A Colorful Image

This page provides access to a our largest image of the Big Boardlittle universe, Version Though it displays just below as a relatively small image, click on it to go to a slightly larger version, then click on that image to go to the much-larger version of it.

Also, please go to this interactive, working version of the Big Board – little universe.  It is not yet quite so colorful!


First principles

Foundations within foundations – it’s just common sense.

What makes us human? …ethical? What gives us hope, depth, perspective?

Deep within the fabric of life there is an energy, an abiding thrust to make things better, more perfect. That is the cornerstone of business, but much more.

Simple logic tells us that there are three forms within functions that define an increasingly perfected state within an experience:

  • The first form that defines our humanity is order and its most basic function, a simple perfection, creates continuity.
  • The second form is a relation and its function creates symmetry.
  • The third form is dynamics and its perfection, a complex function, is harmony.

All scientific and religious assertions that seek to understand and define the universal, begin with the same first principle and evolve within their own understanding and language to the second and third. Yet, the starting point -continuity- necessarily tells us that everything is necessarily related.

This is also the basis of the value chain. The more perfect a moment or an experience is, OR the more perfected a thing or system is, the more valuable it becomes. And, in the deepest sense of the word, it also becomes an expression or manifestation of love.

Thus, we have the beginnings of business (and economics), ethics and morality.

Any assertion that counters life’s evolving perfections is not religion (at best, it’s a cult*);
it is also not business (it’s exploitation or a bad company); certainly it is not good government;
and most often, it is not even good science.

There are scientific endeavors that observe, quantify and qualify that which is fundamentally based on discontinuities or chaos, but these studies require the inherent continuities of mathematics and constants-and-universal to grasp the nature of that discontinuity. –BEC

  1. Introduction & Overview

  2. Big Board – little universe

  3. Just an image of the Big Board – little universe Version

  4. Wikipedia Article, April 2012.

  5. An Unfinished Work, An On-going Study.

  6. 202.34:  the calculations by Joe Kolecki, retired, NASA scientist

  7. Just the numbers

  8. A little story

Belief Systems

Key Question:
Which Belief Systems Accommodate Change and Growth? Which Do Not?

Reflections on belief systems, particularly on those that try to control — this article was written by Bruce Camber   (Initiated November 2012  Last edit: March 29, 2013)

Constants1 and universals2 are inherent throughout all of life.  Based on a combination of logic, mathematics, and consistent measurements, these concepts appear to be true throughout all time and within any space (certainly within defined parameters and boundary conditions). Some people believe these concepts can actually open pathways to understand how it is that there is space and time, and human life and consciousness. It is all so bewildering and the sciences and mathematics around these issues are so complex and seemingly impenetrable, people everywhere yearn for compelling but somewhat easier answers to these big questions about the meaning and value of life.

Many religions3are not very religious and are best understood as a cult.4  Some philosophies also qualify.  Simply stated, cults are the people and their organized set of beliefs that are primarily based on historic footnotes that have been lifted up as the highest principles and concepts around which one can orient their life.  Though it might seem that many types of organizations could be labelled a cult, it is far from the truth. Although basic beliefs within any organization come from their historical  writings,  balanced organizations  give as much, if not more weight and importance to their best scholars’ research, writings and teachings about their historical statements and how these work with the constants understood throughout the sciences and universals understood by the most-respected scholars throughout time.

Cults tend to offer more simplified answers to such questions.  And for their followers, these answers become their Absolute5  framework,  the fundamentals of their belief system.

That definition of a cult is also the beginning of a working definition of fundamentalism.6  It does not matter what the belief system is;  fundamentalists are mostly caught up with the fundamentals that have been defined within a particular space at a particular time. As already observed, these are historic moments. The scientific community is not exempt. It has its own group of fundamentalists among their secularists. These folks stridently proclaim that Atheism is the only true “religion” or system of belief about ultimate things. The radical atheists take what has been given by the sciences, and boldly proclaim,  “We have the only right answers. This is the Way, the Truth, and the Light.”

Mainstream religions and philosophies context their belief system within our known understanding of universals & constants.  Although the focused study of the universals-and-constants is mostly the domain of natural sciences,  other disciplines — logic, mathematics and ethics — also open this world. Our best scientists know that their natural sciences are still young and there are many new worlds and universes yet to explore.

Both types of fundamentalists — religious and scientific — fall short . There are profoundly simple constants-universals that have not been fully explored and generally recognized by the world’s scientific community that could begin to change things. A very simple example was the focus of five high school geometry classes that asked, “How many steps would it take to get to the Planck length using base-2 exponential notation assuming nested geometries all the way?” They found 101 steps to the Planck length going within by dividing each step by 2 and 101.34 steps to the edge of the observable universe going out by multiplying by 2. It appears to be the first time people would see the entire universe from the smallest to the largest, all mathematically notated and necessarily related, on one long board in 202.34 steps or doublings. Perhaps this simple scale based on the Planck length could open new worlds to explore at CERN’s Large Hadron Collider. To the best of our current knowledge, steps 1 to 60 have never been discussed as such. So, this discussion is not a science versus faith discussion.   It is a focus on the ways we approach and interpret both science and religion. It is about exegesis and hermeneutics.   Those two studies are only about the way we interpret religious texts, particularly sacred scripture; however, both can also be applied to the sciences, especially regarding the limitations of science and the edge of discovery. How and why do teachers become fundamentalists?  Where do they go wrong?

Let us start with a focus on Radical Islam and scientific Atheism.


Some attention has already been given to this question within the many studies of the root causes of 9/11. Here is a link to those studies as well as a letter to the Iran’s Grand Ayatollah Ali Khamanei back in 2006. Lumping them with the growing stridency of the today’s breed of Atheists — they, too, are demanding recognition and real power —   all types of fundamentalism really need to be studied, compared and contrasted for their use or lack of use of universals and constants.   Notwithstanding, because Radical Islam and her teachers demand that we acknowledge them (or they will rather arbitrarily blow us up, and they continue to threaten to kill us), we should begin with these two. Surely both are having the penultimate temper tantrum (intolerance to disagreement) so, from here let us compile studies of the most influential among their current groups of teachers. A primary challenge for each of us is to define what is universal and constant within our own life. It is no easy task. The summary in the concluding paragraph of this article is a work-in-progress and the initial work is linked here.7 In 1979 that work included many leading , living scholars.  It had begun to evolve from a study of physics and the sciences, to include religion, logic, ethics, value, and even business.8

A key question to ask is, “What concepts are shared by all of these disciplines?”  Assuming you get a few answers, ask yourself, “What concepts are truly simple?” And also, “What concepts could have a face of perfection?” Those three questions opened the way to this paper’s simple working formula:The form – the function (a face of perfection) and the imperfect quantum world:

Embedded within this little formulation and the statements just above it are links to the first applications of these universals. It goes back to work in 1979 at MIT regarding first principles with 77 leading, living scholars from around the worldIt has stimulated many simple explorations that seem to have been overlooked by the academics and most certainly the religionists and the fundamentalists.

