Amelino-Camelia, Giovanni

Giovanni Amelino-Camelia Screen Shot 2020-05-10 at 5.08.24 PM

Articles: New Scientist, 2005, Big ideas: Relativity
Google Scholar 
Lecture: 1999, PSU: Are we at the dawn of quantum-gravity phenomenology?
YouTube: The Planck Scale II

Most recent email: 8 May 2020 at 4:45 PM

Dear Prof. Dr. Giovanni Amelino-Camelia:

You were one of the first people we asked about our simple little formulation of base-2 notations from the Planck units to the current age of the universe. It came out of a high school, so we were quite unsure what we had done. Just over 202 notations or doublings gave us a simple outline of the universe. After a bit of study, and with a most naive understanding of the issues of cosmology and theoretical physics, we knew we had a very steep learning curve. And, best of all, you didn’t laugh at us!

We are making slow, very slow, progress, but from an outline, today we may have the makings of a model.

Given your involvements with FQXi, I wonder if you might look at our very first time to enter such a sophisticated arena of scholars: 
(The first five pages is the article. There are five pages of references.)

Do you think that it could have some merit?

Thank you.

Most sincerely,


Second Email: March 26, 2018, 11:52 AM

Dear Prof. Dr. Giovanni Amelino-Camelia –

Almost five years have past since our first exchange (below) and three since my last note. 

My work with high school students (and now some college students and professors) continues. Can we attempt to understand fluctuations right down to the Planck-length? Would the rate be one fluctuation per each unit of Planck time? 

Our general work is here:

We have evolved with a simple-but-highly-integrated mathematical chart of the universe based on simple doublings of four Planck units: length/time and mass/charge. In 202 steps we go from the smallest to the largest possible measurements in what appears to be a logical progression:

Would you have a moment to advise us?

Most sincerely,


Second email: Thursday, February 5, 2015 at 10:47 PM

1. Phenomenology of Philosophy of Science:
2. The principle of relative locality: 
3. Quantum-Spacetime Phenomenology:
4. Against spacetime:

Dear Prof. Dr. Giovanni Amelino-Camelia:

You are on my radar again. I have downloaded several more files to my Giovanni folder! Can spacetime be derivative of continuity and symmetry? Might spacetime always require dynamics — harmonics — that are held in tension to create a spacetime moment?

Nonsense questions, but increasingly it seems that our container, finite universe, points in such directions.

In 1957 when Kees Boeke’s book, Cosmic Vision,The Universe in 40 Jumps, was published, he had the attention of prominent scientists. Then, the Eames film, the Morrison’s book, Powers of Ten, the IMAX (Smithsonian) movie (guide), and the Huang’s scale of the universe opened a conceptual door for everyone. They could have a systematic view of the entire universe.

It was a paradigm shift; all the attention was justified.

Most of the world’s people live within their OwnView. Even though subjective and often quite naïve, the solipsistic among us lift it up as the only 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.

Our high school geometry class developed what just may be the second systematic UniverseView. It is quite a bit more granular than Boeke’s work and it was initially based on simple embedded and nested geometries. This work began in March 2011, used base-2 exponential notation and emerged with about 202+ doublings, layers, notations, or steps. It properly began at the Planck Length and Planck Time and then expanded to the Observable Universe and the Age of the Universe respectively. This fully-integrated UniverseView emerged in December 2011 and was officially dubbed, “Big Board – little universe.”

First, we thought our UniverseView with its 202+ notations would be a good container for Science-Technology-Engineering-Mathematics (STEM) education. It put everything within a simple ordering system. Then, in January 2012, in the process of trying to find scholarly references to understand the foundations of our work, we discovered Kees Boeke. In so many ways, it was a vindication that we were not totally idiosyncratic. Somebody had been here before us, but not too, too much was done to extract meaning from the model.

With our simple geometries and math, we thought this was a real ordering system that could help us make sense of the meaning of things within that historic tension to understand who we are, why we are, where we are going, and the meaning and value of life. Given this model has a starting point and an end point, this UniverseView certainly suggests that the universe is finite. To engage the Infinite it appears that we hold the objective and subjective in a creative balance called geometry, calculus and algebra through which we can more fully discover relations.

Boeke’s work is great, but it is not relationally integrated; it just adds (or subtracts) zeroes. It takes base-2 exponential notation (3.333+ times more granular) to begin seeing the simplest continuities, relations and dynamics within each notation. Base-2 is the heart and spirit of cellular division, chemical bonding and complexification (1 & 2). It is here we begin to create a truly relational, integrated and functional UniverseView. It is here that we find the rough-and-tumble within science.

So, although our base-2 UniverseView is the second UniverseView, it seems to hold greater promise. And though we fully recognize that our preliminary models are still just a crack in the doorway, what a sweet and simple opening it is. Kepler should be proud.

We are now just starting to grow our little association with real-and-graciously-open scholars. Just possibly, this work will have as much impact as base-10. Maybe more… if it somehow truly captures the spirit of 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?”

A simple introduction to our simple work is here:

Shall I keep you abreast of any developments? I have three lists: annual, quarterly, or monthly basis. I just placed you on my annual list. Is that OK?

Most sincerely,


First email: July 17, 2013

Dear Prof. Dr. Giovanni Amelino-Camelia:

You never know who is going to be reading those intimate phenomenological words of yours, a most personal exploration of your most recent insights (1999 and 19 slides). Fascinating read.  Just fascinating.

Anything new to add?

I’ll be in Geneva in August when everyone else (but a few) will be traveling!



The earliest universe… the first three of 202 doublings

72 of 202: A grid-matrix-system for everything, everywhere for all time, Not a theory or vision, just math

Editor’s Note: A five-part group of articles, the other four are as follows:
 •  Einstein would have gone even further if he had an integrated view of the universe.
 •  Our Open Letter To You: An introduction to a very simple chart of the universe…
 •  Concepts about which we can say, “These are our first principles.”
 • Big Bang Theory Diffused

Emergence-Growth: Grasping the boundary conditions between the finite and infinite

By Bruce Camber, April 2018


Abstract: To understand the most simple form of emergence and growth still requires engaging a wide diversity of ideation including phase and scaling transitions, initial conditions, and boundary conditions, all between the finite and the infinite. Also, this analysis is set within work that started in December 2011 to define the 202 doublings (or 202 base-2 notations) from the Planck scale (Planck Length/Planck Time and Planck Mass/Planck Charge) through to the current size of the universe, age of the universe, mass of the universe, and charge of the universe. If the “infinitely dense, infinitely hot” conceptual framework can be diffused within a natural inflation, and the dynamics of emergence and growth become self-evident-and-simple, new questions and insights about the very nature of reality are opened.

Is this the first manifestation of space-time?
Are these the first structures of space-time?

Observation. At the time of this writing, only two references came up when googling the words, “boundary conditions between the finite and infinite.”  Both refer to our earlier work. [1] Without those all-important “quotes” as delimiters, there are 242,000 references that discuss how combinations of those words apply to virtually every discipline under the sun. All have a common starting point. Yet, I have yet to become aware of an analysis by scholars of such a starting point between the finite and infinite. [2]

Visualizations of a common starting point. Two initial images open the analysis. Both have been analyzed within many pages of this website. [3] Though not widely affirmed,  it will continue to be argued here, that these initial ratios-then-structures:
(1) create space-and-time,
(2) extend our understanding of the very nature of light,
(3) render the first space-time moment of continuity, symmetry and harmony (infinity), and
(4) create a foundation from which all things as things are made.

Our initial visualization is captured by the two images just above.

