Questions-questions-questions: What is order? How big is this Universe?

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Is There Order In The Universe?
by Bruce E. Camber

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Chart #1

December 19, 2011, New Orleans: Our high school geometry classes created a simple, mathematically and geometrically-ordered view of the known universe. We also found an inherent geometry for disorder.

Yes, rather unwittingly we backed into developing what we now call our Universe View. We used a very simple logic and math. First, we divided the edges of a tetrahedron by 2. By connecting those new vertices, we were able to go deeper and deeper within until we were down in the range of the smallest measurement of a length. We then used that length to multiply those edges by 2 until we were out around the largest-known measurement of a length.

That first object, the tetrahedron, has an octahedron within its center and it too was divided by 2.

Our work began in December 2011. That simple exercise resulted in measurements which opened paths to challenging facts, rather fun concepts, obviously wild-and-crazy ideas, and truly playful speculations. Our first chart was colorful and focused on Planck Length.

Universe Table
Chart #2 One page!

The second chart tried to reduce it to an 8.5 by 11 inch paper!

Throughout this little article there are many references with links. However, there are just nine primary references to other pages. These links are also at the bottom of the page. Also, please be advised, that this project will always be a work in progress.

1. The Power of 2. There are 202 doublings (multiplying by 2) starting at the Planck Length, the smallest conceptual measurement of a length in the universe, out to the Observable Universe, the largest possible length. Within a few years we also did the simple multiplication of the Planck Time, side-by-side with the Planck Length, out to the Age of the Universe. Then on February 11, 2015 we posted our very first draft of a table of the basic five Planck Units (with a most-speculative guess regarding temperature).

The number of notations (also known as doublings, domains, clusters, groups, layers, sets or steps) is a fact established by simple mathematicsReference #1 (below) goes to the initial chart of 2011.  Yes, it is just simple mathematics. And, we were quickly informed that there was a precedence for it… “like Kees Boeke’s base-10!”


In 1957 a Dutch high school teacher, Kees Boeke, used base-10 (multiplying by 10). He found 40 of the 62 base-10 notations. Yet, we believe Boeke’s work is the very first mathematically-driven Universe View. We were unaware of Kees Boeke at that time our work began. Also, we started with (1) embedded geometries, (2) the two measurements, Planck Length and Observable Universe, (3) a simple logic based on the concepts of continuity and symmetry, and (4) multiplying by 2 (base-2 exponential notation). It was not just a process of adding and subtracting zeros. Because base-2 is 3.3333+ times more granular than base-10, it is more informative and natural; the geometries create natural symmetries and levels of imperfection for symmetry-making and symmetry-breaking; and, it mirrors the processes in cellular division, bifurcation theory the dipole nature of chemical bonding, combinatorics, group theory, and complexification (1 & 2).

2. Inherent Geometries. We were studying tetrahedrons and octahedrons, two of the most simple Platonic solids. We started our project by dividing each edge of a tetrahedron in half. We connected those six new vertices and discovered a half-sized tetrahedron in each of the four corners and an octahedron in the middle.

We did that same process with the octahedron and found six half-sized octahedrons in each of the six corners and a tetrahedron within each of the eight faces (link opens a new window). We did that process of going within about 118 times. On paper, in about 50 steps we were inside the atom; and, rather unexpectedly, within another 68 steps we were in the range of the Planck Length.

We then multiplied our two objects by 2 and within about 91 notations or steps, we were in the range of the Observable Universe. Then, to standardize our emerging model, we began at the Planck Length and multiplied it by 2 until we were at the edges of the known universe. We had some help to calculate the number of notations.  We settled for a range from 201 to 205.1  (Reference 2 – See point #4   within those 15 points).


Because we started with a geometry, we learned ways to tile the universe with that geometry.
It is also quite simple. It puts everything within a mathematically-compact relation that over the years has had a wide range of names from the aether (or ether), continuum, firmament, grid, hypostases, matrix, plenum to vinculum. We call it, TOT tilings. The TOT begins with a ratio of two tetrahedrons to one octahedron.  That combination fills three-dimensional space perfectly. Also, there are two-dimensional tilings everywhere within and throughout the TOT tilings! There are many triangular tilings, square tilings, hexagonal tilings and combinations of the three. One of the most simple-yet-fascinating is created by that group of four hexagonal plates within every octahedron. Observing the models, one can readily see how each of those four plates extend as four hexagonal tilings of the universe.  Each is at a 60 degree angle to the other and each group of four shares a common center vertex.

It is all so fascinating, we are now exploring just how useful these models can become.