A simple summary might go something like this:

Continuity defines order. Symmetries and asymmetries in some manner define all relations.

The simple perfection of a relation is a simple symmetry.

And, a harmony defines a perfection of multiple symmetries within a dynamic moment.

One can use religious language, metaphorical language, or scientific language to describe each. And if done well, that language is an observation of one of the faces of the same thing and each language helps to inform the other. When teachers give too much weight to one language over the other, they begin to lose their balance and fall into the trap of thinking that they just may be smarter than all the others.

Over time such conclusions hurt their ability to think-and-reason. Thank you.

1 An evolving analysis of physical constants  within Wikipedia (opens in a new window).
2 An evolving analysis of universals, the problem of universals as well as universal properties, all within Wikipedia (new windows open).
3 An evolving analysis of nature of religion can be found within Wikipedia (opens in a new window).
4 An evolving analysis of nature of cults,   also within Wikipedia (opens in a new window).
5 An evolving analysis of the concepts around The Ultimate, here known as The Absolute,  within Wikipedia (new window).
6 An evolving analysis of the word, fundamentalism,  within Wikipedia (new window).
7 Continuity equations define constants and beg the question, “What is continuity?” Of course, Wikipedia has something to say. In our more speculative moments, we have begun to look for ratios of constants and geometry that create space; and, constants and space to create time.
8 The Wikipedia references to symmetries are helpful. However, to engage this domain with the best living scholars is our current challenge.  Here Roger Penrose and Lisa Randall offer insights.


Background story

Strange things can happen when one is invited to be a substitute teacher for a day, essentially just an assistant for students within five high school geometry classes and for their teacher who is part of our extended family.

My name is Bruce Camber and this is a little story about the unfolding of events that eventually resulted in these pages.

Have you ever seen the entire universe mathematically related and notated on one chart? We had not. In studying the platonic solids and base-2 notation, it seemed to be an interesting task to do the simple base-2 math to create a rather big board that related everything from the smallest measurement to the largest measurement, virtually every part of our universe. So, we created it. The first version of this board was printed down at the Office Max in Harahan, Louisiana on December 17, 2011. It measured 24″ by 120″ which we quickly discovered was a bit big and awkward.  Two smaller charts, 12″ by 60″ were created the next day, December 18, for the classroom discussions on December 19.

The ten-foot board was cut about in half and the top section was put in the front of the class and the bottom section in the back. On the walls on the left and right were the two five foot charts. It all seemed a bit enchanting.

Those  five high school geometry classes were challenged to see the universe using Plato’s five building blocks and base-2 exponential notation. Starting with just the tetrahedron and octahedron, we divided the edges in half, connected those new vertices and keep on going. In just over 101 steps we were in the range of the Planck Length, the smallest measurement in the universe. Then, multiplying by 2, in just over 100 steps we were out to the edges of the Observable Universe.

We found examples of base-10, yet clearly base-2 is more granular.  One divides by 2 or multiplies by 2 instead of by 10.  We also discovered and used a huge history of work done on the order of magnitude and the power of two.  However, it seems that we had a different starting point than most.  At first, we used an imaginary tetrahedron that was 1 meter on its side. Our actual models were 2.5 inches. We expected thousands of steps in either direction.

It was a bit unsettling to discover so few.

We reduced it to a chart with a color wheel as the background and dubbed  it, a Big Board  for our little universe.   Not too much later, we decided to start at the Planck length and just multiply by two. It worked out pretty well and kind of, sort of confirmed our earlier work.  It became  Version and you will find images of it throughout these pages.

This work all started with Plato’s five basic solids and thoughts about basic structure. Though most people do not give it much thought, these basic structures have been studied throughout time, probably starting in ancient Greece with Pythagoras and picked up later by Plato.

For many of the students, this encounter was our second time to explore these five basic solids. The very first time together in March 2011, the students explored models using clear plastic tetrahedrons and octahedrons. Both are pictured within the top rows of our working model of the Big Board. To go inside these models, essentially dividing them in half, requires a little finesse. Simply divide each edge in half and that point becomes a new vertex. With the tetrahedron there are six edges and within the octahedron there are eight edges. Connect all the new vertices and you have the simplest internal structure. Within the tetrahedron are four half-sized tetrahedra in each corner and an octahedron in the middle. Within the octahedron there are six half-sized octahedrons in each corner and a tetrahedron in each of the eight faces.

The students also made icosahedra out of 12 tetrahedrons. It was quite a lot of fun.

The second time with these kids would be more of a challenge. It would be the day just prior to their Christmas break.

The universe in 202.34 -to- 206 steps.  When we began finding simple math errors, the number of notations increased from 206 to 215 (it became our fudge factor). Then a leading astrophysicist said, “There are 206 notations.” Then on May 2, 2012, a NASA physicist made the calculation based on the results of the SDSS-III Baryon Oscillation Spectroscopic Survey  (BOSS).   He reported 202.34 notations.  At that time, looking throughout the web, we could only find bits and pieces of this work.

An earlier history began with the study of perfected states in space time.
Sometime around 2002, at Princeton with geometer, John Conway, the discussion focused on the work of David Bohm, once a physicist from Birbeck College, University of London. “What is a point?  What is a line? What is a plane vis-a-vis the triangle?  What is a tetrahedron?” Bohm’s book, Fragmentation & Wholeness, raised key questions about the nature of structure and thought.  It occurred to me that I did not know what was perfectly and most simply enclosed by the tetrahedron.  What were its most simple number of internal parts?  Of course, John Conway, was amused by my simplicity.  We talked about the four tetrahedrons and the octahedron in the center.

