A polyhedron with 20 faces, one of the five Platonic solids, there are many historical studies. The focus here is on the imperfect, the geometric gaps of the twenty tetrahedrons, extended here (five images below) as three groups of five, a group of four and singlet. In another configuration there are two groups of five and a ring of ten separating them. Notwithstanding, there is rotational symmetry.
Taken all together we tell the school children, “It’s squishy geometry.” And, it is! In the high school, we suggest to the students that it is part of the beginning of quantum geometries and quantum physics. Pictured here on the left is a group of five tetrahedrons bound by red plastic tape. The magnetic balls within each tetrahedron is just to remind us that there is no space that is “empty.” And, I’ll go so far as to suggest that there is also no singularity.
In this second image or five tetrahedrons, bound by blue tape, you can see part of the five tetrahedrons bound by red (above) through the clear plastic. There are in fact 20 tetrahedrons all bound together. They all share a common centerpoint. The next image of five are bound by a dark green plastic tape.
With this image, 15 of the 20 tetrahedrons are displayed. In this image you can see through the clear plastic tetrahedrons, there are reflections of the five bound by red at top and reflections of the blue on the bottom right. That leaves five more tetrahedron to find.
We find a cluster of of four tetrahedrons, here bound by clear tape and you can see wisps of the red, blue and green groups on the edges. This four-and-one configuration (the one being just below) with the three groups of five tetrahedrons harbors a special type of asymmetry and discontinuity. We’ll be studying it closely.
Yes, and finally, very clearly, all by itself, is a solitary tetrahedron. The red, blue and green tape of the abutting tetrahedral sets are readily discerned.
The icosahedral cluster (of twenty tetrahedrons) just maybe considered a transformational nexus and in future articles, that concept will be explored further.
Second email: Wednesday, April 14, 2021, 4:45 PM (Updated a little)
Dear Prof. Dr. Karen Uhlenbeck,
The transition from your mappings of a surface to the circle from bubbles may seem quite natural. Both Wheeler and Denis Weaire are with you, but the high school kids liked the idea of a sphere that could not be divided any further. Our spheres were more simple than the preon or instanton or … Up until recently we rather confidently thought our sphere was defined by Planck’s base units. Frank Wilczek, among others, gave us some assurance that Planck’s constant was properly defined. Today, that’s being challenged so we no longer harbor our earlier confidence.
We are more attracted to Kepler’s simplicity and the concept of cubic close packing of equal spheres. It was there that we learned how the tetrahedron and octahedron could be created by these scale-invariant, infinitesimal spheres.
That was a special day.
We were then attracted to combinatorial geometry and that is a possible direction for some of our students. We also decided that quantum geometry is another direction for others. “Be open to it all.” We actually created models with the tetrahedral gap for the dodecahedron (pentakis) and icosahedron and called it “squishy geometry” because there were so many gaps.
Of course, in our simplicity we thought of quantum fluctuations and even charted consciousness along that grid and allowed for “ontological fluctuations.” We are so idiosyncratic and quite desperate for help.
So, I said when I read your response, “I’ve got to do a little more homework right now… to see if there are connections to those 18 Uhlenbeck entries in ArXiv. I’ve got a long way to go but I am committed to work through all 18!
You are working on rarefied connections discerned over a lifetime of study. We are just trying to understand the most simple connections we can see.
If I get any clues to be able to communicate better to somebody as key to mathematics as you are, I’ll send it along otherwise I’ll not bother you again. Thanks again for the wake up!
Grace & peace, Bruce *
*********************** Bruce E. Camber
First email: Tuesday, 13 April 2021 at 10:30 AM (Updated a little)
Dear Prof. Dr. Karen Uhlenbeck,
Can anything good come out of work within a high school? Certainly,it would be quite naive and filled with many gaps, but just maybe with a bit of coaching…
We were studying the tetrahedron — https://81018.com/tot/ — and the octahedron within it. We could not find any references to the four hexagonal plates within that octahedron. I had asked John Conway about them in and around May 2001. It was a busy day in Princeton and that discussion didn’t get enough attention.
