Sean Carroll

Sean Carroll is a research physicist at California Institute of Technology, the author of many books and articles, and a frequent speaker or expert witness.

Hi Sean,

What a marvelous thing you do with your life!

First, congratulations on being married to Jennifer.
She is such a mensch. I quoted her several years ago
within an article about the size of the universe based
on Hubble measurements (See Footnote 3).

We are just silly high school teachers. Neophytes.
We backed into a strange universe view based on
work in our geometry classes using the tetrahedron
and octahedron.  We divided the edges in half, connected
the vertices to find all the nested tetras and octas
and kept going within 40 times, right by the old fermion

and another 67 times down into the Planck scales.

That was a trip.  When we decided to multiply by 2, we
were out to the edge of the universe in about 90 doublings.
Couldn’t believe it.  It was Boeke on steroids.  He just had
zeros. We had the Planck base units, the Age of the Universe,

all that incredible geometry, and 3.333 times more information.

So what?

Well, we thought it was a good STEM tool, then those

67 notations between the fermion and the Planck scale just

started to tickle us.  What is there?  Nothing?

I don’t think so.  Here’s out latest chart (horizontal scrolling):
We have lots of charts:
And now, after five plus years we are getting bold
and probably very stupid:
Have we dropped off the cliff? Can you help us get back
on terra firma? What’s wrong with our idiosyncratic logic?

Just idiotic?


Most sincerely,

Some of the more far-reaching implications of the Big Board-little universe model

Introduction.  The Big Board-little universe Project uses base-2 exponential notation from the singularity of the Planck base units going out to this present time, Right Now, to encapsulate everything, everywhere, throughout all time1. Though seemingly a bit of  an overstatement, the simple mathematics and logic appear to corroborate such a conclusion. The implications of this fledgling model seem rather far-reaching so five are presented for the discerning analysis and critical review of scholars and thinkers.

Some of the more far-reaching implications of the Big Board-little universe model:

  •  This model begs the question about the finite-infinite relation.  If space and time are derivative, finite, quantized and discrete, then what is infinite? Our working answer is continuity which creates order, symmetries which create relations, and harmonies (multiple symmetries working together) which create dynamics. These are the inherent qualities that define the infinite for science.2
  • There appears to be an ethical bias to the universe. Continuity-order, symmetry-relations, and harmony-dynamics also begin to define a valuation system whereby every notation at every moment has a perceived and dynamic value.3
  • Each notation defines an element of the current universe. Even though time is derivative, it still defines a duration within a single notation. Even though space is derivative, it still defines a length within which particular things have their beingness. In this model each notation has its own particular beingness. The entire universe actively appears to share this length (space) – time infrastructure within the small-scale universe.
  • The structure for homogeneity and isomorphism is defined within the small scale. It is also the bridge between the finite and infinite so renormalization works in quantum electrodynamics and universality works throughout physics on every scale.4
  • This model appears to trifurcate nature. These three seemingly natural domains of this model of the universe appear to be episodic:
    (1) The small scale from notation 1-to-67 could generally be described as ontology and each notation just might manifest again within the human scale and then again within the large scale.
    (2) Perhaps the human scale from notations 67 to 134 could be understood as the domain for epistemology. In some manner of speaking all 67 manifest in the notation for the current time.
    (3) The large scale from notation 134 to 201 is currently considered the domain for cosmology. It begins when the duration (or speed) is less than one second (see notations 142 to 143). Within notation 200 (possibly 201) is the current time.  Its duration is approximately 10.8 billion years.  The duration at notation 134 is within a thousandth of a second.

    Of course, each duration for each notation gets increasingly short as we approach the Planck Time. The duration at notation 67 is 10-to-the-negative-23 seconds (10-23).  Notation 67 is approximately the Planck Length multiple where fermions and protons appear.  There will be many adjustments of these numbers as others help to fine-tune the model.5

All people and things appear to be trifurcated.

One might hypothesize or hypostatize that from the small-scale universe we get our being. Systems are imputed between the 50th and 60th notation; and within systems, the human mind has also been imputed have notations within which to be.

From the human-scale we get our knowing.  Carl Jung called these archetypes.  A special vocabulary will emerge for this part of our self-definition.

Within the Now, this moment, today, there is an integration as a thing, an entity.  All human history, all civilization is within the 200th notation.  Just as an aside, if time travel were to become possible, it will be as an observer. Interactions would also require trifurcating, i.e. simultaneously entering the space and duration of the being and knowing of any given moment in time.

For more, consider these pages:

  1. The Big Board-little universe Project
  2. Top Ten Reasons Top Ten Reasons to give up those little worldviews for a much bigger and more inclusive UniverseView. These are the inherent qualities that define the infinite for science.
  3. Every moment has a perceived and dynamic value.
  4. An analysis of numbers
  5. A Simple View Of The Universe
“In the beginning was the Word, and the Word was with God, and the Word was God. The same was in the beginning with God. All things were made by him; and without him was not anything made that was made.  In him was life; and the life was the light of men.  And the light shineth in darkness; and the darkness comprehended it not.”  John 1:1-5
These words took on new meaning back within the first week of May 2016  — it is a very simple fact that every notation for the entire universe, everything, everywhere, throughout all time, is embedded within light.