I said, “We all should know these things as easily as we know 2 times 2.  The kids should be playing with tetrahedrons and octahedrons, not just blocks.”

“What is most simply and perfectly enclosed within the octahedron?” There are six octahedrons in each corner and the eight tetrahedrons within each face. Known by many,  it was not in our geometry textbook. Professor Conway asked, “Now, why are you so hung up on the octahedron?” Of course, I was at the beginning of this discovery process, talking to a person who had studied and developed conceptual richness throughout his lifetime.  I was taking baby steps, and was still surprised and delighted to find so much within both objects.  Also, at that time I had asked thousands of professionals — teachers, including geometry teachers, architects, biologists, and chemists — and no one knew the answer that John Conway so easily articulated. It was not long thereafter that we began discovering communities of people in virtually every academic discipline who easily knew that answer and were shaping new discussions about facets of geometry we never imagined existed.

Of course, I blamed myself for getting hung up on the two most simple structures…  “You’re just too simple and easily get hung up on simple things.”

My family knows about this curious hang up of mine. They have seen these models on my desk.  We made a pseudo-Rubik’s cube type of game out of the octahedron. One of younger ones in the family is the geometry teacher in the family’s small private high school. “Come in and introduce the kids to Plato’s five basic solids.” That’s about my level. In so many ways, those kids were actually more advanced than me.

During one of my days with them, we made icosahedra with twenty tetrahedrons in each. It was not perfect geometry; the tetrahedrons had to be taped together and you could feel them moving. I called it squishy geometry, but told them that I have yet to find a good discussion about it under quantum geometry or imperfect geometries, “…but when I find it, I’ll report in.”

At first, our dodecahedron was a simple paper thing. We were trying to think of its simplest number of parts… “Could it be twelve odd objects coming into a center point, each with a pentagonal face and three triangular sides?” It didn’t seem like it would readily be extensible.  On my desk was a “Chrysler logo” made up of five tetrahedrons. There was always a gap — squishy geometry — but it trigger a thought, “What would a pseudo-dodecahedron look like if it were made of twelve of those pentagonals (each made up of five tetrahedrons)?”  Very quickly we had a model. A few hours later we were filling it with Play Doh to see what was within it.  And just within, we found an icosahedron waiting.

Now that was fascinating to us, but is it? Is it common knowledge among all the best-of-the-best within mathematics, chemistry, and physics? We did not know and it was put on our list of things to do.

In thinking about a sequel class to that earlier time together, we began focusing on exponential notation. Having learned a little about base-2 notation  — my first time over these grounds — we published pages deep within our business website to begin to share it with a wider audience:   http://smallbusinessschool.org/page883.html

If you find it of some interest, there are links to more background pages from both.

Can Plato’s five most basic objects in some way hold each progression together in a mathematical relation? Is it meaningful in any way? We would all enjoy hearing from you.
Please drop us a note!   –  BEC

March 2013 analysis

Possibly the most simple, internally-consistent view of the universe, all mathematically notated, necessarily interrelated, within 202.34 doublings from the Planck Length, the smallest measurement of a length, to the Edges of the Observable Universe, the largest.

Might our universe and this world be more simple than we have ever thought?

Have you seen the entire universe mathematically related and notated on a single chart? We couldn’t find it on the web so we did it ourselves.  In just 202.34 steps this chart goes from the smallest measurement to the largest. It all started in a high school geometry class so it is relatively straight forward and easy to understand, yet it opens some mystery as well. It is difficult to figure out how to interpret and work with the first 60 steps (or doublings, or layers, or notations). These are extremely small and, to date, have not been addressed as such by the academic community. Yet, these steps may open a way to understand our universe and ourselves in new ways.

Let’s take a look (You can open it here within a new window).

At the very top of the chart there are two rows of the most basic three-dimensional figures. The top five are named after Plato and are simply referred to as the five Platonic solids. It seems curious that only a very select group of people ever look inside these figures. If children did, this simple view of our universe would be second nature. Take any of those figures and divide each edge in half and connect the points. Keep doing it. In just 101 steps, you will be approaching what most scientists believe is the smallest possible measurement of a length in this universe (the Planck Length). A contemporary of Einstein, Max Planck formulated that measurement in 1899 and 1900. His most basic measurements have been around for awhile and today are generally considered to be among the the natural units or fundamental constants of our universe.

To make things a little easier we should start at the bottom of the left three columns of the chart at the Planck Length, 1.616199(97) x 10−35 meters.  There are many others who use the simple figure, 1.616×10−33  centimeters.

The next step, multiplying each result by 2, is called base-2 exponential notation. Now let’s move up the chart. At step 101 at the top of those columns on the left, we emerge with the width of a fine human hair. Multiply that by two and you are at width of a typical piece of paper, step 102 on the right.

Now go down those three columns on the right side of the chart. Continue to multiply by two. In just over 101 steps you will have gone out past the Sun, then exited the Solar System and then the Milky Way, and quickly pushed out to be in the range of the edges of the observable universe.

We wanted to give this chart a highly-descriptive name so we called it, Big Board – little universe.

Big Board – little universe:

From the Planck Length to the Edges of the Observable Universe.

Yes, this project started back in December 2011 in River Ridge, Louisiana just a few miles up river from New Orleans. Within a few hundred feet of the river is the John Curtis Christian School. Though well-known for football, their academics are very good. In the geometry classes they had been studying the platonic solids. Strange things can happen when one is invited to be a substitute teacher, essentially just an assistant for students and their teacher, Steve Curtis, who is part of our extended family.

December 19 was the last day before the Christmas break. What a day to be a substitute! One quickly asks, “How do you keep their attention? What could catch their imagination?” For example, “How could one make that simple dodecahedron (pictured) a bit more interesting?” The first and only other time with these students was used for model building so they could explore the inside structures of the basic five. The dodecahedron was not part of that effort, so to make it more accessible, we asked, “Why not make each face of that dodecahedron out of five tetrahedrons (pictured)?”

That makes the familiar strange and the strange a little more familiar.