December 19, 2011, New Orleans: It wasn’t until my few days with the kids in high school that we did a three-dimensional Zeno walk down inside the tetrahedron and octahedron. We didn’t know that we were blazing a new trail. Dividing the edges by 2, connecting the next vertices, down inside we went. It was like a rabbit hole, that became a wormhole, and then an infinitesimal, ever-smaller path within…
After getting comfortable with the concepts, we went all the way down into Planck’s scale and then out to the age of the universe following the doublings of Planck Time and Length. It mapped out: https://81018.com/chart/ There were just 202 base-2 notations from the start until Now!
Now that was five years ago (2016). Now what do we do? We were splashing pages all over the web so collected them in one place: https://81018.com The homepage is always our latest stab in the dark.
If you do not have time to go to those pages, might you consider the questions just below that are on the homepage today? Your answers would answer our plead for a little guidance from somebody with the fewest number of gaps in this walk between physics and math. Thank you, thank you.
************************ Questions: 1. Might there be fundamental units of length and time, as well as mass and charge (similar to, but more accurate than the Planck base units), that are among the parameters that define the first moment or instant of the universe? Answer: Yes | No | Maybe Comment: ____________ 2. Might an infinitesimal sphere be a first manifestation of such base units? Answer: Yes | No | Maybe Comment: ____________
3. Might sphere stacking and cubic-close-packing of equal spheres be among the.first functional activities to define the universe? Answer: Yes | No | Maybe Comment: ____________ 4. Might the rate by which spheres emerge be determined by a fundamental unit of time which would be one sphere per unit of a fundamental length? For example, we used Planck Time. That computes to 539.116 tredecillion spheres per second given the value of Planck Time is 5.39116(13)×10-44 seconds. Answer: Yes | No | Maybe Comment: ____________ 5. Might base-2 notation be applied to create an ordering schema for these spheres? If that fundamental unit of time were Planck Time, approximately 436,117,076,900,000,000 seconds would pass to get to the current time which would be within the.202nd.doubling (base-2). Answer: Yes | No | Maybe Comment: ____________ 6. Might there be a range of perfection from the earliest notations and prior to any kind of quantum fluctuation, be it ontological or physical? Answer: Yes | No | Maybe Comment: ____________ 7. Might these spheres: ___(a) be defined by continuity-symmetry-harmony (which redefines infinity)? ___Answer: Yes | No | Maybe Comment: ___(b) …become the basis to define the aether? ___Answer: Yes | No | Maybe Comment: ___(c) …be the reason for homogeneity and isotropy? ___Answer: Yes | No | Maybe Comment: ___(d) …and, be the essence of dark matter and dark energy? ___Answer: Yes | No | Maybe Comment: ____________ 8. Might you be open to receive another eight questions about foundational concepts no sooner than eight months from today? Answer: Yes | No | Maybe Comment:
History. The crescendo of violence within the history of our past twenty generations (about 400 years) should stop us cold. It’s insanity. Who do we think we are? The nastiest side of all of us must be profoundly rooted within the deep chemistries of life. Evidence goes back to 4000 BCE. It’s nothing new. What is new is that we are now beginning to understand the actual geometry and physics of violence. Just maybe! Our naive hope is that as people shine light on the very nature of violence, perhaps we’ll begin to get a handle on it, and we’ll learn how to grow out of it. Yes, the antithesis of violence-and-killing is continuity-symmetry-harmony. Either we have growth and prosperity or chaos and dystopia.* -BEC
Introduction: Many of the concepts by which we live our life are incomplete. Some of them are wrong. And, we all know that that we could do better, but we don’t. First we have to unlearn what is wrong, affirm what seems to be correct, and then learn new concepts.
Some of those concepts will not seem new, but we’ll be looking at them in new ways. Basic concepts should give us a sense of value. The actual nature of values also should be a substantial part of all our discussions. So first, let me ask, “What is the origin of new concepts, emotions, ideas and values?” Nobody really knows, but let’s work on it!