The explanation is rather straightforward and, although there may be a few new concepts here, it is actually all very simple.

There are three charts, all based on multiplication by 2, that will help to explain. This first chart (opens in new tab or window), first printed up in December 2011, begins at the Planck Length (the smallest possible length) and goes out to an approximation  of the size of the universe which is called, the Observable Universe. The February 2015 chart begins at the Planck Time and it uses 13.8 billion years as the Age of the Universe, the endpoint of this simple mathematical progression.  Within this chart all five of the Planck base units are tracked to their current expressions.  Again,  there are just over 201 base-2 exponential notations from that first moment of creation to this day. These two charts are vertically-scrolled.

The third chart, developed in April 2016, is  horizontally-scrolled.  It is a very different experience to follow the progressions of the top rows where the Planck Time and Planck Length scale up to the current time (Age of the Universe) and to the Observable Universe. Here is the key:

At any notation along the scale, one can take the number from Planck Length scale and divide it by the corresponding number from the Planck Time scale and result will approximate the speed of light.

Every notation reflects the speed of light.  Yes, let there be light. In every notation let the light shine. And, may it shine in all the dark places on this earth and beyond. -Amen

On Constructing the Universe From Scratch

UPDATED: SUNDAY, MARCH 20, 2016   Commentary/Reflections (new tab)

An Early Draft


“I have learned that many of the Greeks believe Pythagoras said all things are generated from number. The very assertion poses a difficulty: How can things which do not exist even be conceived to generate? But he did not say that all things come to be from number; rather, in accordance with number – on the grounds that order in the primary sense is in number and it is by participation in order that a first and a second and the rest sequentially are assigned to things which are counted.”
Theano, On Piety (as reported by Thesleff, Stobaeus, and Heeren)


Using the model of the universe generated through the Big Board – little universe Project where there are just over 201 base-2 notations from the singularity of the Planck base units (particularly from Planck Time) to the Age of the Universe, the question to be addressed is, “Which numbers come first and why?” Mathematical logic calls out the most simple-yet-powerful numbers that can be used to build and sustain a highly-integrated universe. Our other assumptions are here. Each of these key numbers and number groups are introduced; each will then become the focus of additional study, further analysis, and the basis for a more-in-depth report about each number. Our initial numbers are:
(1)   3.1415926535897932384626433+  or π or Pi
(2)  74.04804896930610411693134983% or the Kepler Conjecture
(3)  0, 1 where the numbers are: zero and one
(4)  7.356103172453456846229996699812° called the Pentastar gap
(5)  1:1.618033988749894848204 or the Phi ratio
(6)  4.6692016091029906718532 which is a ratio called the Feigenbaum constant
(7)   110 of Stephen Wolfram’s rules
(8)  6.6260709×10−34 J·s or Planck constant plus all related numbers
(9)  Groups of dimensionless constants, all known mathematical and physical constants
(10) 13.799±0.021 billion years, the Age of the Universe

Please note: Links inside the body of the article most often open a new tab or window within a Wikipedia page. For those occasional inks that  do not open new windows, please use the back-arrow key to return to the referring page. All links within the Endnotes will eventually go to source materials if posted on the web.

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Most of us know the universe is infused with numbers. It seems nobody really knows how all these numbers are organized to make things and hold it all together.

In our work with high school students there is a constant demand that our numbers be intellectually accessible. Simplicity is required.  So, it is rather surprising that we ended up engaging the Planck Length (and the other Planck base units) very early in our study of the platonic solids. We also started to learn about base-2 notation and combinatorics. We had to do it. We had divided our little tetrahedron in half so many times, we knew we were in the range of that limit of a length, and we wanted to find a place to stop. Eventually, to get more accurate, we started with the Planck Length, used base-2 exponential notation, and multiplied our way out to the Observable Universe.1 It took just over 201 doublings.  What?  Huh?

That little fact is as unknown as it is incredible  (even as of January 2016 when this article was first posted).

In December 2011 we could find no references to the 201+ notations in books or on the web. We did find Kees Boeke’s 1957 work with base-10 notation. It was a step in the right direction, but it had no lower and upper boundary, no Planck numbers, and no geometry. It had just 40 steps amounting to adding zeroes.

We were looking for anything that could justify our “little” continuum. We didn’t know it at the time, and we later learned that we were looking for those deep relations and systems that give us homogeneity and isotropy, a cosmological constant, and an equation of state. Though we already had put everything, everywhere throughout all time in an ordered relation, we had no theoria, just the praxis of numbers. We tried to set a course to go in the direction of a theory that might bind it all together.

The first 60+ doublings constitute a range that scholars have been inclined to dismiss over the years as being too small;  some say, “…meaninglessly small.” Yet, being naive, it seemed to us that the very simple and very small should be embraced, so we started thinking about the character of the first ten (10) doublings. Trying to understand how to “Keep It Simply Simple,” we were pleasantly surprised to discover that there was so much work actively being pursued by many, many others throughout academia and within many different disciplines to develop the logic of the most simple and the most small.