Indeed. That object is known as the Pentakis Dodecahedron. We filled the inside cavity (pictured) with Play Doh. In a few days, that unusual object was removed and the obvious pieces were carved out.  It was in this process when the key evocative question was asked, “How many steps within would we have to go to get to the Planck length?” We assumed thousands and found just over 100. Flummoxed! “Why haven’t we used this before? Could it be that it’s just too simple?”

It was a straightforward task to do the simple base-2 math to create the first draft of what would become a rather big board. On December 17, the first draft was printed at Office Max in Harahan, Louisiana. Their widest paper for this kind of thing was 24 inches. “Let’s do it.” The resulting chart measured ten feet long. It didn’t take long to agree that it was too big and awkward so on the next day, two smaller charts, 12″ by 60″ were printed.

We put the two small charts on the left and right side of the class and cut the biggest in half and put the top section in the front and the bottom in the back. The setting was magical.

Now, there is a huge history of work that has already been done using base-10 exponential notation. Kees Boeke, a high school teacher, started that work in 1957 in Holland and it has become a staple of the classroom to study orders of magnitude. Although the big board is quite analogous to it, it has a very different sense of itself. Instead of multiplying and dividing by simply adding or subtracting a zero (0), we were much more visceral, emulating natural cellular growth, by using base-2 exponential notation.

That Big Board – little universe became Version

Not too much later, we decided to start at the Planck length and just multiply by two. It worked out better and kind-of-sort-of confirmed our earlier work. That became our next version which you see here.

What does it mean and what can be done with the data?

1. The universe in 202.34 steps. This chart is a simple tool to help order information. When we began finding simple math errors within Version 1, we turned to the professionals. A leading astrophysicist said, “There are 206 notations.” Then on May 2, 2012, a retired NASA physicist, Joe Kolechi, made the calculation based on the results of the Baryon Oscillation Spectroscopic Survey (BOSS). He reported 202.34 notations. We trusted him so we are using his calculation.

2. The Planck Length, the first step and the next 60 steps. We have thought and thought about the Planck length. It is an elusive concept defined by three fundamental physical constants: the speed of light in a vacuum, Planck’s constant, and the gravitational constant. Yet, what is it? For over 100 years, people have attempted to define it more richly than 1.616×10−35 meters.

It was time to engage some of the students in some speculative thinking, “Let’s do a series of thought experiments.”

First, of course, we will have to assume that Max Planck was right. Second, even if the Planck Length is a dimensionful or a dimensionless number, it is an actual measurement of a physical unit and it can be multiplied by 2. And third, it can be understood to be a very special case of a simple point. It is anybody’s guess if it defines some kind of special singularity.

Even as a simple point, when multiplied by 2, there are two points. Freeman Dyson, physicist-exemplar with the Institute for Advanced Studies of Princeton, New Jersey argues that when we multiply by two, we should actually be multiplying by three, one for each dimension of space. I would counter that each point exists in three-dimensions but each is still a singular point. It doesn’t much matter anyway; there are plenty of points to go around.

Within ten steps, multiplying by 2, there are 1024 points. Within twenty steps, there are over a million points. Within 30 steps there are over a billion, in 40 steps over a trillion, in 50 steps over a quadrillion (1000-trillion), and at 60 over a quintillion (1,152,921,504,606,846,976).

This all started with Plato’s five basic solids and thoughts about basic structure. Though most people do not give it much thought, it has been studied throughout much of our history, seemingly formalized by Pythagoras and extended by Plato. Our working concept was that the basic structure of the five platonic solids in some way permeates every subsequent layer (notation, doubling or step). And, if this simple-yet-idiosyncratic worldview can hold water, then in a substantial way, these five figures would, in very special ways, become the backbone of our constants and universals.

Constants and Universals. How do we go about defining what is truly universal and constant? Never an easy task, most often based on a combination of logic, mathematics, and consistent measurements, the constants have proven true throughout all time and within any space. The universals are constants understood by the most-respected scholars throughout time and they have generalized and extended these constants in meaningful ways. Some people believe these concepts open pathways to understand how it is that there is space and time, and human life and consciousness. Today, what has been rigorously dependent on the study of physics and then the other sciences, has evolved to include religion, logic, ethics, value, and even business.

With that as a platform, a key question to ask is, “What concepts are shared by all of these disciplines?” Then we ask, “What concepts are the most simple?” And also, “What concepts could have a face of perfection?” Those three questions opened the way to a very simple container, a generalized model within which to work. It is emergent, internally-dependent form – function (the faces of perfection) and the imperfect quantum world:

• Order – Continuity and discontinuity
• Relations – Symmetry and symmetry-breaking
• Dynamics – Harmony and not-harmonic, dissonant, discord

This work dates back to 1979 at MIT regarding first principles with 77 leading, living scholars from around the world but that work went nowhere until the encounter with the geometry kids of Steve Curtis’s classes at John Curtis Christian School in River Ridge, Louisiana.

It is difficult to know if a set of ideas is worth pursuing. The first challenge after that class was to do a literature search. We found all kinds of supportive information but nothing using base-2 exponential notation. The next step was to test the ideas with friends and family. It is embarrassing to be naïve and wrong at the same time, so some caution was exercised.

By March 2012, we had no serious detractors, yet no deep confirmation that the Big Board was really useful. To push the judgment and to have a foundation for collaboration, we wrote it all up in the style of Wikipedia for Wikipedia. When the first draft went up in April, it quickly found several protesters who said, “This is original research. It needs scholarly review before we will trust its efficacy.” By the first week of May, it was taken down. It had a very short run, but it was good theater.

On a personal note, in my very early days of study, the chairman of the MIT physics department, Victor Weisskopf, helped me with an invitation to visit with John Bell at CERN Laboratories. Bell’s inequality equations as applied to the Einstein-Podolsky-Rosen thought experiment of 1935 had rendered most enigmatic experimental results. Though way over my head, I knew enough to ask a few questions about the nature of information, the nature of thought, and the very nature of a thing, particularly of a photon, the force carrier of electromagnetism. That was 1977. This domain of inquiry has been swirling around over many years.

So now, with this rather skeletal model of the Big Board as our working construct, it was easy to wonder, “Have we come full circle? Are we back looking at the same questions that we were asking in throughout the ’70s, particularly in 1979?” So, to get properly oriented, based on that simple construct, order-continuity, relations-symmetry, and dynamics-harmony, are there particular questions that could be asked to clarify a path? For example, how is it that there is continuity between layers? What precipitates discontinuity? When is there symmetry-making and symmetry-breaking? What algorithms and formulas might make these simple interior models begin to cohere and function in such a way as to explain the phenomena within theoretical physics and quantum theory?