The 202 base-2 notations. In this study each of the 202 doublings (or notations) is taken separately to begin to discern the manifold and unique possibilities (diversity) of emergence and growth. [4] Within each doubling the logic from the prior doubling unfolds-and-enfolds in the next doubling. Here, my intuitions tell me for the first time that number theory becomes shape theory which becomes dynamic theory.

Pi. Given the special class that pi has among dimensionless constants, its simplicity and ubiquity call out for special recognition as the first manifestation within our continuity equations. [5]

Taken as a given. The Planck scale defines the first instance of a physical manifestation and the first illustration (above left) seems to be the best possible visualization of the first three steps; and the tetrahedral-octahedral image (above right) gives us clues for the next few steps.

Couplet. This tetrahedral-octahedral complex is currently called a couplet in the spirit of a literary expression, whereby two lines of verse, usually in the same meter and joined by rhyme, form a unit. It instantiates as a range of dynamics even within this most simple configuration and has a simple thrust.

Planckspheres. The concept of a Planck sphere (PlanckSphere or plancksphere) has been used by physicists and geometers who in some manner engaged the 1899 work of Max Planck to define fundamental natural units for length, mass, time and energy. [6]  It all begins with the generation of one circle-sphere.

This it would seem is a key action of the first notation.

With each successive sphere, a progressive concept of space-time emerges. Within this model of the universe, spheres are being generated one at a time, just as imaged above, making the space-time, mass-energy that defines an expanding universe.

Can you picture this universe quite literally being constantly (the expansion of the universe) filled with indivisible spheres that have a very small energy and mass and are themselves so small and fast they seemingly penetrate everything instantly? In this model of the universe, here is the edge of creation, dark matter, dark energy, once known as the ubiquitous aether. Now, can we picture them organized by the most basic geometries? Could it be that these are the first three, always-active, always-present, notations that are constantly defining our universe?  Could this be Neil Turok’s perpetual bang?

Formulas. The formulas that create this moment are many. The first, of course, is the ratio that defines the dimensionless constant known as pi. In this model that exquisite, never-ending, never-repeating ratio bridges the finite and infinite and it is first defined by Planck Length (the embedded link is an early attempt to understand it — we now read there are some who have defined Planck Length without the use of the gravitational constant) and Planck Time.  Within this model complexification comes very quickly; and within this website, it will take awhile to unravel our earliest understanding of it. [7]  Also, other dimensionless constants appear to be present. I believe in some manner of speaking, Euler’s identity is at work and that the universe is foundationally defined by exponentiation.

Speculations.  The sole action of the first notation is to generate basic spheres; for now, these spheres are called planckspheres  The piling on as imaged within that sequence defines the foundational notational activity of the #0 notation.

Being entirely speculative, the first doubling is the form of structure whereby the first lines and triangles and tetrahedrons are created and there is an emergence of projective geometries.

The second doubling (second notation) is the form for the tetrahedral-octahedral couplet and the earliest beginnings of Euclidian geometries.

Yes, it would seem that these are the footings for number theory, geometry, and rest of mathematics. Here are the foundations for the Langlands programs. Here are the makings for fractals. And, of course, I could go on and make even more outrageous statements, yet let us save that for another day.

These first three steps are quite enough to get a discussion going!

See A0, A1, A2, A3 for more.  Why not? Why now?

Please note as of 7 May 2018: Many more scholarly papers are to be distilled regarding period-doubling bifurcation and boundary conditions. This page will continue to be updated as a result of incorporating the insights and knowledge of those scholars.

Endnotes & Footnotes:

[1]  Both references are the same article; one is within this website and other in LinkedIn:
____ •  Simple Logic & Math – Our Universe In 202 Doublings:
____ •  Simple | Bruce Camber | Pulse | LinkedIn

This discussion continues [a][b], [c], [d][e].

[2] This work unwittingly jumps out of the current flows of the scholarly community. It is idiosyncratic. Yet, being even more speculative, it seems to follow that these spheres are the foundations for the finite-infinite transformation, the homogeneous and isotropic nature of things, the historic struggles to define the aether, a grid, the matrix, or the plenum; and cosmic microwave background radiation, dark energy, and dark matter. There are triangular, square, and hexagonal plates the are created within that tetrahedral-octahedral complex that literally tile and tessellate the universe. You have to observe it to believe it.

[3]  An image of the tetrahedral-octahedral cluster (on the top, on the right) was used in the original chart in December 2011. The students made many models like this one, however, the very first models like it emerged in 1997.  The dynamic image on the left, using the Graphics Interchange Format (GIF), was first used in January 2016 within an article about numbers. There are many discussions about it.

[4] One of the goals of this project is to analyze each doubling (notation, step, group) and to link it to scholarly work that defines the notation better than any other.  Even though the work started in December 2011, this project is still within its earliest stages of development.

[5] There are many analyses of pi within this website, however, our page for Pi Day on March 14 will be continually enhanced. It is not a static or historic page. Other pages include:
•  Pi Day and Stephen Hawking
•  Pi Day today
•  More Pi Day
•  Our original pi page

The four primary continuity equations begin with Planck Length / Planck Time and Planck Mass / Planck Charge and these will be tracked here: and here.

[6] Planckspheres.  This link goes to a working reference page, nowhere close to a first draft, but a composite of source materials (hopefully with links to those materials), to be read, re-read, then ground up, churned, and ingested to see if some sense can be made of the work of many people who have struggled with these issues!

The first person to whom I turned is John Wheeler and his concept of quantum foam.

[7] Complexification. This work will open a new avenue to complexification as understood within mathematics as given by this link to Wikipedia’s work. The argument for simplicity and beauty is relatively short lived within this scale of the universe.

There are many articles within this website to help open these discussions.

Key pages to open questions/insights about the nature of reality:

Continued research for this article:

One usually knows when an article is complete. This article is not. It is hardly considered a first-draft. Here are just a few of the articles that are being reviewed:

•  Uwe C. TäuberCritical Dynamics  (Cambridge University Press, 2014), Phase Transitions and Scaling in Systems Far From Equilibrium (Annu. Rev. Cond. Matter Phys. 2017. 8:1–27  doi: 10.1146/)

•    Heating of the intergalactic medium by the cosmic microwave background during cosmic dawn by Matias Zaldarriaga and, Tejaswi Venumadhav, Liang Dai, and Alexander Kaurov.

•   Shape dynamics

•  Richard Fitzpatrick, Teaching notes from  2006-03-29

•  Reviewing our history: Physics in the Twentieth Century, Victor Weisskopf, MIT Press, 1972, 1979

•   The physics of reality: space, time, matter, cosmos Proceedings of the 8th symposium honoring mathematical physicist Jean-Pierre Vigier,  August 2012 Editors, Richard L. Amoroso, Louis H. Kauffman, Peter Rowlands   Editor’s note: In 1980 Camber studied with JP Vigier at the L’Institut Henri Poincaré (IHP) in Paris.

  • Good science is not easy.
  • Good science requires integrity and humility.
  • False starts are probably a given.
  • Nobody likes “your idea” more than you.
  • For more… there is always more.

Richard Feynman constantly reminds us all:  “The first principle is that you must not fool yourself — and you are the easiest person to fool.” from his Cargo Cult Science.  Also published in Surely You’re Joking, Mr. Feynman! (p. 343)

Today’s key question: How do we market simplicity, real deep simplicity that raises complexity? Hawking and Newton started with the abstracted and complex and have confused us all for centuries.

On opening our definitions of the infinite

HOMEPAGES: JUST PRIOR|1|2|3|4|5|6|7|8|9|10|11|12|13|14|15|ORIGINAL


“…never-ending, never-repeating,
always the same, always unique.”

By Bruce Camber, February 2018  The Next Homepage: Which Model Works Best 
International Space Station
Is our universe  infinite?