Within the studies of combinatorics, cellular automaton, cubic close packing, bifurcation theory (with Mitchell Feigenbaum’s constants), the Langlands program, mereotopology and point-free geometry (A.N. Whitehead, Harvard, 1929), the 80-known binary operations, and scalar field theory, we found people working on theories and the construction of the simple. Yet, here the concepts were anything but simple.2

It is from within this struggle to understand how all these numbers relate, we began our rank ordering of all possible numbers. This exercise helps to focus our attention.

Planck Length and Planck Time. One might assume that we would put the five Planck base units among the most important numbers to construct the universe. As important as each is,  it appears at this time that none of them will be among the Top 5. Although very special, the Planck numbers are determined by even more basic and more important concepts and numbers. At the very least, all those numbers will come first.


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First Principles. The work to find the Top Numbers was preceded by an end-of-year report after four years of studying and using the Big Board-little universe charts. That report titled, Top Ten Reasons to give up little worldviews for a much bigger and more inclusive UniverseView 3, was done with comedian David Letterman in mind. He often had a Top Ten on his show.

“#10” for us it is, “Continuity contains everything, everywhere, for all time, then goes beyond.” One of the key qualities to select our most important numbers is the condition of continuity and discontinuity starting with the simplest logic and simplest parts.

A Quick Review of the Top Ten Numbers in the Universe.

Because many scholars have addressed the question, we did a little survey.

Scholars and thought leaders. Our limited survey began with leading thinkers in the academic-scientific community and then thoughtful people from other disciplines:

Base-2 notation. Yes, our work with base-2 notation originated from within a high school. We have no published scholarly articles and there has been no critical review of our emerging model. Nevertheless, we forge ahead with our analysis of numbers and systems.


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Goals. Our singular goal is to try to construct our universe using mathematical logic. We begin with the magic of the sphere. Our #1 number is Pi (π).

#1 = π
Numerical constant, transcendental and irrational all rolled into one

For us Pi (π) seems to be a very good starting point. Non-ending or continuous, it is also  non-repeating which is discontinuous. This most simple construction in the universe requires just two vertices to make the sphere. How does it work? It appears to give form and structure to everything. Using dimensional analysis and scaling laws, this progression of the first 20 notations shows the depth of possibilities for constructions when multiplying by 8. Our open question: In what ways do the  Feigenbaum constants within (bifurcation theory) apply?4

B2 Vertices Scaling Vertices (units:zeroes) Bifurcation* Ratio*
0 0 0 N/A N/A
1 2 8 vertices 0.75 N/A
2 4 64 1.25 N/A
3 8 vertices 512 1.3680989 4.2337
4 16 4096 (thousand:3) 1.3940462 4.5515
5 32 32,768 1.3996312 4.6458
6 64 262,144 1.4008286 4.6639
7 138 2,097,152 (million:6) 1.4010853 4.6682
8 256 16,777,216 1.4011402 4.6689
9 512 134,217,728 1.401151982029 4.6689
10 1024 1,073,741,824 (billion:9) 1.40115450223 4.6689*
11 2048 8,589,934,592
12 4096 68,719,476,736  *This bifurcation and
13 8192 549,755,813,888  ratio columns come
14 16,384 4,398,046,511,104 (trillion:12)  from a Wikipedia article
15 32,768 35,184,372,088,832  about Feignebaum’s
16 65,536 281,474,976,710,656  constant.
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(18)

(discussion begins on the next page)
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Discussion.  Pi still holds many mysteries waiting to be unlocked.  Among all numbers, it is the most used, the most common, and the most simple but complex.  We assume, that along with the other mathematical constants, pi (π) is a bridge or gateway to infinity. We assume it is never-repeating and never-ending.  It is “diverse continuity.”  There are enough scaling vertices within ten doublings to construct virtually anything.  So, to analyze a possible logical flow, any and all tools that have something to do with pi (π) will be engaged. Spheres2Again, among these tools are combinatorics, cellular automaton, cubic close packing, bifurcation theory (with Mitchell Feigenbaum’s constants), the Langlands program, mereotopology and point-free geometry (A.N. Whitehead, Harvard, 1929), the 80-known binary operations, and scalar field theory.   Perhaps we may discover additional ways to see how  pi gives definition — mathematical and geometric structure — to our first 60-to-67 notations. What are the most-simple initial conditions?

More Questions. What can we learn from a sphere? … by adding one more sphere? When does a tetrahedral-octahedral couplet emerge? When do the tessellations emerge? At the third notation with a potential 512 scaling vertices, surely dodecahedral and icosahedral forms could emerge. Within the first ten notations with over one billion potential vertices, could our focus shift to dynamical systems within the ring of the symmetric functions?