The first 60 notations, steps, doublings or layers. To date, the first 60 are only measured with colliders like the Large Hadron Collider at CERN labs. These colliders begin their work at the 66th notation and it is anybody’s guess as to how many notations have been utilized and articulated. The results from the colliders render a lot of data, but very little about the interface between information and the deepest structure of physicality. So, if nothing else, the imposed structure of base-2 notation could provoke new insights. For example, because there is an assumed inherent correspondence between layers, perhaps there are also analogical constructions within known notations and with information theory itself.

Speculations just might open a path for thought experiments.

Consider the work of the International Organization for Standardization (ISO) on the Open Systems Interconnection (OSI). They use seven abstraction layers to define the form and function of networking, a rigorous communications system. If all 202.34 layers of the universe in some way use an analogous construct, then as the first steps toward a thought experiment, we might simply force the OSI model over the first 60 layers as a starting point for rather free-associations and speculations. For example, perhaps 1-to-10 in some way perform like the physical layer, 10-to-20 like a data link layer, 20-to-30 like the network layer, 30-to-40 like the transport layer, 40-to-50 like a presentation layer, and 50-to-60 are like the beginnings of the application layer.

We have just begun to explore the OSI analogue. When the “thought experiment” door is opened, all kinds of wild and crazy notions begin to flow.

Just to get a feel for the numbers, we documented the climb up the 202.34 steps and put all those numbers on the web. An old acquaintance from MIT (and one of the world’s more rigorous-yet-speculative thinkers in combinatorial mathematics), Ed Fredkin suggests that it is akin to numerology. Perhaps. But new ideas have to start somewhere. If we suspend our harshest judgments that close doors and open ourselves to a new insights, by walking around in the chaos-confusion-and-the-unknown, sometimes new ideas and thoughts begin to catch a trace of coherency, and then rigorous, coherent thinking can follow.

If you look at the first column on the left of the Big Board, and go all the way down to the first 40 notations, you’ll notice there are over one trillion points at the 40th notation. In the left-most column at step 34 is the word, SPECULATIONS. Below it is “Quantum State Machine.” At this point in time, there are over 140,000 references in Google. Assuming that even .1% are of interest, there are 140 references to research and consider. The Modulus for transformation opens even more research to consider the question, “What is the transformation from one notation to the next?” Perhaps Theta-Fushian functions address the issue. How do cubic functions – cubicities — apply? With just four clustered points, the tetrahedron emerges. Within eight points the five-tetrahedron cluster (pictured above) emerges. Perhaps within such simplicity and with its imperfect binding (there is up to a 1.5 degree gap between faces), here is the beginning of an energy wheel that acts and works like quantum fluctuations. That gap is extended within the icosahedron and Pentakis dodecahedron. And here, between these structures we could be a heartbeat away from opening a new foundational study within physics-chemistry-biology, epistemology-and-mathematics, and cosmology.

There is so much more to consider and ponder. On a somewhat whimsical note, I concluded back in January 2012, in defense of the pursuit of this study, the following:

  • Each notation (step or doubling) can be studied to discern relations first within itself, then to the other two notations — “within” and “going out” — knowing ultimately that everything is related to everything.
  • It begins to envision every academic study in a necessary relation, one to another. Academic silos are so yesterday!
  • It re-introduces the platonic solids as a structural form for the study of continuity conditions within a complex enfolding of symmetries. Someday we may actually know what that means!
  • It opens symmetry groups to a much wider study in other disciplines beyond material science and theoretical physics.
  • It could open an exploration of imperfect geometry (or quantum geometries) whereby transcendental, imaginary and irrational numbers in some manner of speaking are discerned within the transformations from the perfect to the imperfect.
  • We might discover a form/function that aligns all 202.34 notations such that we are able to discern the Planck length as a truly standard measurement unlike the meter or inch-foot–yard.

Any one of the above statements is worthy of exploration.  Perhaps in so doing, new knowledge will open up.  That would be novel; so, of course, there’s more to come.

Thank you.


Regarding the title. In discussing this construction of the universe with physicist John Baez, (University of California – Riverside) he commented, “Well, it’s an idiosyncratic view of the universe.” I said, “That’s it. The initial title for this emerging paper.” Yet, to advance the concepts, we needed a more challenging, less self-effacing title. And until we are quite readily and intelligently challenged, it shall remain today’s new title.

Footnote: Perhaps the universe is nested in ways that we cannot measure or discern with physical instruments outside the mind. If you find it of some interest, let us know. Please share your thoughts. Can Plato’s five most basic objects in some way hold each progression together in a mathematical relation? Is it meaningful in any way? Could geometry ultimately be the Janus-face of numbers? We would all enjoy hearing from you.