Off track long, long ago. The geometer/philosopher, Plato, died in 348 BCE. His most famous student, Aristotle 1 became so respected and so astute about most things, when he made a claim about geometry that was wrong, people believed him anyway. He was such a genius and so sure of himself, it took over 1800 years to catch his biggest mistake.2 And even today, nobody is sure about the implications. The geometry of his error is still not well understood.
Aristotle believed the universe could be perfectly filled with tetrahedrons. It_can’t. With just the tetrahedron, there are geometric gaps.3 In our studies of the most-infinitesimal, there are no gaps.4 Tetrahedrons with octahedrons fill space perfectly. Yet, it’s assumed that infinitesimal gaps occur considerably before the particle-wave duality within physics.
We project that these gaps have everything to do with consciousness, the mind, identity, and creativity. We think of them as highly-refined, but much-much smaller synapse.5 Yes, the claim is made that synaptic functions have analogues in the infinitesimal and here is the beginning of consciousness.
If in some measure true, this facet of this domain will open a new path to explore one of our great mysteries. Even today our measuring devices can not size up anything much smaller than the particle-wave duality. Quite random but entirely consistent, the largest of these gaps is called a quantum fluctuation,6 yet scholars admit that they do not know what it is or why it is.
In these times, we need to know more about this most basic motion. It’s fundamental.
I know it sounds unlikely, but please bear with me. I project that it is within the gaps in the deep composition of our universe that formulas and ratios build relations and make new things happen. I project that it is within these gaps that we each get our unique identity. It is within these gaps that we find our creativity. Here is the indeterminant. And, here too is our madness. Here, too, is our debauchery and it is all called free will.
This first clue that we have missed over and over again, this gap opens up one’s sense of the potential you, but then Newton comes along and puts us squarely in a box and closes the lid and we begin to forget about tetrahedrons and octahedrons.
Space and time are derivative, finite, and quantitative. But we do not give up on those absolutes. Plus, there has been no simple alternative to Newton’s work so it has persevered.
And, we’re not highly motivated. Newton’s concepts agree with us. His concepts became our commonsense view of the universe. Absolute space and time give us a certain sense of independence. It gives us our own space and time, a sense of privacy. Ultimately it also gives us our ego, our own world and worldview, and that’s comforting. We can all live separate lives. It’s okay to be either the subject or the object. Things can be things. You are over there and me, I’m over here.
We’ve become entirely comfortable within Newton’s ways, so we’ll first have to break through his sense of our commonsense. Isaac Newton, though clearly a genius, did not have all the answers. He had quite a few good ones, but his biggest idea was also his worst. When he thought about space and time, he needed both to be absolutely everywhere, behind-within-and-throughout everything — the container of all that is. He really thought it was true. His arguments were convincing, so he was believed within his time, and eventually we all believed him. “It is just commonsense.” So, rather unwittingly our parents believed him, and most everybody in the family going right back into the 1700s, believed him (even though they may not have even known his name). Notwithstanding, he was wrong.
And, we got stuck, no, imprisoned within that simple body of data. We realized that the world is very large and we’re just a speck. And, now we’ve discovered that the universe is so much larger and we’re even more removed and less important.
The truth is that Newton’s wishful thinking is all a facade that we create in our minds.
A better idea. The truth of simple mathematics is that we are all profoundly and intimately related and this universe is not as big as it looks.
A new synthesis. Here spacetime is derivative and finite. Everything is necessarily building and evolving all the time; what we do effects not only our health and the health of others, it immediately effects the health of the nations and even the universe.
The conclusion. Everything you think-say-and-domakes a difference.
Space-time has a beginning, a starting point. And, the endpoint is right now, Today. According to our best scientific measurements, it all began about 13.81-to-14.1 billion years ago. Now that seems long ago but it’s a bit like yesterday, so let’s dig deeper.