#2 = Kepler’s Conjecture


Not a very popular topic, one might ask, “How could it possibly be your second choice?” Even among the many histories of Kepler’s voluminous work, his conjecture is not prominent. To solve a practical problem — stack the most cannon balls on the deck of a ship —  he calculated that the greatest percentage of the packing density to be about 74.04%. In 1998 Professor Thomas Hales (Carnegie Mellon) proved that conjecture to be true.  By stacking cannon balls, all the scholarship that surrounds cubic close packing (ccp) enters the equation.  The conjecture (and Hales 1998 proof) opens to a huge body of current academic work.5 There we found this animated illustration on the right within Wikipedia that demonstrates how the sphere becomes lines (lattice), triangles, and then a tetrahedron. With that second layer of green spheres emerges the tetrahedral-octahedral couplet.

Sphere to tetrahedron-octahedron couplet
Attribution: I, Jonathunder

This image file (right) is licensed under the Creative Commons Share-Alike 2.5 Generic license.

Revisions. As we find experts to guide us within those disciplines where pi has a fundamental role, undoubtedly sections of the article will be substantially re-written and expanded. Our goal has been to find the most logical path by which all of space and time becomes tiled and tessellated. Perhaps there is a new science of the  extremely small and  the interstitial that will begin to emerge. These just might be foundations of foundations, the hypostatic, the exquisitely small, the ideal.6  We plan to use all the research from Kepler to today, particularly the current ccp (hcp and fcp) research from within our universities, in hopes that we truly begin to understand the evolution of the most-simple structures.


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#3 = 0, 1

Circle to CoordinatesThe numbers, zero (0) and one (1) begin the mapping of pi to Cartesian coordinates. Beginning with a circle, each sphere is mapped to two-or-three dimensional Cartesian coordinates.  It is the beginning of translating pi to sequences and values. The first iterative mapping  is a line,  then a triangle,  then a tetrahedron, then an octahedron.  When we focus solely on this subject, with experts to guide us, perhaps we can engage the study of manifolds that are homeomorphic to the Euclidean space.6

#4 = Pentastar gap = 0

0.12838822… radians

7.356103172453456846229996699812179815034215504539741440855531 degrees

PentastarThe little known 7.356103 degree gap is our fourth most important number, the possible basis for diversity, creativity, openness, indeterminism, uniqueness and chaos.7 That Aristotle had it wrong gives the number some initial notoriety; however, it is easily observed with five regular tetrahedrons which would have eight vertices.  It appears to be transcendental, non-repeating, and never ending. Where the tetrahedron with four vertices and the octahedron with six have been been whole, ordered, rational, and perfect, tessellating and tiling the entire universe, the potential for the indeterminate which has the potential to become the chaotic resides somewhere deep within the system. We believe that place just may be right here.

Within this infinitesimal space may well be the potential for creativity, free will, the unpredictable, and the chaotic.  Here may well be the basis for broken symmetries. Of course, for many readers, this will be quite a stretch. That’s okay. For more, we’ll study chaotic maps and the classification of discontinuities.

#5 = phi = φ = The Golden Ratio

Phi-formula = φ = 1:1.618033988749894848204586

Of all the many articles and websites about the golden ratio and sacred geometry, our focus is on its emergence within pi and within the platonic solids.  Phi is a perfection.  It is a mathematical constant, a bridge to infinity. We are still looking to see if and how phi could unfold within the tetrahedral-octahedral simplex. Could that answer be within Petrie polygons? The magic of the golden ratio does unfold with the dodecahedron, the icosahedron, and the regular pentagon. Within this listing, phi has bounced back and forth with the Pentastar gap. Which manifests first? Is it manifest if it is inherent?

Starting with this article, we have begun an active study of Phi and its relations to pi and the Platonic solids.  Although there are many, many papers about phi, none are from our special perspective of 201+ notations.



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#6 = Feigenbaum constants

δ = 4.669 201 609 102 990 671 853 203 821 578

We are the first to admit that we are way beyond our comfort zone, yet to analyze and interpret the processes involved within each of the doublings, each an exponential notation, requires tools. This Feigenbaum constant gives us a limiting ratio from each bifurcation interval to the next…. between every period doubling, of a one-parameter map. We are not yet sure how to apply it, but that is part of our challenge.

It gives us a number. It tells us something about how the universe is ordered. And, given its pi connection, we need to grasp its full dimensions as profoundly as we can. We have a long way to go.

#7 = Rule 110 cellular automaton

Wolfram Rule 110There are 255 rules within the study of elementary cellular automaton.  Rule 110 was selected because it seems to define a boundary condition between stability and chaos.   All 255 rules will be studied in light of the first ten notations to see in what ways each could be applied. Any of these rules could break out and move up or down within this ranking. Steve Wolfram’s legacy work,  New Kind of Science (NKS) is online and here he lays the foundations for our continued studies of these most basic processes within our universe.8