That tiling is a perfection, however, imperfections were readily discovered. Using just the tetrahedron, we found that not all constructions fit together perfectly. For example, the simple pentastar, a five-tetrahedral cluster, cannot perfectly tile space; it creates gaps.

Those gaps have now been thoroughly documented; yet to the best of our knowledge, Frank & Kaspers were the first to open this discussion in 1958.  Englishman F.C. Frank was knighted in 1977 for his lifetime of work.

Using simplicity as our guide, we concluded that here is one of the early beginnings of an imperfection. This shape is created with just five tetrahedrons and seven vertices. We refer to this object as a pentastar. It has a gap of about 7.36° (7° 21′) or less than 1.5° between each of the ten faces.

There is a quite fascinating warping and weaving between the perfect and imperfect.

By adding just one more tetrahedron to that pentastar cluster, a 2D perfection is created by the hexagonal base of six tetrahedrons.  Then, by adding more tetrahedrons it can become the 20 tetrahedral cluster known as the icosahedron, and then out to the 60 tetrahedral cluster, the Pentakis Dodecahedron.

We dubbed these imperfect figures, squishy geometry; the constructions have considerable play. Yet in more temperate moments, we call this category of figures that do not fit perfectly together, quantum geometry.  At that time, we did not know there is actually a disciple within geometry and theoretical physics defined as such.

3. Numbers and Potential Geometries Gone Wild. By the 10th doubling there are 1024 vertices. Assuming 1 for the Planck Length, there are then 2, 4, 8, 16, 32, 64, 128, 256, 512, and 1024. The simple aggregation of all notations up to 10th would be 2000+ vertices. Within just the 20th doubling (notation) there are over 1-million vertices, within just the 30th notation over 1-billion, the 40th notation over 1-trillion, and the 50th over a quadrillion vertices. By the 60th notation, a quintillion more vertices are created and that measurement is still below the range of our elementary or fundamental particles.

Imagining all the possible hidden complexities has become a major challenge!

Although this rapid expansion of vertices within each doubling is entirely provocative, it became even greater when we finally followed the insights of Freeman Dyson (Reference #3 – point #11). Dyson is Professor Emeritus, Mathematical Physics and Astrophysics at the Institute for Advanced Studies in Princeton, New Jersey. He said, “Since space has three dimensions, the number of points goes up by a factor eight, not two, when you double the scale.”  On the surface, it is straightforward, yet we are now trying to get the deepest understanding of scaling laws and dimensional analysis to most fully work with Dyson’s comment. Also, we believe that scaling symmetries are necessarily involved with the transitions from one doubling (domain, layer, notation or phase) to the next.

4. Driving Concepts. The simple mathematics provides a basic order and continuity that we have imposed on the universe. The simplest geometries provide a robust range of symmetries and relations. Add time and put these objects in motion, folding and enfolding within each other like a symphony, and we can begin to intuit very special dynamics and a range for harmony (Reference #4).  When those concepts were first written up back in the 1970s, it seemed to describe a perfected state within space and time, but it was too vague. It needed a domain or container within which to work and it seems that this just may be it (opens new window/tab).

5.  Big Board-little universe and the Universe Table (Reference #5). By September 2013, a class of sixth grade students got involved and a core group of about 40 high school students continued to study this formulation. First, it seemed like an excellent way to visualize the entire universe in a systematic way and on a single piece of paper. Second, as a simple ordering tool, it placed most of the academic disciplines in the right sequence. Mathematics, logic, philosophy, theology and ethics seemed to apply to every notation. An interdisciplinary study called STEM (for Science-Technology-Engineering-Mathematics) seeks a deeper and more vibrant exploration of all four. This chart readily did that and more. Our chart was developing a special traction. It was working for us.

We then began observing some very simple correlations between notations and let our imaginations work a little overtime.

6. Keys to humanity are in the middle of the Known Universe. Within our range from 201+ to 205+ notations, human sperm is within notation 100, human hair within 101, the thickness of paper (upon which we record our history) 102, and the human egg 103 (Reference #6).

That seems like a concrescence of meaning.

We are just starting to parse the 201+ notations in thirds, fourths, fifths… using musical notation as the analogue and metaphor.

7. The first 60+ notations, doublings, or layers are unchartered. We asked, “What could possibly be there?” To get some ideas, we started going back throughout history and philosophy. We placed Plato’s Forms (Eidos) within the first ten notations. Aristotle’s Ousia (Essence or structure) became the next ten from 11 to 20. Substances were 20-30, Qualities from 30-40, Relations 40-50 and then Systems 50-60. Within Systems we projected a place for The Mind (Reference #7), from the most primitive to the most developed.