Please drop us a note! – BEC

Big Board – little universe

Tetra86Octa86Hexa86Dodeca86 Icosa86 Tetrahedral-Octahedral Tetrahedral chain
Introduction: Starting in the center-left column below, a unit of measurement based on the Planck Length is divided-by-two 101 times until that measurement is the Planck Length, generally considered the smallest unit of measurement within space and time. In the center-right column, the same measurement is multiplied by two. In 101+ steps we are out to the edges of the observable universe. Assume that the simplest three-dimensional form defined by the fewest number of vertices is the tetrahedron.  Assume that the nesting of the basic Platonic structures within each other necessarily interrelates all structure of every manifestation within the known universe. The blanks are for students to find answers from examples within their studies, especially biology, chemistry, physics, astronomy and astrophysics and also to correct mistakes.  Go to the general overview…
Basic Questions, Basic Structures, and Form-and-Function: Could all structures be in some way derivative of the five basic solids discussed by Plato and the Greeks in and around 360 BC? If that concept is taken as a given, then questions about form and function could be re-engaged. Perhaps base 2 exponential notation is a place to start.Though apparent throughout the sciences, these five basic solids have not been used to develop an integrative model for human knowledge. Perhaps this is a step in that direction. Most academics today cannot tell you what is most simply contained within a tetrahedron or octahedron (by dividing the edges in half and connecting the vertices). Pictures below illustrate some answers. It seems that the simplest mathematical operations can still open new paths and logic to explore.
Go to the image file of this board
 Tetra2-98Octa2 Cubocta84 PentakisDodeca2 Icosa2 TOT2
GENERAL DISCIPLINES  (and Scale) PLANCK NUMBER EXAMPLES  (within ±50%) DECREASING IN SIZE Get smaller, divide by 2  (Center left column)
INCREASING IN SIZE Get larger, multiply by 2 (Center right column)
HUMAN SCALE 101. Range: Human Hair 40.9755356 microns Around 40 microns 101. Thicker Hair HUMAN SCALE
BIOLOGY 100. Sperm cell diameter 20.4877678 microns 81.9510712 microns 102. Thickness of paper MANUFACTURING
Cytology 99. Diameter of average human body cells 10.2438839 micronsor 1.02438839×10-5m .163902142 millimeters or 1.63902142×10-4m 103. Egg cell diameter ___
Microbiology 98. Diameter of average human capillary 5.12194196 microns orabout .0002 inches .327804284 millimeters 104. This period. Got it? ___
97. Red blood cells~2.4 µm 2.56097098 microns (µm) .655608568 millimeters 105. Large bacterium Bacteriology
Bacteriology 96. Rather small bacteria and red light (1.28 µm) 1.28048549 microns or 1.2804854×10-6 m 1.31121714 millimeters or 1.3112171×10-3m 106. Large grain of sand ___
NANO- TECHNOLOGY 95. Range of visible light ~ 400 to 1000 nm 640.242744 nanometers 2.62243428 millimeters 107. A small ant Myrmecology
___ 94. Nanoparticles ~ 100-to-10000 nm 320.121372 nanometers 5.24486856 millimeters (around a quarter inch) 108. Very small objects that we can still handle PHYSICS
___ 93. Thickness of gold leaf ~125 nm 160.060686 nanometers 1.04897 centimeters or 1.04897375×10-2m 109. Often parts of common small objects CHEMISTRY
___ 92. Nanowires 80.0303432 nanometers 2.09794742 centimeters 110. Rather small things ELECTRONICS
___ 91. Semiconductor chip 40.0151716 nanometers 4.19589484 centimeters 111. A spoonful TECHNOLOGY
Virology 90. Virus range 20.0075858 nanometers 8.39178968 centimeters 112. Anything 3.3 inches! BIOLOGY
___ 89. Thickness of a cell wall is around 10 nm 1.00037929×10-8 meters or 10 nanometers 16.7835794 centimeters or 1.67835794×10-1m 113: Small living and manufactured things ZOOLOGY
Immunology 88. Insulin molecule 5.00189644×108 meters 33.5671588 centimeters 114. Objects we handle BOTANY
___ 87. DNA helix ±2 nm 2.50094822 nanometers 67.1343176 centimeters or 19.68 inches 115. Agricultural and manufactured things ANTHROPOLOGY
Chemistry 86. Glucose molecule and Fullerenes diameter (Buckyballs) range: ~1.1nm 1.25474112 nanometers 1.3426864 meters or 52.86 inches 116. A child or other smaller animals SLEEP & VISIONS
Genetics 85. Distance between base pairs within DNA ±340 pm .625237056 nanometers or 6.25237056×10-10 meters 2.6853728 meters or 105.723 inches 117. A bed, a little stable or place to rest INSIGHTS & IDEAS
HUMAN SCALEPN 75 to 150 84. Diameter of a water molecule ±280 pm .312618528 nanometers or 3.12618528×10-10 meters 5.3707456 meters 118. A small bedroom PSYCHOLOGY
Picometrespm 83. Diameter of a carbon atom ±70 pm .156309264 nanometers or 1.56309264×10-10m 10.7414912 meters, 35.2411 feet 119. A home, a small barn or shop SOCIOLOGY
 ___ 82. Helium atom diameter 7.81546348×10-11 meters . 21.4829824 meters 120. Property FAMILIES
___ 81. Hydrogen atom ±25 pm 3.90773174×10-11 meters 42.9659648 meters 121. Larger properties RETAIL
___ 80. ____ 1.95386587×10-11m 85.9319296 meters 122. Complex systems CONSTRUCTION
___ 79. Use Huang scale 9.76932936×10-12m 171.86386 meters or about 563 feet 123. Big buildings or a little neighborhood GEOLOGY
___ 78. Wavelength of an X-ray 4.88466468×10-12m 343.72772 meters or about 1128 feet 124. A huge complex or a neighborhood ARCHITECTURE
___ 77. Diameter of florine ion 2.44233234×10-12 m 687.455439 meters 125. Farms and large complexes AGRICULTURE
___ 76. Gamma wavelength 1.22116617×10-12m 1.37491087 kilometers 126. Very small towns SMALL POLITICAL SYSTEMS
BEGINNING OF 75. Use Falstad scale 6.10583084×10-13m 2.74982174 kilometers 127. Smallest states TRANSPORTATION
