solution of the Einstein equations as the radius of the singular surface determined by the gravitational potential given by the time-time component of the covariant, symmetric, 2nd rank, gravimetric tensor -- (formula 1) -- which relates any mass m to a length ls (m) at the singularity by (formula 2),thus relating mass linearly to length, while the reduced Compton wavelength relates any mass unit to a length unit by Compton wavelength."

Formula 1:    g44 = -----------------------
            ( 1 - (G0/c2) (m/r) )

Formula 2: ls(m) = (G0/c2) m

Key words:

1. The Schwarzschild radius