Yes, this website is about the finite-infinite relation. Little understood, there are five primary transitions in our thinking that might help to open the discussion about its very nature:

1. What is finite?  The 202 doublings from the Planck units to the Observable Universe create a container universe whereby space and time are observed to be derivative, finite and discrete. We have a sense of time; yet, nothing is truly past. All notations are always active and interdependent, and function constantly to define the whole as well as to define itself uniquely. From the CERN-scale of particles (and all the other particles of the Standard Model)  to the Planck-scale, there are as many as 67 notations. Considered a domain for strings, the claim by many scholars is that it is too small for anything else.

But, just consider 64 of those doublings.  We know what defines the Planck units. What more might be defined within each of those 64 doublings (domains or notations)?

Let us return to the Chessboard & Wheat story.

Each doubling provides more than enough space for complexification and a very gradual, systemic definition of strings. At first, this progression is mathematically-defined, just numbers, with no apparent physicality that can be measured, each doubling builds on the prior, from the most-simple to the complex. And yes, all 64  are obviously infinitesimal.

See Notations 1-10  Deeper  More

2. Observations of a natural inflation. These doublings define a natural inflation and quiet expansion; it directly challenges the Big Bang theory. An analysis of six groups of numbers somewhat evenly spaced across all the notations begins to follow the logic/research that define their cosmological epochsMore…

3. What is infinite? In 1925, the great mathematician, David Hilbert wrote, “We have already seen that the infinite is nowhere to be found in reality, no matter what experiences, observations, and knowledge are appealed to.”  Even today, many scholars would agree, but perhaps Hilbert and those scholars are mistaken. Consider the non-ending and non-repeating numbers such as pi, Euler’s equation (e), and possibly all the other dimensionless constants. If something is never-ending and never-repeating, how can it be finite?  If we take these numbers as they are, in the most simple analysis, aren’t these evidence or a manifestation of the infinite within the finite? Isn’t this a deep continuity?

All are dimensionless constants. Never-ending and never-repeating. If you can, try to empathize with those words,  never-ending and never-repeating.  How could that ever be finite?  Our historic problem is that we try to impute too much into the infinite.  We tend to drag all of history with us with all the suspicions and problems.

Yes, I believe access to begin to understand the infinite is to be found in the primary dimensionless constants where the number being generated does not end and does not repeat. As a necessary part of the definition of the Standard Model of Particle Physics, there are as many as 26 such numbers given by John Baez and 31 by Frank Wilczek (and others). There are over 300 such numbers defined by the National Institute for Standards and Technology (NIST).  And, then there is Simon Plouffe; he has identified, through algorithmic programming, 11.3 billion mathematical constants (as of August 2017) which includes pi, Euler’s number, and more.  This use of “never-ending, never-repeating” as the entry to the infinite will be challenged. If it can be defended, then there are more connections betweeen the finite and infinite than David Hilbert and scholars ever anticipated.  More

The most-used and best-known dimensionless constant is pi. Pi is everyone’s pi, our single best connection to the infinite. And, every equation that uses pi qualifies!  More

Sphere to tetrahedron-octahedron couplet 4. Doublings. The simple doubling of the Planck base units appear to generate lattice through the cubic-closed packing of the spheres such that triangles, then the tetrahedron, then the octahedron, are manifest. Perhaps part of the thrust of Planck Charge is the emergence of numbers within every dimensionless constant. With tetrahedrons and octahedrons all space and time can hereinafter be tiled and tessellated (100% filled). More

5. Faces of the infinite.  If not absolute space and time, within these studies, the infinite may be known for three  characteristics within dimensionless constants and basic symmetries that define the finite:

•   Continuity.  There is the continuity of numbers, first within the dimensionless constants, never-ending, never repeating, and then in creating the perfections that are simple geometries. Here is the logic of number theory and logic itself.

•   Symmetry.  The symmetry of the spheres, then the symmetry within the tetrahedrons and octahedrons, constantly evolve from simple to complex symmetries.

•   Harmony.  When two symmetries actively interact in a moment of time, there is a moment of harmony. Musicians and audiences often hear that perfection. Perfections are places and times when the infinite intersects with the finite.

Often I say the the infinite is the qualitative face of continuity (order), symmetry (relations) and harmony (dynamics) and the finite is the quantitative face of the same. These facets of the infinite are keys to more fully understand the finite and vice versa.


Our scholars struggle in a different way:

A chart, The Quantum Structure of Space and Time, editor David Gross, Solvay 23, 2005

A chart, Dimensionless constants, cosmology and other dark matters Wilczek-Aguirre-Reese-Tegmark article, 2006

These two charts are over ten years old, yet still summarize current scholarship.


Related pages. First, the URL for this page is

Results related page:

A study group based on this page:

This discussion began as a 2016 homepage:


Earlier discussions:



The next step is to attempt to bring the dimensionless constants, the simplest geometries, and all those ratios together within a transformation nexus.


Scholars: Many people have been asked to provide some feedback about our work. Until we have heard, “That’s wrong,” we will continue to try to learn as much as possible from current scholarship. Because there are so many fundamental intellectual problems that scholars have been struggling with for over 50 years (Big Bang Theory), over 100 years (Planck units, special & general relativity),  over 300 years (nature of space and time), over 2300 years (the nature of bounds and the boundless, i.e. Euclid’s elements), and over 30,000 years (nature of infinity), we will ask, “In what ways might our insights address those historic problems?” knowing that this effort began in a high school geometry class.   Thank you.


From Planck Charge…


The Thrust Of The Universe

by Bruce Camber  First Draft Initiated: June 5, 2017 Last update: September 2018 

Précis. Let us try to open a door to wider postulations and analyses of three primary thrusts of the universe. The first is light and its dynamic definition of time, length, mass, and charge.  The second is the infinite-finite generation of never-ending, never-repeating numbers. Some of those numbers may also be referred to as the incommensurables, transcendentals, or irrationals, as well as fundamental physical constants or mathematical constants.  And, the third thrust is the mechanism of each actual doubling. It is the power of two. It is the geometry of interiority. It is all about base-2 exponential notation and period doubling and emergence.

Observations and Speculations. These are our initial four faces of thrust:

1. Light. Of course, the keynote of Einstein’s famous equation, light is the primary thrust of the universe and binds the universe as a simple whole. It is the magic within the definition of each base unit.

Planck charge. This charge is one of the first mathematically-defined faces of thrust. Though a difficult analysis for most of us because it becomes so complicated so quickly, it will require a concerted study by many to begin to grasp all the “fine structure” and details. Planck charge is defined by c (the speed of light in a vacuum), ℏ (hbar, the reduced Planck constant), ϵ (the permittivity of free space), e (the elementary charge) and α (alpha, the fine structure constant).

All these values are primary finite-infinite transformations. Following just the doubled number of the Planck charge as it progresses up each notation is easy; understanding (1) the processes involved and (2) the manifold potential for exponential growth is not.

Planck Charge

Although each successive doubling of Planck Charge is a natural extension of the many finite-infinite transformations, there are many opportunities for complexification. Max Planck defined Planck Charge as:

Certainly it is a very small number, yet the charge of up, charm and top quarks is smaller and the down, strange and bottom quarks yet smaller (See the Orders of Magnitude-Charge). By the 67th notation, which we call the CERN-scale, the charge has gone from that very small number at the Planck Scale, 1.875×10-18 coulombs to 276.7891 Coulombs, now equivalent to the discharge of a few average lightning bolts. Every facet of charge is also a facet of thrust. Notwithstanding, the International System of Units ( SI ), the coulomb (C) is the preferred unit of electric charge quantity and we will try to grasp all the facets of thrust found within it.