#8 = Max Planck numbers

We have been working on our little model since December 2011. Over the years we have engaged a few of the world’s finest scientists and mathematicians to help us discern the deeper meaning of the Planck Base Units, including the Planck Constant. We have studied constants from which the Planck numbers were derived, i.e. the gravitational constant (G), the reduced Planck constant (ħ), the  speed of light in a vacuum (c)the  Coulomb constant, (4πε0)−1 (sometimes ke or k) and the Boltzmann constant (kB sometimes k). This engagement continues. We have made a very special study of the  Planck Base Units,  particularly how these numbers work using base-2 exponential notation and with the Platonic solids.  We had started with the Planck Length, then engaged Planck Time.  Finally in February 2015, we did the extension of Planck MassCharge, and, with a major adjustment to accommodate simple logic, Temperature.  We have a long, long way to go within this exploration.  Essentially we have just started.9

Notwithstanding, there is a substantial amount of work that has been done within the academic and scientific  communities with all the Planck numbers and those base numbers that were used to create the five Planck base units.  Perhaps chemistry professor, C. Alden Mead of the University of Minnesota began the process in 1959 when he first tried publishing a paper using the Planck units with serious scientific intent. Physics professor Frank Wilczek of MIT was the first to write popular articles about the Planck units in 2001 in Physics Today (312, 321, 328)From that year, the number of articles began to increase dramatically and experimental work that make use of these numbers has increased as a result.                                                                             


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 #9 = Mathematical & physical constants

Given we started with pi (π), it should not be surprising that we are naturally attracted to any real data that shows pi at work such as the Buckingham π theorem and the Schwarzschild radius.

We will also bring in Lord Martin Rees “Six Numbers” as well as the current work within the Langlands programs,  80 categories of binary operations, scalar field theory, and more (such as the third law of thermodynamics and zero degrees Kelvin).

In studying the functionality of these many numbers, especially those among the dimensionless constants, we believe this list will evolve and its ordering will change often. In searching the web for more information about about dimensionless constants, we came upon the curious work of Steve Waterman and an emeritus chemistry professor at McGill University in Montreal, Michael Anthony (Tony) Whitehead.  I showed their work to a former NIST specialist and now emeritus mathematics professor at Brown University, Philip Davis.  He said, “There are always people who wish to sum up or create the world using a few principles. But it turns out that the world is more complicated. At least that’s my opinion.   P.J.Davis”  Of course, he is right; Einstein did a good job with e=mc2.  Because claiming to find all the physical constants derived by using pi, the isoperimetric quotient, close cubic packing and number density is not trivial10, we’ll be taking a second look. Perhaps they are onto something!  We have brought their work out in the open to be re-examined and in so doing we will re-examine over 140 physical and mathematical constants. This work is also ongoing.

#10 = The Age of the Universe

13.799±0.021 billion years

This number is important because it creates a boundary condition that is generally recognized for its accuracy throughout the scientific and academic communities. Though it may seem like an impossibly large number of years, it becomes quite approachable using base-2 exponential notation.  Without it, there is no necessary order of the notations.

Although there are many different measurements of the age of the universe, for our discussions we will use 13.799±0.021 billion years. The highest estimate based on current research is around 13.82±0.021 years. Also, within this study there are some simple logic problems. In 2013, astrophysicists estimated the age of the oldest known star to be 14.46±0.8 billion years.

Notwithstanding, using base-2 exponential notation all these measurements come within the 201st notation. At the 143rd notation, time is just over one second. Within the next 57 doublings, we are out to the Age of the Universe. So, with the Planck Time as a starting point and the Age of the Universe (and our current time) as the upper boundary, we have a container within which to look for every possible kind of doubling, branching and bifurcation. We can study hierarchies of every kind, every set, group or system. Eventually we can engage holomorphic functions within our larger, ordered context, i.e. the seen-and-unseen universe.11                    


 This article was started in December 2015.  It’s still in process. Your comments are invited.

Page 9 of 10                                                                                                                    EARLY DRAFT.

Endnotes about our open questions, plus a few references:

Our Initial exploration of the types of continuity and discontinuity: Continuous-discrete, continuous-quantized, continuous-discontinuous, continuous-derivative… there are many faces of the relations between (1) that which has a simple perfection defined in the most general terms as continuity yet may best understood as the basis of order and (2) that which is discrete, quantized, imperfect, chaotic, disordered or otherwise other than continuous.   These are the key relations that open the gateways between the finite and infinite.

Questions:  What is a continuum?  What is a discrete continuum?

2  We are simple, often naive, mathematicians. We have backed into a rather unique model of the universe. To proceed further we will need to understand much more deeply a diverse array of relatively new concepts to us; we are up for the challenge.  We have introduced just a few of  those many concepts that attempts to define the very-very small and/or the transformations between the determinant and the indeterminant. There will be more!

3  Of the Top Ten Reasons, the first three given are our first principles. We know it is an unusual view of life and our universe. The sixth reason advocates for a Quiet Expansion of our universe whereby all notations are as active right now as they were in the very earliest moments of the universe.  When space and time become derivative, our focus radically changes.  It opens a possible place for the Mind down within the small-scale universe.  Our current guess is between the 50th and 60th notations.  The archetypes of the constituents of our beingness are between notations 67 (fermions) to notation 101 (hair) to notation 116 (the size of a normal adult).  Then, we live and have our sensibility within notation 201, the current time, today, the Now.  So, this unusual view of the universe has each of us actively involved within all three sections of the universe: small scale, human scale, and large scale.  To say that it challenges the imagination is a bit of understatement.