Within these first sixty notations, it seems we just might be seeing the basis for isotropy and homogeneity within our little universe. As the domains (doublings, layers, notations, steps) approach he Planck Units, the number of vertices become smaller, and the everything in the universe increasingly shares  some aspect of the systems, relations, qualities, substance and structures, and perhaps everything shares all aspects of the forms. Here is the pre-structure of structure.  Of course, we are just being speculative.

It’s great fun to be speculative, yet we will try not to be too reckless!

“It seems that the cellular automata (of the Wolfram code) belong right within the Forms.” Of course, that’s also a simple guess. We continued, “And within Systems, we have all those academic subjects that have never had a place on a scientific grid or scale of the universe.”

We dubbed this domain “the really-real Small-Scale Universe.”

8. Einstein-Rosen Bridges, Wormholes & Intergalactic Travel The imagination can readily get ahead of facts, yet bridges and tunnels appear everywhere in nature. So, when we partitioned our known universe in thirds, we discovered that elementary particles and atoms began to emerge in the transition area from the first-third, our Small-Scale Universe, to the second-third, our Human-Scale. Well then, what happens in the transition to the third-third, from the Human-Scale to the Large-Scale Universe?

We decided to be wildly speculative.

In the grand scheme of things, the transition from the second-third begins with notations 134 to 138. At Notation 134 you could up on the International Space Stations,  just 218 miles above the earth’s surface. At Notation 137, you would be about 1748 miles up and at Notation 138, about 3500 miles up.

What happens? “Einstein-Rosen!” was the charge. “It’s the beginning of wormholes!”

That raised a few eyebrows. After all, we surely need a shortcut to explore the Large-Scale Universe. So, now we are calling on our leading space entrepreneurs, especially Elogn Musk of SpaceX, “Go out looking, but don’t go inside any of those wormholes yet. We all need to be thinking a bit more about their structure.” If we take it as a given that space is derivative of geometry (symmetries), and time derivative of number (continuities), we begin to see the universe quite differently.

Of course, we have far more questions than we have insights so we truly welcome yours.


9. A system for value, thinking, logic, reasoning and more. As you can see, our evolving Universe View was quickly becoming a structure for a rather idiosyncratic style of thinking, reasoning and logic (Reference #9).

The concept of a perfected moment in space-and-time was pushing us to think about order, relations and dynamics in new ways. Continuity, symmetry and harmony were becoming richer than space and time. This marks our first attempt to begin writing about this perception of our interior universe where our numerical-geometrical structure of the universe became its own inherent logic. It wasn’t long before we began thinking about how this structure could also be applied to thinking itself, then reasoning, and so much more. A mentor and friend from long ago, John N. Findlay, might call it an architecture for the thrust or zest for life.

This system seems to have within it many possibilities for seeing wholeness where today information and systems do not cohere, so we are glad to share these skeletal models (including the one just to the left) for your inspection. We hope you find it all as challenging as we have, and that you have enjoyed taking this rather quick tour through this work.

We are in the very early stages of this journey and we welcome your insights, your comments, and your questions. Thank you.


Endnotes and footnotes:

  1. The URL for the very first chart of our simple math:
    These pages were to support our attempt to publish a Wikipedia article about base-2 exponential notation from the Planck Length. That article was published in April 2012 but their specialists led by an MIT mathematician deemed it “original research” and it was removed early in May 2012. That was our truly first indication that our simple logic-math-and-geometry had been overlooked by the larger academic community.
  2. An analysis of 15 key points:
  3. Prof. Dr. Freeman Dyson, Professor Emeritus, Mathematical Physics and Astrophysics of the. Institute for Advanced Studies (IAS), Princeton, New Jersey since 1953., author (among hundreds of article and dozens of books)  of Interstellar Transport (Physics Today 1968), Disturbing the Universe (Harper & Row, 1979). This link opens within the IAS website.
  4. The first principles based on the concept of perfection:
  5. our first tour of the Big Board-little universe and the Universe Table in 2012 and in just 10 steps:
  6. Space Entrepreneurs to Star Wars VII:
  7. Belief systems:
  8. The circular chart just above.

More analysis: All these writings are in process. Here are our initial drafts:

There will come an invitation to participate, then perhaps a collaborative exploration of these questions:

Keys to this page, order

• This page became the homepage, June 4, 2014.
• The last update was Friday, August 19, 2022.
• This page was initiated in May 2014.
• The URL for this file is
• The headline for this article: Is the Order in the universe.
• First byline is: Questions-questions-questions: What is order? How big is this Universe?