By the 137th notation, now part of the large scale universe, the coulomb’s value has become the very substantial 3.2677×1023 within a universe that is still a fraction of a second old with a density of a neutron star (3.7920×1032 kilograms) in the range of 2815.8174 km (or 1749.67 miles).

2. Infinite-finite transformations. Is the most basic continuity/order that defines the infinite within the non-repeating, non-ending numbers? Is its most simple expression the circle-line (diameter) ratio known as pi?

What is pi?

In all our work, continuity is the first face of that which is infinite. Continuity gives us some confidence that the the universe is fundamentally defined by order, and constants-and-universals. Yet, the non-repeating, non-ending numbers also give us a sense that a certain kind of uniqueness is inherently part of everything, everywhere throughout all time. Within our most simple geometries we see how this uniqueness becomes even statistical and even more unpredictable. These qualities appear to be  profoundly ingrained within who we are and what we are. That data set appears directional; however, it may be wise to leave that issue open for further discussion. Within this model, it is the beginning of thrust and the thrust of life.

IBM Research – Zurich: Nanographene molecule, Olympicene, 2010

3. Doublings — the power of two — and the geometries of interiority. The most-studied, entirely obvious thrust is cellular development, especially from inception to birth and then throughout life. Our approach is to begin to open up the mechanisms of these doublings assuming that these are analogues of a more fundamental doubling within the notations that precede cellular development. The first expression of cellular doublings is within notations 97 (red blood cells). The DNA helix is within notation 87. In this range prime numbers include 89 (cell wall thickness), 97 (sperm cell), 101 (hair cells), and 103 (egg cell).

The first 67 doublings or notations. Prior to the Big Board-little universe project, we found no discussions or articles about the first 67 notations, i.e. the Planck scale to the CERN-scale. If the concept of continuity means anything at all, there should be a bridge between the two. A most-simple, logical bridge could be constructed using base-2. Perhaps it is purely mathematical. There can be no measurement of physicality. So, in that light, among our more speculative postulations is the concept of the long-disputed dark energy (perhaps 68% of the total energy in the universe) will never be measured by standard processes today which are just below the 67th notation.

David Bohm
David Bohm

Quantum fluctuations. If that which cannot be measured by standard processes today is dark energy, perhaps David Bohm described it best in 1957 (Causality & Chance in Modern Physics pages 163-164): “Thus, in the last century only mechanical, chemical, thermal, electrical, luminous, and gravitational energy were known. Now, we know of nuclear energy, which constitute a much larger reservoir. But the infinite substructure of matter very probably contains energies that are as far beyond nuclear energies as nuclear energies are beyond chemical energies. Indeed, there is already some evidence in favour of this idea. Thus, if one computes the “zero point” energy due to quantum-mechanical fluctuations on even one cubic centimetre of space, one comes out with something of the order of 1038 ergs, which is equal to that which would be liberated by fission of about 1010 tons of uranium.”

Other analogies. There are many thrusts described within mathematics and many of these are summarized on line 11 within the chart of the universe.

Search. The words, “thrust of the universe“, were entered (with the quotes) into Google on Sunday, June 4, 2017; only seven (7) results were returned. One was an article on cosmological inflation that I wrote on July 12, 2016. The first entry was by David Birnbaum of NYU. In time each of those six other entries will be examined in depth.

Screen Shot 2017-08-29 at 7.23.56 PM
John Findlay

Personal history. In and around 1975, I first heard the term, thrust of the universe within a classroom with John Niemeyer Findlay, a Rhodes scholar and expert on both Plato and Hegel. On occasion he lectured about the fabric of life, an energy and direction, an abiding thrust, to make things better. Although I enjoy my memories of that ever-so-dear professor with a small smile and all-knowing twinkle in his eye, most philosophy does not shape, texture and extend their concepts of thrust.

Center for Science Information (CSoI). At one time, this center made “thrust” a focal point of their work. Sponsored by the National Science Foundation (NSF), this consortium of universities and scholars is based within Purdue University and it was here that I was introduced to the work of Princeton’s William Bialek. He is the John Archibald Wheeler/Battelle Professor of Physics. Like John Wheeler whose chair he holds, Bialek is one of those scholars who searches for abiding principles. I thought that he just might find our work using base-2 exponentiation to be of interest so I was learning as much as possible about his research, particularly about his work to develop a mathematically-sophisticated introduction to the natural sciences. He was carrying on the Wheeler spirit and tradition; one of his projects resulted in a special Freshman class seminar called Integrated Science. Wheeler once said, “Behind it all is surely an idea that is 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?”1

Footnote #1: John Archibald Wheeler, 1911-2008, physicist, How Come the Quantum? from New Techniques and Ideas in Quantum Measurement Theory, Annals of the New York Academy of Sciences, Vol. 480, Dec. 1986 (p. 304, 304–316), DOI: 10.1111/j.1749-6632.1986.tb12434.x

When I first discovered the Center for Science of Information, they had a clear focus and mandate from the NSF to grapple with thrust from large-scale to small-scale and throughout all of life. According to Bob Brown, the Managing Director of the center, in this period of their work (July 2017), none of their 50+ scholars are currently focused on the cosmological thrust of the universe. That’s unfortunate. There are many possibilities to consider to find the deepest sources of thrust within this universe. Because this article will be under constant pressure to be updated with the latest and greatest insights within emergent scholarship, we will consider it to be an open working document. We will continue to write to anybody and everybody making scholarly contributions within this subject area and we’ll update this article appropriately. Of course, one of our earliest emails was sent to William Bailek and Bob Brown.

Looking ahead. We been been trying to reconcile the Planck Length multiples with each of the other Planck units. Assuming the concept of continuity is meaningful, one might conclude that Notations 1 to 67 define a truly small-scale universe. Notations 67 to 134 define the human scale. And, notation 134-to 202 define the large scale. However, Planck Time is necessarily defined by Planck Length and the speed of light within every notation. Given we live and die within Notation 202, it is as if Carl Jung was right about archetypes and all these human scale notations are in some manner of speaking archetypal. All appear to be deeply inter-related; and in some measure, each of us constantly participates within multiple notations. Can we further develop the concept of non-locality, both quantum nonlocality and the nonlocal Lagrangian, to define a concept that spans multiple notations? Perhaps our mind and sleep are within notations 40 to 50. Hypothetically our functional parts for perception and beingness may be defined by their Planck Length notation, an instantaneous systems integration brings together all the notations within the Now of the 202nd notation.  Does that make any sense to you?

Ratio analysis. It surely seems that the thrust of the universe is found in all the ratios. And, if these ratios are the truly real or really-real, and entities-and-things are derivative, we all need to reconsider our subject-object logic.

Can we “go inside” Max Planck’s formula for charge?  Can we so totally identify with each facet of his formula, we feel the results?PlanckCharge How does this combination of ratios work together to define the base unit for charge? Ratios, a dynamic tension, are the immediacy of the infinite within every facet of the finite.

Point process theory: Nagel and Mecke The dimensionless constants within the context of the first 67 notations just might become a new full-scale science. The first simple bridge between the finite and infinite may well be with their non-ending, non-repeating numbers. Are all dimensionless constants never-ending and non-repeating? That area of research is another vibrant area of scholarship and worth our time to consult with the experts.

Naive Assumptions. Within our initial analysis, it seems that our fundamental assumptions are:

1. The infinite is the source of the finite, its structure and energy (thrust).

2. The primary bridges between the infinite and finite are the dimensionless constants.

3. A well-studied mechanism for thrust can be found within cellular division.

4. The structures of the small-scale universe give rise to the CERN-scale, human-scale, and large-scale structures. 5. The mathematics of period doubling bifurcation, pi, the coupling constant, point processes hold clues for further study. Yes, all the mathematics introduced within line 11 of the horizontally-scrolled chart are part of this study. The scaling vertices (line 9 just above two rows above) are called construction vertices for our use within pointfree geometries. All five assumptions will be woven into the fabric of this open study. One of our most simple-but-complex places to start is within pi.

