Let’s take a look (Pictured on the right, 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 those vertices (opens in new window). Each little circle is a vertex. Keep doing it. In just 101 steps, you will be approaching what most scientists believe is the smallest possible measurement 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  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)x10^-35 meters. Others use the simple figure, 1.616×10^-35 meters or 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 the width of a typical piece of paper; that is 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 (NOLA) and just downriver from the NOLA airport. 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 the students and their teacher, Steve Curtis, who is part of our extended family.

December 19, 2011 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 begin to explore the inside structures of the basic five.

The dodecahedron was not part of that effort, so to make it more simple, we asked, “Why not make each face of that dodecahedron out of five tetrahedrons (pictured)?”

That makes the familiar strange. Instead of a simple dodecahedron, this one had 60 external faces!

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 then cut that ten-foot board 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 Boeke’s work, it has a very different sense of itself. Instead of multiplying and dividing by simply adding or subtracting a zero (0), we begin with exacting measurements given to us from Max Planck.  Second, we have all of our geometries with us.  So, our chart is much more visceral; it has 3.3333+ times more notations. It emulates natural cellular growth and chemical bonding. Now, that was enough to get us going, yet we knew along the way we would find many other foundational reasons.

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 2.0.0.1 which you see here.

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

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.

Thought experiments anybody?

Perhaps it is time to engage some of the students in some speculative thinking.  I have asked among the most-curious of them,  “What is the next step? Can we do a series of thought experiments?”  The questions continued, “Could we just start by constructing simple models within the first ten steps and then become increasingly complex?  Could this study be a pre-science or hypostatic science where we begin to see the interface between perfection and imperfection?”  So where do we begin?

First, of course, we will have to assume that Max Planck was right and his concept is a good place to begin. Second, even if the Planck Length  is a dimensionful or a dimensionless number, it is still 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 vertex, some might say a point. It is anybody’s guess if it defines some kind of special singularity.

As a simple vertex, when multiplied by 2, there are two vertices.  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 vertex exists in three-dimensions but each is still a singular vertex.  It doesn’t much matter anyway; there are plenty of vertices to go around.

Within ten steps,  multiplying by 2, there are 1024 vertices.  Within twenty steps, there are over a million.  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).  One could do very complex geometries with all those vertices.

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, layer 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.

Attempting to Set This Work With Constants and Universals

How do we go about defining what is truly universal and constant?

Certainly not 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 in part based on those constants as 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 most-complex chemistry, 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 platform, 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

In practice, we therefor assume that there is continuity from the smallest to the largest measurement.  We assume that there is a deep-seated symmetry, even if it can not be observed,  from the smallest to the largest measurement.  And finally, that within every type of measurement, there are possibilities of transformations that account for all dynamic actions within our universe.

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.

From family to Wikipedia and back again to the family

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 had been taken down. Though it had a very short run,  it was good theater.

I learned early that idiosyncratic ideas are not much tolerated within the academy.

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. Yet, scholars demand informed questions, so, there were times I appeared naive. Always there was more to learn about the nature of information, the nature of thought, and the very nature of a thing.  What is a photon?  In what ways  is it a  carrier of electromagnetism?  Although that was way back in1977,  those domains of inquiry still swirl with questions.

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?

To get perspective on it all, a group at the high school is focusing on it.  The Argonne National Laboratory has sent us fifteen highly-exacting photographs from the work of their scientists within the small-scale world and the students have been challenged to take each photograph and assign it to a notation.  Nikon’s Small World photographs from their annual calendar and contest are also being used.  I have confirmed a comment by Prof. Dr. John Baez about this construct being idiosyncratic, and by asking questions of leading scholars around the world, have become the personification of idiosyncratic.

From ideas, to theories, to constructs, to mathematics, I have often heard and read that the simple models are more elegant than the complex and that simplicity has a special elegance and  beauty.  So, here within this paragraph will be the links to discussions and meetings with people, from our finest scholars to our most fresh-and-open children, when and where we have used this construct to explore the meaning and value of  life.

The next steps: The first 60 notations, steps, doublings or layers.

To date, the only possibilities for measurement of any of those first 60 can is within 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.

Highly-Speculative ideas that 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.

It seems a bit silly to explore the OSI analogue, but within analogies are possibilities of making the strange familiar and the familiar strange. When the “thought experiment” door is opened, all kinds of wild and crazy notions just might 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 vertices 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 a cluster of four vertices, the tetrahedron becomes possible. With five, two tetrahedrons. With seven vertices the five-tetrahedron cluster (pictured above) could emerge.  Using Chrysler’s description of their logo, we call it a Pentastar. 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 more whimsical note, I concluded back in January 2012, in defense of the pursuit of this study, the following:

1. 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.
2. It begins to envision every academic study in a necessary relation, one to another. Academic silos are so yesterday!
3. 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!
4. It opens symmetry groups to a much wider study in other disciplines beyond material science and theoretical physics.
5. 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.
6. 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. That would be novel; so, of course, there’s more to come.

Thank you.

-Bruce

Footnote.  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.”  It became 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, the current title shall carry this project forward.

Perhaps the universe is nested in ways that we cannot measure or discern with a physical instrument other than the mind. If you find it of some interest, let us know. Please share your thoughts. It appears that we all need to re-examine the simplest concepts and parameters more closely. Could Plato’s five basic solids 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