4 Open Questions. There are many open questions throughout this document. It is in process and will surely be for the remainder of my life. All documents associated with this project may be updated at anytime. There should always be the initial date the document was made public and the most recent date it was significantly updated. Although the Feigenbaum constants are our seventh number selected (and there are more links and a little analysis there), we will attempt to find experts who can guide us in the best possible use of these two constants within our studies. Bifurcation, it seems, has an analogous construct to cellular division, to chemical-and-particle bonding, to cellular automaton (especially Rule 110,)  and to the 80 categories of binary operations.

5 Wikipedia, ccp, and genius. Jimmy Wales is the founder and CEO of Wikipedia. His goal is to make the world’s knowledge accessible to the world’s people. He has a noble vision within precarious times. In order to be published within Wikipedia, the material has to have its primary sources of information from peer-reviewed publications. As a result, Wikipedia is not where “breakthrough” ideas will first be presented. Blogging areas like WordPress are a more natural spot and Google quickly indexes all those blogging areas. It took only a day before they found this article. So with a little ingenuity one can quickly find many new references to new ideas and then go to Wikipedia to find the experts on that subject. Prior to this research, we had barely scratched the surface of ccp. We did not know about the Feigenbaum constants or Kepler’s conjecture. For sure, we had never seen the cannonball stacking illustration that helped us to visualize the process by which a sphere becomes a lattice, becomes a triangle, and then becomes a tetrahedron. We are quite confident that our first four numbers are the right selections possibly even within the right order. If you believe otherwise, of course, we would love to hear from you.


Page 10 of 10                                                                                                               IN PROCESS

6 A hypostatic science. Our small-scale universe, defined as the first 1/3 of the total notations, ranges from notation 1 to just over 67. It is established only through simple logic and simple mathematics. Because it cannot be measured with standard measuring tools or processes, validating its reality requires a different approach. Because it cannot be measured with standard measuring tools or processes, validating its reality requires a different approach. Our first indication that it may be a reality is found between notations 143 and 144 at exactly one second where the speed of light “can be made” to correspond with the experimental measurement of the distance light travels in a second. Currently it appears to be one notation off which could be as brief as just one Planck Time unit.

One of our next tasks is to carry that out to a maximum number of decimal places for Planck Time and Planck Length, and then to study the correspondence to a Planck second, a Planck hour, a Planck Day-Week-Month, a Planck Light Year, and finally to the Age of the Universe and the Observable Universe.

Our goal is to determine if this is the foundational domain for the human scale and large-scale universe. We are calling this study a hypostatic science because it is a study of the foundations of foundations.

7 From SUSY to Symmetry Breaking and Everything In Between. One of the great hopes of the Standard Model and many of the CERN physicists is that supersymmetries will be affirmed and multiverses will wait. Within the Big Board-little universe model, their wish comes true. Plus, they gain a reason for quantum indeterminacy and embark on a challenge to apply all their hard-earned data acquired to embrace the Standard Model to the most-simple, base-2 model.

Here are four of our references through which we learned about the heretofore unnamed pentastar gap.  The Lagarias-Zong article (#4) is where I learned about Aristotle’s mistake.
1. Frank, F. C.; Kasper, J. S. (1958), “Complex alloy structures regarded as sphere packings. I. Definitions and basic principles”, Acta Crystall. 11. and Frank, F. C.; Kasper, J. S. (1959), and “Complex alloy structures regarded as sphere packings. II. Analysis and classification of representative structures”, Acta Crystall. 12.
2.  “A model metal potential exhibiting polytetrahedral clusters” by Jonathan P. K. Doye, University Chemical Laboratory, Lensfield Road, Cambridge CB2 1EW, United Kingdom, J. Chem. Phys. 119, 1136 (2003) Compete article, as a PDF:‎
3. “Polyclusters” by the India Institute of Science in Bangalore, illustrations and explanations of crystal structure. PDF:
4. “Mysteries in Packing Regular Tetrahedra” Jeffrey C. Lagarias and Chuanming Zong. 

8 Cellular Automaton.  Although the discipline is intimately part of computer science, its logic and functions are entirely analogous to mathematical logic, functions, and binary operations. We have just started our studies here with great expectations that some of this work uniquely applies to the first ten notations.

9 The Planck Platform.  All the numbers associated with the generation of the Planck Constant and the five Planck base units, plus the Planck units unto themselves are grouped together until we can begin to discern reasons to separate any one number to a notation other than notation 1.

10 The Magic Numbers.  Mathematical constants, dimensionless constants and physical constants are studied in relation to the isoperimetric quotient, close-cubic packing, number density and to bifurcation theory and to the 80 categories of binary operations. We will working with the processes developed by geometer, Steve Waterman, and chemistry professor, Michael Anthony Whitehead and the generation of the 142 physical constants.