Power of Pi. The most commonplace, best known, and longest studied of all the dimensionless constants, there is a special place for pi in this website. Since the very first writing about this project in December 2011, there have been several discussions about pi ( π ). The focus will be to build on just two of those discussions: (1) an early-draft article from January 2016 about numbers and (2) a very current working article about visualization.


Resource pages for this article:

• Chart of the Universe • Measuring an Expanding Universe Using Planck Units

Helen Keller

“Walking with a friend in the dark is better than walking alone in the light.” ∼ Helen Keller

helenkeller“The relation is what is primarily real.”

Continuity (the first face of order) and symmetries (the first face of relations) are principles deep within the foundations of all foundations. These two define the foundations of the infinite and when unfolding within the finite world of space and time, a perfection of symmetries creates harmonies (the first face of dynamics).

Helen Keller was blind and deaf and it took the loving and patient heart of Anne Sullivan to pull her out of the darkness.  She had more than a visceral sense of darkness, yet she learned sequences, the order, the nature, and the meaning of such continuity.  Then she learned relations, particularly that relations are more real than the subject that is experiencing and the object that is providing the inputs for the experience.

A first principle of our project:  There are deep and abiding geometries (relations) that bind everything, everywhere, for all time, within those 201+ notations.

Explanation: Help to uncover the deepest and most-compelling symmetries throughout the universe by working with just our first continuity equation, from the Planck Length to the Observable Universe. In the first notation we doubled the Planck Length. Still so small, it is thought by most to be meaningless. Yet, when we follow scaling laws and dimensional analysis and assume the Planck Length is a vertex, we multiply by 8 so the numbers of vertices increase, 1, 8, 64, 512, 4096, and so on. Complexification could occur rather quickly. By the 20th notation, there are over a quintillion vertices with which to work. There are at least 40 more doublings (or notations or groups or sets) before there is an expression of physicality within measurable space and time. A point-free geometry (Whitehead) and bifurcation theory is assumed (hypothesized, hypostatized and/or instantiated) to give structure and definition to those first 60+ notations. The only evidence of such structure is given within the homogenity and isotropy throughout our little universe (and within the supersymmetries of particle physics). Thereafter, manifestations of symmetry are actually seen throughout the universe.

10 (as in the top ten reasons … to adopt an integrated UniverseView)

Big Bang Cosmology Homepage  Top Ten overview   Other key pages: Quiet Expansion  Floods  Implosion  Hawking
 Posted: 19 December 2015 Updated: December 2016 Comments are encouraged. Big Board-little universe Project

Top Ten Reasons to give up those little worldviews
for a much bigger and more inclusive UniverseView

“Number 10”

Continuity contains everything, everywhere, for all time, then goes beyond.

The first principle of The Big Board-little universe Project is that the infinite is defined by three basic concepts:
1. continuity that is order,
2. symmetries that are relations, and
3. harmonies that are dynamics.
The finite is defined by space and time. And, here our entire universe begins to be blanketed with simple equations, numbers and logic that immediately become quite complex.

Explanation: Help to drag the largest continuity equation out of the closet. It starts at the smallest possible measurements of a unit of length and time. That is the Planck Length and Planck Time; and, it goes out to the Observable Universe and the Age of the Universe respectively. It is most simple math and logic. These two continuity equations, created by multiplying by 2 (base-2 exponential notation) have just over 200 notations or doublings. It is all quite approachable and it goes right to the heart of our commonsense logic which is the legacy of Sir Isaac Newton’s infinite sense of space and time. Within this project, space and time are discrete, quantized, derivative-yet-dynamic; each has a start and a current state. And, that changes everything.

More... And even more...
Please note: Links within each paragraph go to Wikipedia pages and most often open a new tab or window. Links from “More” at the end of each item, go to some of our earlier discussions within the project.

“Number 9!”

Symmetries structure everything, both the seen and the unseen.

Also a first principle of our project, there are deep and abiding geometries that bind everything, everywhere, for all time, within those 201+ notations.

Explanation: Help to uncover the deepest and most-compelling symmetries throughout the universe by working with just our first continuity equation, from the Planck Length to the Observable Universe. In the first notation we doubled the Planck Length. Still so small, it is thought by most to be meaningless. Yet, when we follow scaling laws and dimensional analysis and assume the Planck Length is a vertex, we multiply by 8 so the numbers of vertices increase, 1, 8, 64, 512, 4096, and so on. Complexification could occur rather quickly. By the 20th notation, there are over a quintillion vertices with which to work. There are at least 40 more doublings (or notations or groups or sets) before there is an expression of physicality within measurable space and time. A point-free geometry (Whitehead) and bifurcation theory is assumed (hypothesized, hypostatized and/or instantiated) to give structure and definition to those first 60+ notations. Evidence of such structure is given within the homogenity and isotropy throughout our little universe (and within the supersymmetries of particle physics). Thereafter, manifestations of symmetry are actually seen throughout the universe.

More...   Even more...  And, even more...

“Number 8”

Harmony creates a fleeting moment of perfection that inspires us and guides us.

Harmony is also a first principle of this project. All those symmetries that bind everything, now begin their movements across all time (within just over 200 notations) from the structure of the first moment to the structure of the current time, today, right now.

Explanation: Empower symmetries to interact with other symmetries within each notation and across notations. The old concepts related to the harmony of the spheres take on new meaning. By adding the continuity equations defined by Planck Time, symmetries have more than a form; they have a function that moves within a notation and throughout the notations, creating a momentary perfection or perfected state within space-time. Also, we begin to understand the very natural antithesis, chaos and indeterminacy, especially by looking at the most basic structures with the first 60 or so notations…

More... More to come... (pages about harmony being developed)

“Number 7”

You can re-engage all those mysteries in Science-Technology-Engineering-Mathematics (STEM) and start to understand it all.

Every aspect of education is touched. Our goal is to use this project as a STEM resource, first within 10 high schools, then a pilot of 100, then 1000 schools…

Explanation: Everything everywhere starts simply. This new order of things where space and time are derivative of continuities and symmetries (and occasional harmonies) will empower deeper insights into fabric and applications of STEM. One of the initial aims of our first attempt at a Kickstarter program was to engage no less than 100 high schools to use the Big Board-little universe model to re-ignite a love for science and mathematics and all their derivatives. We can see and feel a burst of creativity that begins to address the old questions and opens up a huge boat of new questions.

More to come... (STEM work for schools)

“Number 6”

Calm your soul within a Quiet Expansion of the universe.

Could the Big Bang theory be deflated by the Big Board-little universe’s Quiet Expansion (QE)? Could a QE become a source of order and inspiration within education… for the general public?

Explanation: We will build upon an article that was written in September 2014; we were thinking about the Big Bang theory while working with the chart of just the Planck Length. It seems that one could easily impute those first 60 doublings to be the foundation of pervasive, unseen structural geometries that finally give rise to the physicality of particle physics. It seemed that the first 67 notations would have been a rather quiet expansion of creation.

More... (first questions about the big bang in

“Number 5???

#5. You are the center of it all.

Humanity and so many facets of our human population has been marginalized by science, religious beliefs and arrogance. Within this emerging model of the universe, people are key.

Explanation: Within the 202+ notations at 100 is the human sperm cell and at 103 is the human egg. In the middle of the Planck Length doublings is the vibrancy that gives us modestly intelligent life. A molecule of water, the source of continuity within life, is just 280 picometers, within the 80th notation, has a special symmetry with so many other chemical and biological processes within the next 20 notations. If you are the center of it all, we say, “Sit up, and be sure to drink a lot of water.”