SMALL SCALE 74. ___ 3.05291542×10-13m 5.49964348 kilometers 128. Towns AERONAUTICS
PN 1-TO-75 73.___ 1.52645771×10-13m 10.999287 kilometers or within 6.83464 miles 129. Small cities, or large towns JUDICIAL SYSTEMS
NUCLEAR PHYSICS 72. Average range of the size of atom’s nucleus 7.63228856×10-14m 21.998574 kilometers 130. Large towns LOCAL POLITICS
___ 71. Gold atom nucleus 3.81614428×10-14 m 43.997148 kilometers 131. Large cities ___
___ 70. Aluminum atom 1.90807214×10-14m 87.994296 kilometers 132. Small states ___
___ 69. Electron diameter 9.54036072×10-15m 175.988592 kilometers or 108 miles 133. Very small countries or anything within 100 miles NATIONAL POLITICS
___ 68. Helium atom diameter 4.77018036×10-15 m 351.977184 kilometers or 218 miles 134. Within the orbital range: International Space Station SPACE POLITICS
Femtometresfm 67. Neutron diameter Hydrogen – 1.75±×10-15m 2.38509018×10-15m 703.954368 kilometers 135. Countries ___
  66. Diameter of a proton or fermions (femtometre ) 1.19254509×10-15m 1407.90874 kilometers or about 874 miles 136. Larger countries ___
65. 36+ quintillion vertices 5.96272544×10-16 m 2815.81748 kilometers 137. Regions of earth ___
THEORETICAL PHYSICS 64. Neutrinos, quarks 2.98136272×10-16m 5631.63496 kilometers 138. Largest countries ___
Attometers 63. ___ 1.49068136×10-16m 11,263.2699 kilometers or about 7000 miles 139. Diameter of the earth ___
am 62. ___ 7.45340678×10-17m 22,526.5398 kilometers 140. GPS Satellite Altitude ___
61. ___ 3.72670339×10-17m 45,053.079 kilometers 141. ___ ___
VERY-SMALL 60. Over a quintillion vertices 1.86335169×10-17m 90,106.158 kilometers 142. ___ ___
SCALE UNIVERSE 59. Quarks 9.31675848×10-18m 180,212.316 kilometers (over 111,979 miles) 143. ___ ___
PN 40-to-60 58. ___ 4.65837924×10-18m 360,424.632 kilometers 144. Distance: Earth to Moon ___
___ 57. ___ 2.32918962×10-18m 720,849.264 kilometers 145. ___ ___
___ 56. ___ 1.16459481×10-18m 1,441,698.55 kilometers 146. Diameter of the sun ___
Zeptometers 55. ___ 5.82297404×1019m 2,883,397.1 kilometers 147. ___ ___
zm 54. ___ 2.91148702×10-19m 5,766,794.2 kilometers 148. ___ ___
___ 53. ___ 1.45574351×10-19m 11,533,588.4 kilometers 149. ___ ___
___ 52. ___ 7.27871756×10-20m 23,067,176.8 kilometers 150. ___ BEGINNING OF
___ 51. ___ 3.63935878×10-20m 46,134,353.6 kilometers 151. ___ LARGE SCALE
___ 50. Over a quadrillion vertices 1.81967939×10-20m 92,268,707.1 kilometers 152. ___ PN 150-to-202.34
___ 49. ___ 9.09839696×10-21m 184,537,414 kilometers 153. Range: Earth to Sun ASTRONOMY
___ 48. ___ 4.54919848×10-21m 369,074,829 kilometers 154. To go to Ceres asteroid ___
___ 47. ___ 2.27459924×10-21m 738,149,657 kilometers 155. Range: Jupiter-to-Sun ___
___ 46. Pati Preons 1.13729962×10-21m 1.47629931×1012m 156. Range: Saturn-to-Sun ASTROPHYSICS
Yoctometers 45. ___ 5.68649812×1022m 2.95259863×1012m 157.Range: Uranus-to-Sun Terametres (Tm)
ym 44. ___ 2.84324906×10-22m 5.90519726×1012m 158. Range: Pluto-to-Sun LARGE SCALE
___ 43. ___ 1.42162453×10-22m 1.18103945×1013m 159. ___ UNIVERSE
___ 42. ___ 7.10812264×10-23m 2.36207882×1013m 160. 24 hour light travel ___
___ 41. THE CHALLENGE: 3.55406132×10-23m 4.72415764×1013m 161. ___ ___
VERY-VERY, 40. Over 1 trillion vertices 1.77703066×10-23m 9.44831528×1013m 162. ___ ___
SMALL-SCALE 39. 549 billion vertices 8.88515328×10-24m 1.88966306×1014m 163. 7-day light travel ___
UNIVERSE 38. 274 billion vertices 4.44257664×10-24m 3.77932612×1014m 164. ___ ___
PN 20-to-40 37. 137 billion vertices 2.22128832×10-24m 7.55865224×1014m 165. ___ ___
36. 68 billion vertices 1.11064416×10-24m 1.5117305×1015m 166. ___ Petametres (Pm)
35. 34,359,738,368 vertices 5.5532208×10-25m 3.0234609×1015m 167. ___ ___
SPECULATIONS: 34. 17,179,869,184 vertices 2.7766104×10-25m 6.0469218×1016m 168. In the range of one light year (ly) (9.4×1015) 1 parsec ~ 31 trillion km or 19 trillion miles
Quantum State 33. 8,589,934,592 vertices 1.3883052×10-25m 1.20938436×1016m 169. ______ 1 parsec ~ 31 trillion km or 19 trillion miles
Machines (QSM) 32. 4,294,967,296 vertices 6.94152599×10-26m 2.41876872×1016m 170. Go to Proxima Centauri (39.9 Pm) 1 parsec (3.26 light years, 30.8 Pm)
(QSM) 31. 2,147,483,648 vertices 3.47076299×10-26 m 4.83753744×1016m 171. Distance to Alpha Centauri A & B (41 Pm) ___
___ 30. Over 1 billion vertices 1.735381494×10-26 m 9.67507488×1016m 172. Distance to Sirius (81 Pm, 8.6 ly) ___
Modulus for 29. 536,870,912 vertices 8.67690749×10-27 m 1.93501504 ×1017m 173. Distance to Tau Ceti (110 Pm) 100 Petametres or 11 light years (ly)
transformations (Mt) 28. 268,435,456 vertices 4.3384537×10-27m 3.87002996×1017m 174. Diameter of Orion Nebula (350 Pm) ___
27. 134,217,728 vertices 2.16922687×10-27m 7.74005992 ×1017m 175. Distance to Regulus star (730 Pm) ___
Mt 26. 67,108,864 vertices 1.0846134×10-27m 1.54801198×1018m 176. Omega Centauri diameter (1.6 Em) Exametre (Em): 110 light years (ly)
___ 25. 33,554,432 vertices 5.42306718×10-28 m 3.09602396×1018m 177. Thickness of our Milky Way (2 Em) Our Galaxy
___ 24. 16,777,216 vertices 2.711533591×10-28m 6.19204792×1018m 178. Distance to Helix Nebula (6.2 Em) ___
___ 23. 8,388,608 vertices 1.35576679561×10-28m 1.23840958×1019m 179. Distance to the Orion Nebula (13 Em) 12.38 Em
QSM 22. 4,194,304 vertices 6.778833978×10-29m 2.47681916×1019m 180. Horsehead Nebula (15 Em) ___
___ 21. 2,097,152 vertices 3.3894169890×10-29m 4.95363832×1019m 181. ___ ___
EXTREMELY 20. 1,048,576 vertices
1.69470849×10-29m 9.90727664×1019m 182. ___ ___
SMALL-SCALE 19. 524,288 vertices 8.47354247×10-30m 1.981455338×1020m 183. Small Megellanic Cloud diameter in Milky Way (150 Em) 198.1 Exametres
UNIVERSE 18. 262,144 vertices 4.2367712×10-30m 3.96291068×1020m 184. To the center of our galaxy (260 Em) ___