11 The first 67 notations. Given the work of CERN and our orbiting telescopes, we can see and define most everything within notations 67 to just over 201. The truly unseen-unseen universe, defined only by mathematics and simple logic, are: (1) the dimensionless constants, (2) that which we define as infinite, and (3) the first 60-to-67 notations. It is here we believe isotropy and homogeneity are defined and have their being. It is here we find the explanation for the most basic cosmological constant. It is here the Human Mind takes its place on this grid which claims to include “everything-everywhere-for-all-time.”


Please note: We are still working on this article. A running commentary is being developed within the LinkedIn blogging area for Bruce Camber. Besides editing the overall document, we’ll are still working on the end notes using some of these reference materials.

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?” 


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.”

Just what’s happening here?

For over 100 years, the Planck Length was virtually ignored.  That length was so small, it seemed meaningless.¹  Nothing and nobody could measure it.  It was just a ratio of known constants.  Yet, it created a conceptual limit of a length which gave a New Orleans high school geometry class a goal or a boundary beyond which they did not have to go Recent measurements from the Hubble telescope provided the upper limit so this class could define the number of base-2 exponential notations from the smallest measurement of a length to the Observable Universe, the largest.

Within that continuum everything can be placed in a mathematical and geometric order.  Everything.  That is, everything in the known universe. The most remarkable discovery was that it took no more than 205.1 base-2 exponential notations.  It would be our very first view of an ordered universe. And, it readily absorbed all of our worldviews.

That was December 19, 2011.  Formally dubbed, “The Big Board – little universe,” we then asked, “What does it mean?  How do we use it?”  When we engaged the experts, they appeared a bit puzzled and seemed to be asking, “Why haven’t we seen this chart before?” Those who knew Kees Boeke’s 1957 book, Cosmic Vision,  asked, “How is it different from Boeke’s work using base-10 exponential notation?”   That was a challenge. Our best answers to date – it’s more granular, it mimics chemical bonding and cellular reproduction; it’s based on cascading, embedded, and combinatorial geometries – were not good enough.  In April 2012 even the Wikipedia  experts  (Steven Johnson,  MIT) protested.  He classified our analysis as  “original research” and within a very short time our Wikipedia article was taken down.  Others called it idiosyncratic (John Baez, UC-Riverside), but they did not tell us what was wrong with our analysis.

“Let’s just make as many observations as we can to see what can we learn?”  A NASA senior scientist and a French astrophysicist helped us with our calculations.  Their results gave us a range; the low was 202.34 notations and the high, 205.11.  We could identify many things between the 66th notation and the 199th notation.  But, there were blanks everywhere so we got busy speculating about them. The biggest group of empty notations was from 2 to about 65. We asked, “Conceptually, what could be there?”  Max Planck may have given us a clue when in 1944, in a speech in Florence, Italy; he said, “All matter originates and exists only by virtue of a force which brings the particle of an atom to vibration and holds this most minute solar system of the atom together. We must assume behind this force the existence of a conscious and intelligent mind. This mind is the matrix of all matter.” (The Nature of Matter, Archiv zur Geschichte der Max-Planck-Gesellschaft, Abt. Va, Rep. 11 Planck, Nr. 1797, 1944)  Matrix is a good word. Throughout history others have described it as the aether, continuum, firmament, grid, hypostases, plenum and vinculum.

We made two columns and within the top notations, 100-to-103, we found humanity.  That seemed politically incorrect until we discovered the cosmological principle that the universe is isotropic and homogeneous.  So, if it is true for us, it would also have to be true for “everybody” anywhere in the universe.

This is high school.  We had been following embedded geometries, particularly the tetrahedron and octahedron.  We observed a tetrahedral-octahedral-tetrahedral chain.  In no more than 206 layers everything in the universe is bound together.  We learned about tilings and could see that the four hexagonal plates we discovered within the octahedron also created tiles in every possible direction.

“What is this all about?  Just what’s happening here?”

We knew we were imposing a certain continuity and order with our mathematics (base-2 exponential notation), and we were also conveying certain simple symmetries and relations with our geometries.  That wrapped our work within a conceptual framework that was quite the opposite of the chaotic world of quantum mechanics.  Our picture of the known universe was increasingly intimate and warm; it was highly-ordered and had immediate value. And the more we looked at it, the more it seemed that all of science and life had an inherent valuation structure.   Here numbers became the container for time, and geometries the container for space.  How each was derived became our penultimate challenge. Ostensibly we had backed into a model of the universe and somehow we began to believe that if we could stick with it long enough, it just might ultimately give us some answers to the age-old question, “What is life?”

We had strayed quite far from those tedious chapters in our high school geometry textbook.  Yet, we also quickly discovered how little we knew about basic structure when we attempted to guess about the transitions from one notation to the next.  We asked, “How can we get from the most-simply defined structure, a sphere, to a sphere with a tetrahedron within it?”  We needed more perspective.