More... (new tab or window; a tour of the Planck Length chart)

“Number 4”

#4. Your Mind is actually on the grid after all.

The brain-mind debate is centuries old. The mind has had no place to be… no place to rest. Unwelcomed and questioned, here we hypostatize its place on the grid between notations 50 and 60.

Explanation: Nothing can be measured by an instrument that fits into the first 60 notations or doublings of the Planck Length, so by engaging systems philosophy, we begin to construct an ideal (the same challenge that faced Lawrence Krauss with his book, A Universe from Nothing) starting with Plato’s forms. The doublings quickly become complexity. From Forms, to Structures, to Substances, to Relations, and finally to Systems, this is an idealized structure where the Mind seems to naturally reside within Systems.

More to come... (a page about the Mind is being specially developed)

“Number 3”

#3. You’re in at least three places at the same time.

We get our most basic definition within the Human-Scale Universe, yet we live and have our being within the 201st notation where we find the current time. And, just perhaps, our minds exist within a Systems construct of the universe between notation 50 and 60. Surely our sleep, which we say is “resting in the arms of God”, is within the small-scale universe, 1-67.

Obviously there are huge intellectual challenges ahead!

Explanation: Each of the three major divisions of the Universe — the Small Scale, the Human Scale, and the Large Scale — has about 67 notations. As we study our many charts of the five Planck base units, we can see ourselves in at least three places at the same time. Perhaps we are manifest with a bit of every notation. We’ll now begin thinking how all three scales work coterminously to create personal identity. As a result, we hypostatize new functionalities for space-time junctions, historically called wormholes, for the mind and the archetypal definition of human throughout the Human Scale Universe. Hopefully, along the way we may even begin to grasp the very nature of a blackhole. To that end we now return to Hawkings’s 1975 paper, Particle Creation by Black Holes.

More...  (goes to our first chart based on the Planck Length)

“Number 2”

#2. You can get your act together.

We have been constrained by worldviews that ignore the larger universe and our role within it. We have been constrained by a science that has difficulty penetrating those notations below the 67th.

Explanation: An Integrated Universe View changes everything. Simplicity is the starting point. Continuity takes its place at the centerfold. Symmetries within it capture our eye and imagination. There appears to be goodness in the universe and the place of asymmetries, indeterminacy, questions and openness, and the very nature of the self begins to thrive as a creative nexus! If only my Mom were still alive, she might just understand what motivated her to say so often to me, “You’re cruisin’ for a bruisin’.”

More... (goes to a page about ethics)

“Number 1”

#1. Religions & science can get together again.

Our worldviews can no longer ignore the constants and universals that define and give structure to the universe and our role within it.

Explanation: Our very-young and often naïve UniverseView is a call to all religions of every flavor. We all need a touch of humility to engage the openness of those constants and universals so we can see how each is used within the sciences and how these create bridges with the metaphors and analogies within religious history. My father was an Episcopalian turned Unitarian Universalist. As we were growing up, he would often say, “You can’t legislate morality.” He was right. It is much deeper than legislation and rules. Religion is part of the very fabric of ethics, universals, and constants. Their metaphors can be richer and more robustly textured. The Universe is so simple, so marvelous, so intricate, and extraordinarily complex all at the same time, we all should celebrate every hour of every waking moment.

More...    And more...

You are invited to be involved.Twitter

What Did We Ever Do Without Our Universe View?

by Bruce Camber, November 30, 2015

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 Morrison’s 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?” 

15 Key Questions About Life, The Universe, And Who We Are

Editor’s note: This page was written before there was a website for this work. Many links go back to experimental pages that were initially incorporated into our business website for our weekly television show on PBS stations throughout the USA.

Prepared by Bruce Camber for five classes of high school geometry students and a sixth-grade class of scientific savants.

There are no less than 15 concepts reviewed here. All have been explored within a high school yet have been virtually ignored by the larger academic community. It begs the questions, “Are any of these concepts important? Which should we keep studying
and which should be deleted?” And, of course, if we delete any, we need to know why.tetrahedron copy 2

Students have been known to ask a rather key question, i.e., “Can’t you make it easier to understand?”  So, in light of the universal pursuit for simplicity, beauty and wholeness, our geometry classes just may have stumbled onto a path where we begin to see all the forces of nature/life come together in a somewhat simple, beautiful, yet entirely idiosyncratic model.

It feels a bit like Alice-in-Wonderland — the entire known universe in 201+ notations or doublings — all tied together with an inherent geometry, an ever-so-simple complexity. The students ask, “Can this somehow include everything, everywhere?”

#1 Key Question: Is there a deep-seated order within the universe?

Geometry 101: From the Planck Length to the Observable Universe
December 19, 2011: Defining our Parameters and Boundaries

octahedronOver 120 high school students and about twenty 6th graders have divided each of the edges of a tetrahedron in half.
• They connected the new vertices to discover four half-sized tetrahedrons in each of the corners and an octahedron in the middle. They did the same with that octahedron and observed the six half-sized octahedrons in each of the corners and eight tetrahedrons, one in each face.

We continued this process mathematically about 112 times until we were in the range of the Planck length. We eventually learned that this process is known as base-2 exponential notation. When we discovered, then compared our work to that of Kees Boeke (Cosmic View, Holland, 1957), we thought base-2 was much more informative, granular, and natural (as in biological reproduction and chemical bonding) than Boeke’s base-10. Plus, our work began with an inherent geometry, not just a process of adding and subtracting zeros.  More… (opens in new tab/window).

#2 What are the smallest and largest possible measurements of a length?

Doublings and Measurement
December 2011: Getting More Results

SDSS-III-BOSSWe had taken those same tetrahedrons with their embedded octahedrons and multiplied them by 2. Within about 90 steps (doublings), we thought we were in the range of the recently-reported findings from Hubble Space Telescope and the Sloan Digital Sky Survey (SDSS III), Baryon Oscillation Spectroscopic Survey (BOSS) measurements (opens in new tab/window) to bring us out to the edges of the observable or known universe. It appeared to us that this perfect conceptual progression of embedded tetrahedrons and octahedrons could readily go from the smallest possible measurement to the largest in less than 205 notations. We decided at the very least it was an excellent way to organize the data in the entire universe.

More questions:  What are the most-simple parameters with which to engage the universe?  Do the geometries (relation/symmetry), base-2 (operations of multiplication or division), and sequence (order/continuity) provide an operational formula for expansion of the operand?

#3  Do these charts in any way reflect the realities within our universe?

 Big Board – little universe and our first Universe Table

2011 -2012

We had also develop a big board (1′ by 5 ‘) upon which to display this progression so we could begin inserting and updating examples from the real world within each notation (domain, doubling, or step).  The very first, very rough board (December 2011) started showing up within blogs (May 2012):  To simplify the look and feel of those listings, we started working on a much smaller table (8.5″ x 11″) in September 2012. That Universe Table was based on the big board:

Another question: What are the necessary relations between adjacent notations?

#4 How do we prioritize data (calculations), information and insight?  What is wisdom?

202.34 to 205.11: From Joe Kolecki to Jean-Pierre Luminet

May 2012: Getting Some Professional Insight and Confirmation

1aWe consulted with Joe Kolecki, a retired NASA scientist involved with the education of school children. He did a calculation for us and found about 202.34 notations from the smallest to the largest (based on the age of the universe).

We had also consulted with Jean-Pierre Luminet, a French astrophysicist and research director for the CNRS (Centre National de la Recherche Scientifique) of the observatory of Paris-Meudon. He calculated 205.11 notations: See footnote 5

The nagging question: What are the necessary relations between adjacent notations (or doublings, layers or steps)?

#5 How does each notation build off the prior notation? What is a notation?  Is it geometrical?

