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 1.0.0.1.

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?

**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.

-Bruce

**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