PN 10-to20 17. 131,072 vertices 2.1183856181504×10-30m 7.92582136×1020m 185. ___ ___
16. 65,536 vertices 1.05919280907×10-30m 1.58516432×10921m 186. Go to Large Magellanic Cloud 1.5 Zettametre: 150,000 ly
15. 32,768 vertices 5.2959640453×10-31m 3.17032864×1021m 187. Small Magellanic Cloud (2 Zm) 3 Zettametres: 310,000 ly
14. 16,384 vertices 2.6479820226×10-31m 6.34065727×1021m 188. ___ ___
13. 8192 vertices 1.3239910113×10-31m 1.26813145×1022m 189. ___ ___
Note: Theta-Fushian functions 12. 4096 vertices 6.61995505672×10-32m 2.53626284×1022m 190. Distance to the Andromeda Galaxy 24 Zm
See: Models 11. 2048 vertices 3.30997752836×10-32m 5.07252568×1022m 191. ___ ___
1.65498876928×10-32m 1.01450514×1023m 192. (Fill in a blank) 101 Zettametres
Cubicities 9. 512 vertices
8.2749438464×10-33m 2.02901033×1023m 193. Go to Centaurus A Galaxy (140 Zm) ___
Primary QSM 8. 256 vertices
4.1374719232×10-33m 4.05802056×1023m 194. (Fill in a blank) ___
Primary Mt 7. 128 vertices
2.0687359616×10-33m 8.11604112×1023m 195. ___ ___
Nested Geometries 6. 64 vertices
1.0343679808×10-33m 1.62320822×1024m 196.  ___ Yottametre (Ym)
Primary cubicities 5. 32 vertices
5.171839904×10-34m 3.24641644×1024m 197. Length of the Great Wall (4.7 Ym) ___
Strings & Knots 4. 16 vertices
2.585919952×10-34m 6.49283305×1024m 198. Distance to the Shapley Supercluster (6.1 Ym) ___
Primary knots 3. 8 vertices
1.292959976×10-34m 1.29856658×1025m 199. Length of Sloan Great Wall (13.7 Ym) 12.98 Ym
Cubicity or string 2. 4 vertices
6.46479988×10-35 2.59713316×1025m 200.___ ___
Primary String 1. 2 points 3.23239994×10-35 5.19426632×1025m 201.___ ___
Singularity 0. 1 point 1.616199(97)x10-35m 1.03885326×1026m 202. EOU at 202.34 ___
Synopsis: This Big Board-little universe is to order data in a way to open a discussion about our basic assumptions — the universals and constants — that guide our thinking and work. An initial focus is Max Planck’s calculation in 1900 of the Planck Length.Very Brief History: The work began by attempting to find new starting points for creative thinking, new insights, even breakthroughs, regarding the very nature of space and time. In the 1970s the following first principles were formulated as preconditions for a space-time moment at the zero-point defined by Planck, Stern and Einstein.First principles: Deep within the fabric of life there is an energy, an abiding thrust to make things better, more perfect. That is the cornerstone of business, but much more. Simple logic tells us that there are three forms within functions that define an increasingly perfected state within an experience:
1. The first form that defines our humanity is order and its most basic function, a simple perfection, creates continuity.
2. The second form is a relation and its function creates symmetry.
3. The third form is dynamics and its perfection, a complex function, is harmony.These three — continuity, symmetry and harmony — just might be the precursors of a space-time moment.
A Working Project:A Big Board of our little universe. This work is copyright by three groups, all of River Ridge, PO Box 10132 New Orleans, LA 70123 USA Pentakis Dodecahedron Illustration 3.Pentakis
32 external vertices or points, 60 external tetrahedra, a layer of 46 asymmetrical
tetrahedra and an icosahedron in the center.
The challenge of four simple concepts:
1. A universal scale created by doublings. A simple scale that starts with a point at the Planck Length (PL),  assumes Planck’s logic and mathematics are OK and that the PL singularity, an actual measurement, can be doubled. At each step there is a physical measurement. It takes 202.34 doublings to go from the PL to the Edges of Observable Universe (EOU). See all of the above.  
 Icosahedron Illustration 2. Icosahedron20 tetrahedrons
13 points
with shared center point,
1.54 steradians
 2.  Nested geometries.  The first doubling renders two points and the second doubling four points. With four points a tetrahedron could be rendered; it is the simplest three-dimensional form defined by the fewest number of points. The third doubling renders eight points. With just seven of those points, a pentagonal cluster of five tetrahedrons can be inscribed (Illustration 1). With the fourth doubling, now sixteen points, the icosahedron with its thirteen vertices (points) can be created. (Illustration 2). A tetrahedron within the pentagonal cluster (Illus. 1) can inscribe four smaller tetrahedra and an octahedron within itself with just six of those points (and by dividing each edge in half).  More
This project was initiated for the geometry classes of Steve Curtis at The Curtis School in River Ridge, LouisianaVersion FiveTetrahedrons Illustration 1.Five Tetrahedrons,7 points7.36° gap or deviation

Gaps between faces are less than 1.5°

3.  Facts and Guesses. Simple math renders simple facts. What can be done with these numbers, images and forms? What functions can be intuited? Perhaps a challenge to students could be to use buckyballs and the basic Platonic solids to build a most primitive kind of machine.   More to come… This quest is a thought experiment that begins at the PL and proceeds with facts and guesses to edge of the observable universe.4.  Non-commutative geometry, irrational numbers…Another idiosyncratic application to number theory, non-commutative geometries, irrational numbers, and dimensionful numbers is to see all of these as the results of a modulus of transformation and gaps between faces of less than 1.5° (as seen in the seven-point, five-regular tetrahedra when each shares an edge).  Much more to come

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