An Encounter with Wikipedia

April-May 2012: Grasping the New Realities

We wrote it up for Wikipedia to have a place to collaborate and build out the document with other schools and even universities. But, in May 2012, their review group told us that it was original research. Though there was a clear analogue to base-10 notation from Kees Boeke from 1957, an MIT professor, Steven G. Johnson (he reviews entries for Wikipedia) said that it was “original” research. We begrudgingly accepted his critique:
The simple math:

#6 What is perfect and what is imperfect?

Pentastar, Icosahedron, Pentakis Dodecahedron

December 2011 to December 2012: One Year of Insights

We then observed some curious things. First, geometries can get messy very quickly. We were using the five Platonic solids. Starting with the tetrahedron, we quickly discovered that these objects rarely fit perfectly together. The pentastar, five tetrahedrons clustered tightly together, do not perfectly tile space, but leave a gap. This gap has been thoroughly documented yet to the best of our knowledge it was first written up by two mineralogists, Frank & Kaspers, in 1958. In its simplicity, we concluded that this was the beginning of imperfections and it extended out to the 20 tetrahedron cluster also known as the icosahedron, and then out to the 60 tetrahedron cluster (just the outer shell), which is called a Pentakis Dodecahedron. We dubbed these figures, “squishy geometry” because you could actual squish the tetrahedrons together. In a more temperate moment, we dubbed this category of figures a bit more appropriately, “quantum geometry.”

#7   What is the Planck Length? Is it a legitimate concept?

Frank Wilczek, Encouragement from an Authority

December 2012

We consulted Prof. Dr. Frank Wilczek (MIT) regarding his many articles in “Physics Today” about the Planck Length. He assured us that it was a good concept and that the Planck Length could be multiplied by 2. We titled our next entry, “Everything Starts Most Simply. Therefore, Might It Follow That The Planck Length Becomes The Next Big Thing? The current state of affairs in the physics of CERN Labs is anything but simple. We figure if we built things up simply, we might gain a few new insights on the nature of things.

#8  Is life a ratio?  Does it begin with Pi and the circumference of a circle?

Steve Waterman’s polyhedra and mathematics
March 2014: Discovering Others Searching the Boundaries

In December 2013, I sent a note around to an online group of mathematicians, mainly geometers; and of those who responded, Steve Waterman had done some truly original, rather-daunting, work that had certain similarities to Max Planck’s work a century earlier. It was not until a lengthy discussion in April 2014 that I began to understand the simplicity and uniqueness of his extensive work. He had emerged with many, if not most, of the 300+ NIST constants, the gold standard of the sciences. He had used constants in a similar way that Max Planck used the speed of light and the gravitational constant to begin his quest for the Planck Length. Waterman provokes the ratios of known constants to come ever so close to the NIST measurements. His math implies an inherent universal wholeness and he does it with a series of “what if” questions. It took me awhile to grasp his fascinating, far-reaching results:

#9 Is there anybody doing mathematics in any way related to these notations?

 Edward Frenkel and his book, “Love & Math: The Heart of Hidden Reality”

In October 2013, Edward Frenkel’s book, “Love & Math: The Heart of Hidden Reality” became part of our picture. Perhaps this remarkable mathematician can shed light on those areas where we all are weakest. We let him know we had his book and would be reading it to answer simple questions, “Why doesn’t anybody care about this construction? What are we missing? Why are people so sure that the fermion and its extended family represent the smallest-possible measurement of a length, especially in the face of the Planck Length? Why shouldn’t we attempt to think of the Mind and mathematics as representations of those steps between the Planck Length and those within the particle families?”

Through Frenkel’s work we have begun to discover the Langlands Program and its progenitors (i.e. Frobenius) and the current work in areas like sheaves, the categorifications of numbers, and the correlation functions. We have begun to learn about the work of other remarkable mathematicians like Grothendieck, Drinfield, Witten, Kapustin, and so many more.

The most important first-impression was that we could begin to discern the transformations from one notation to the next and possibly even discern the very nature of a vertex.

#10   What is a vertex? Are there primary vertices that establish the Planck Unit measurements and secondary scaling vertices?

Over a Quintillion key vertices within just the 60th notation using base-2 exponentiation

Throughout these past 2+ years, we have discerned other simple-yet-interesting mathematical facts.  First, we decided that we should not refer to the Planck Length as a point because it is a rather exact length, so we are giving each vertex a special status and believe we might learn more by understanding Alfred North Whitehead’s concept of pointfree geometries introduced within his book, “Process and Reality.”

Within just the 10th doubling there are 1024 vertices. The simple aggregation of all notations up to 10 would be 2046 vertices. Within just the 20th doubling (notation) alone there are over 1 million vertices. In just 30th notation alone, another one billion-plus vertices are created. Within the 40th notation another trillion-plus vertices. With just the 50th notation, you’ll find over a quadrillion vertices. By the 60th notation, a quintillion more vertices are created. Imagine all the possible hidden complexity!

The expansion of vertices within each doubling has been a challenge for our imaginations and conceptual limitations. Yet, it could be an even greater challenge and far more complex if we were to follow Freeman Dyson’s suggestion. Using base-4 notation for the expansion of the tetrahedrons and base-6 notation for the expansion of the octahedrons, at the 60th notation, there would be a subtotal of 1.329228×1036 for the tetrahedrons and 4.8873678×1046 vertices from the octahedrons. Using simple addition that would be:


+                    1,329,228,000,000,000,000,000,000,000,000,000,000,000,000


The base-2 exponentiation is the “simple math” starting point. It is a simple focus on the process called doubling and only accounts for number of times the original Planck Length has been doubled for each notation. If the focus is on objects, after the fourth doubling, there are four expansions to track, base-1 for the sole octahedron within the tetrahedron, base-4 for the tetrahedrons within the tetrahedrons, and base-6 for the octahedrons within the octahedron and base-8 for the tetrahedrons within the octahedron.  Addressing that schema and the results are:

Base-8 tetrahedrons:   1.5324955×1054 units

Base-6 octahedrons in the octahedron:  4.8873678×1046

#11 What are scaling laws and dimensional analysis?

Freeman Dyson

Mon, Oct 22, 2012

Freeman Dyson, in an email to me (for which he gave me permission to share), suggested the following: “Since space has three dimensions, the number of points goes up by a factor eight (scaling laws and dimensional analysis), not two, when you double the scale.” Of course, we felt we had more than enough vertices with which to contend, so we just multiplied by 2, using the simple analogue from biology or chemistry. Yet, we readily acknowledge that his advice could readily open even more doors for new explorations, so this question is raised and another dimension of our work has been set out before us by a sage of our time!

#12 Key Question: Is the inherent structure of the first 60 notations shared by everything in the universe?

January 2013 to today


January 2013:

Speculations about the first 60 notations

With our simple logic, it seems that with the diversity of particles and the uniqueness of identity, that the structure could continue to expand right up to the 201+ notations.

However, below that emergence of measurable particles, and their aggregate structures, a simple logic would tell us that there is a cutoff point as you go toward the Planck Length where a deep-seated Form (perhaps notations 3-to-10) and Structure (perhaps notations 11-20) might somehow be shared by every thing in the known universe. With vertices rapidly increasing with every doubling, options begin to manifest for types of Substances (possibly notations 21-to-30), then types of Qualities (perhaps notations 31-to-40), then types of Relations (possibly 41 to 50), and finally types of Systems (possibly 51-to-60). What does that mean? How are we to interpret it? It is on our list to continue to ponder.