Who is doing this kind of work?”  We began our very initial study of the Langlands Program and amplituhedrons. Then, we walked back through history, all the way to the ancient Greeks and we found strange and curious things all along the way. There were the circles of Metatron that seemed to generate the five platonic solids. “How does that work? Are there experts who use it?  How?”  We still do not have a clue.  All the discussions about infinitesimals seemed to come to a crescendo with the twenty-year, rancorous debate between Thomas Hobbes and John Wallis.  It was here that we  began to understand how geometry lost ground to calculus and algorithms.

The Big Board-little universe was awkward to use.  It was five feet tall and a foot wide.  Using the Periodic Table as a model, an 8½-by-11 chart was created and quickly dubbed, The Universe Table.  It would be our Universe View into which we could hopefully incorporate any worldview.  It was an excellent ordering and valuation system.

Frank_WilczekThough the Planck Length became a natural unit of measurement, a limit based on known universal constants, it wasn’t until Frank Wilczek of MIT opened the discussion did things really begin to change. In an obscure 1965 paper by C. Aldon Mead, his use of the Planck Length was pivotal. In 2001 Wilczek’s analysis of Mead’s work and their ensuing dialogue was published in Physics Today. Wilczek, well on his way to obtaining a Nobel Prize, then began writing several provocative articles, Scaling Mt. Planck.  Even his books were helpful. In January 2013 he personally encouraged us on our journey.

In 1899 Max Planck began his quest to define natural units.  At that time he took some of the constants of science and he started figuring out natural limits based on them. There are now hundreds that have been defined. Each is a ratio and each can be related to our little chart and big board.  The very nature of a ratio seems to be a special clue. It holds a dynamic tension and suggests that the relation is primary and all else is derivative.

We have a lot of work in front of us!  And, we are up for the challenge.

Who would disagree with the observation that our world has deep and seemingly unsolvable problems?  The human future has become so problematical and complex, proposals for redirecting human energies toward basic, realizable, and global values appear simplistic.  Nevertheless, the need for such a vision is obvious. Rational people know that there is something profoundly missing. So, what is it? Is it ethics, morality, common sense, patience, virtues like charity, hope and love?  We have hundreds of thousands of books, organizations and thoughtful people who extol all of these and more.  The lists are robust.  The work is compelling, but obviously none of it is quite compelling enough.

First, it has to be simple.  Our chart is simple.

Second, it has to open up to enormous complexities. Using simple math, by the tenth notation there are 1024 vertices. We dubbed it the Forms or Eidos after Plato. The 20th notation would add a million vertices; we called it Structure. The 30th adds a billion new vertices. We ask, “Why not Substances?” The 40th adds a trillion so we think Qualities. The 50th adds a quadrillion vertices. We speculate Relations. By the 60th notation there are no less than a total of 2 quintillion vertices with which to create complexity. We speculate Systems and within Systems there could be The Mind. As if a quintillion vertices is not enough, the great physicist,  Freeman Dyson, advises us that really we should be multiplying by 8, not by 2, so potential complexity could be exponentially greater.

Three, it should be elegant.  There is nothing more elegant than complex symmetries interacting dynamically that create special harmonies.  We can feel it. And, we believe the Langlands program and amplituhedrons will help us to further open that discussion.

What is life?  Let us see if we can answer very basic questions about the essence of life for a sixth grade advanced-placement science class and for very-average, high-school students.  These are our students.  The dialogue is real.  The container for these questions and answers is base-2 exponential notation from the Planck Length to the Observable Universe.  To the best of our knowledge, December 19, 2011 was the first time base 2 exponential notation was used in a classroom as the parameter set to define the universe.  Though our study at that time was geometry, this work was then generalized to all the scientific disciplines, and more recently it was generalized to business and religion.  So, as of today, readers will see, and possibly learn, the following:

1.  See the totality of the finite, highly-ordered, profoundly inter-related, very-small universe where humanity is quite literally back in the middle of it all.

2.   Engage in speculations about the Infinite and infinity whereby the Creative and the Good take a prominent place within the universal constructs of Science.

3.   Extend the scale of the universe by redefining the Small Scale and engaging in speculations about the deep symmetries of nature, giving the Mind its key role within Systems, and demonstrating the very nature of homogeneity and isotropy.

4.   Adopt an integrated universe view based on Planck Length and Planck Time such that Science, Technology, Engineering and Mathematics are demythologized,  new domains for research are opened, and philosophies and religions are empowered to be remythologized within the constraints of universals and constants.

People ask, “Aren’t you getting ahead of yourself?  Isn’t this a bit ambitious?”  The concepts of space and time raise age-old questions about who we are, where we have come from, and where we are going.  With our little formulation, still in its infancy, we are being challenged to see life more fully and more deeply.  And so we reply, “What’s wrong with that?”


1   Physics professor, Joe Wolfe (Australia), says, “Nothing fundamentally changes at the Planck scale, and there’s nothing special about the physics there, it’s just that there’s no point trying to deal with things that small.  Part of why nobody bothers is that the smallest particle, the electron, is about 1020 times larger (that’s the difference between a single hair and a large galaxy).

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