February 2013

Literature survey

We’ve thought about this very, very small reality from the first notation to the 60th. Perhaps it is what Frank Wilczek (MIT) calls the Grid and Roger Penrose (Oxford) calls Conformal Cyclic Cosmology. We just call it the Small-Scale Universe. Actually, in deference to one of my early mentors, we call it the “really-real” Small-Scale Universe. And, because we started with simple geometries, our imaginative notions of this part of our universe appear to be historically explored yet relatively unexplored as a current scientific framework. First, we turned to our six sections: Forms (Eidos), Structures (Ousia), Substances, Qualities, Relations, and Systems (The Mind).

Also, picking up on a suggestion by Philip Davis (NIST, Brown), that the sphere is more fundamental than the tetrahedron, we start with a one-dimensional length, the Planck Length. When it doubles, it becomes a two-dimensional sphere. When it doubles again (4), it becomes a three-dimensional sphere with a tetrahedron within it. When it doubles again (8), we see the octahedron within the tetrahedron. When it doubles again (16), we begin to see the four hexagonal plates within the octahedron. We are projecting all these forms-structures, substances-qualities, relations-and-systems are complexifications of the first two vertices within the first doubling. We further project that there is a transitional area between each of the three scales, Small-Scale Universe, Human-Scale Universe, and Large-Scale Universe and each would include somewhere between 67-to-69 notations.


May 2014

Discovering Quanta Magazine

Amplituhedrons, Euler, and geometries mixing within necessary relations with geometries

We discovered the writings of Natalie Wolchover within Quanta Magazine, quantum geometries, and on the work of Andrew Hodges (Oxford), Jacob Bourjaily (Harvard) and Jeremy England (MIT). We believe these young academics are opening important doors so our simple work that began in and around December 2011 has a larger, current scientific context, not just simple mathematics. Within the excitement and continuing evolution of the Langlands programs, we perceive it all in light of defining a science of transformations between notations. We are now pursuing all the primary references for people working within quantum geometries.

The simplest, smallest, largest experiment, albeit a  thought experiment based on logic, the simplest mathematics (base-2 notation and platonic geometry), and the base Planck Units, quickly opened doors to look at this data in a radically new way. It will slowly become the basis for many new science fair projects.  The question is asked, “Could This Be The Smallest-Biggest-Simplest Scientific Experiment?”


October 2013 to February 2014: A National Science Fair Project

Some students wanted to take the project further. Here was an initial entry of one of our brighter students:


January 2012: Is there a concrescence in the middle?

Is the ratio, 1:2, somehow special? Approximately between 101 and 103, clustered in the middle by the width of a hair, are paper upon which we document our history and the human egg. Perfectly human representations in the middle of this scale became a source for some reflections.



October 2013: Considering the Thirds, 1:3

Between Notation 66-to-67 and from 132-to-134:

The significance of the first third, particularly the transformation from the small scale to the human scale, was obvious — particles and atoms. The last third, the human scale to the large scale, we played with ideas, then made an hypothesis. In a most speculative gesture, turning to the Einstein-Rosen bridges and tunnels, we posited that range as a place to begin looking for wormholes.

Also, there are the simple multiples by 3: Notations 6, 9, 12, 15, 18, 21, 24, and so on.

Music. We are now studying the fourths, fifths, sixths and sevenths… wondering in what ways are there parallels to music. How do things combine, mix, and move together to create a specific thing or a new thing? We began studying the notational ranges defined by simple mathematics and music to see what we could see from a natural division, first the notation to the whole, and then simple multiples of each initial notation.

Notational range for The Fourths: 50.6 – 51.3, 101.17 – 102.6, and 151.7 – 153.8 and finally 202.34 – 205.11

Also, there are the simple multiples by 4: Notations 8, 16, 20, 24, 28, 32, and so on.

Notational range for The Fifths: 40.47 – 41.2, 80.94 – 82.4, 121.41-123.6, 161.86 – 164.8…

Also, there are the simple multiples by 5: Notations 10, 15, 20, 25, 28, 32, and so on.

Notational range for The Sixths: 33.72 – 34.35, 67.44 – 68.70, 101.17 -102.6, 134.89 – 136.95, 168.61 – 171.30…

Also, there are the simple multiples by 6: Notations 12, 18, 24, 30, 36, and so on.

Notational range for The Sevenths: 28.62 – 29.30, 57.24 – 58.60, 86.46 – 87.90, 114.48 – 117.20…

Also, there are the simple multiples by 7: Notations 14, 21, 28, 35, 42, and so on.

Notational range for The Eighths: 28.62 – 29.30, 57.24 – 58.60, 86.46 – 87.90, 114.48 – 117.20…

Also, there are the simple multiples by 8: Notations 16, 24, 32, 40 and so on.

As we go up the notational scale, the only numbers left are the prime numbers. Here we posit their relation to all other numbers is through base-2 yet also let us explore how these primes could be responsible for the next level of complexification of mathematics.

To date, our very cursory, initial observations have not opened up more wild-and-crazy speculations! However, the obvious parallel to music has us thinking about the nature of chord, half notes and ratios (July 2014).

#14 Who are we and where did we come from?
1971-1973: Synectics, Polymorphs, Colloquiums, and more
Continuity-Order, Symmetry-Relations, Harmony-Dynamics

We are products of our experience. In 1971, when I (Bruce Camber) was just 24 years old, though active in the radical-liberal political community, my longstanding intellectual curiosity was the nature of creativity, the processes for problem-solving, the nature of a paradigm, and the stuff of scientific revolutions. At a think tank in Cambridge, I focused on interiority, analogies, empathy, and processes to open pathways to a deeper sense of knowing and insight. Within a Harvard study group, the Philomorphs, I studied basic geometric structures with Arthur Loeb. At Boston University, I was deeply involved with the weekly sessions of the Boston Studies in the Philosophy of Science with Robert S. Cohen, chairman of the Physics Department. It was within this mix, that the form-and-function of a momentary perfected state in space and time was engaged (continuity-order, symmetry-relations, harmony-dynamics). For many years, that formulation drove my studies to the point of ignoring all else. Now, years later, that work continues.

#15 Where are we going?  What is the meaning and value of life?

2015 and beyond

The Derivative Nature of Space and Time

Some of us have come to believe that space is derivative of geometry and time derivative of number… and all things as things are unique ratios between the two. Of course, we continue to ask ourselves, “So? What does that mean and what do we do with it?” And, as you might suspect, we have far more questions than we have insights. We are way out on the edges looking for new meaning in this universe. The inquiring minds of our most inquisitive students, want to go further,”Maybe we can find a path to a multiverse! “

Let’s develop a community of people and schools who are working on this simple structure.

Tiling and Tessellating the Universe: A Great Chain of Being

Initiated: December 1, 2014  Very minor update: Sunday, January 28, 2018

Tetrahedral-Octahedral-Tetrahedral (TOT) clusters and couplets tile and tessellate the universe1 (opens in new tab or window). In earlier writings, we have observed how the universe could be tiled in 202 exponential notations (or steps, layers, doublings, or domains).

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

Tetrahedron-Octahedron-Tetrahedron is TOT.The 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 tot1 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 and throughout all 202 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,8 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.Tet-Oct.png

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. tetrahedrons159Again, 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 and we were surprised to find most of her tilings within the model.

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. Dr. Francis Collins. 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 had sent me a review copy of his new book, The Language of God, and she and I spent several hours discussing it. 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.  More…

3. Tetrahedron-Octahedron-Tetrahedron: 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.

4. The First Chart: 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.

5. Where is the Good in Science, Business and Religion. 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.

6. 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.

7. 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.

8. 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.

9. 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.

10. 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.

11. 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.

12. 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.

13. 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.

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

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
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 202 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 202 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 (Please see the references and endnotes). 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 (open in a new tab or window).

The simple mathematics (original Wikipedia article in 2012) 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) (opens in new window or tab)?

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.