Learning a new model.8 Back in 2016, while studying the charts that we had been working on since 2011 in our high school geometry classes, it became clear that this new model was mostly about the very earliest universe. We laid out each notation in a continuous line. Unusually long for the internet, that page scrolls right and left, or side to side. We started with that first unit of time defined mathematically in 1899 by the fellow who helped Einstein get out of that Swiss patent office in Bern, go back to school, and get his Nobel Prize (1921). Yes, that was Max Planck.
Max was a force unto himself; but, Einstein and World War I & II over-shadowed him. His most-seminal work of 1899 was virtually ignored. Even Max ignored it. But in 2001, an MIT professor, Frank Wilczek, opened the door and turned on the lights, and now Max’s work is part of the recognized foundations of physics.
Yet, at the same time, our physics community had 100 years of very fertile, imaginative work. Some of it is idiosyncratic and it will continue to be idiosyncratic until new concepts9 build bridges to it. Our scholars force-fit their work around the dominant, infinitely-hot, big bang paradigm. Though they’ve had some success, there were always concepts that didn’t fit well.
They tried to create a string theory, but it’s still balled up. They tried creating multiverses but couldn’t sell them to the public. They tried pulling math and physics together with what are known as Langlands programs, yet that bridge is barely a string over a huge chasm.
When we began to engage all these studies, we were naive. We’re still naive. But, at no time in our brief history did the infinitely-hot, big bang metaphor resonate because when we began trying to learn about it, we were also going deep inside the universe to the Planck scale chasing tetrahedrons and octahedrons. Once acclimated at a particular level, we would cut an edge in half and go down to the next smaller level. In 45 steps from our classroom, we were down among the particles. In 67 more steps we were into Planck’s scale.
We were lucky; we had Max Planck to instruct us. We learned a bit about our 112 steps down to the Planck scale. To be a bit more consistent, we used his numbers, particularly Planck’s length to go back up to our classroom, just multiplying the edges by 2. Just like you’d expect, in 112 steps we were back in the classroom, but then in just another 90 steps, doubling those numbers each step of the way, we were out to this day and the size of our universe.
You read that correctly. Just 90 additional doublings from our classroom! That’s a total of just 202 base-2 doublings to encapsulate the universe — everything, everywhere, for all time.
Back in 2011 we were having a good time discovering the universe when we were told that we were entirely idiosyncratic. A little concerned because we were nobody from nowhere special, it took nine years to learn how idiosyncratic everybody else was as well! So now, with a shade more confidence, we believe these 202 base-2 notations are mathematically logical and real. And, they actually seem to want to share more key facts about our universe.
Within this construct, the first claim became self-evident because of its simple logic. It’s a fact: “The universe can be parsed by 202 base-2 notations from the very first moment of time to this exact time, today.” To date, there are nineteen more insights that follow.12 That the universe is foundationally exponential is the 20th claim. Each of those claims opens up our universe for deeper explorations.
Surely if base-2 notation accurately describes a natural doubling mechanism of the universe, why not explore base-3, base-5, base-7… all the other possible prime number bases? When we do so, we will begin to find other functional mechanisms. Why not? If all notations are always dynamic, there is thrust, a natural expansion underway.
The universe seems quite opportunistic and seems willing to try every-and-any equation to find what works best. So, yes, these prime number notations are begging to be explored further.
I think here we will find openings to John Wheeler’s wormholes,14 a rather special mathematics and geometry to facilitate shortcuts throughout the universe. If so, as small as it is within 202 notations, the universe will get substantially smaller.
Among all the missing puzzle pieces that Planck gave us in 1899, the first 67 notations from the Planck base units to the wave-particle duality are new, unique, and important. Although still unknown to most of our scientific and academic community, these notations change the way we see our universe and ourselves. Here is a very simple beginning, the core connectivity that pulls everything closer together.
Numbers. One of Max Planck’s mysterious numbers was not infinitesimally small. It was grotesquely large. Planck Temperature was enigmatic at the get-go and it still is today.15 Though aware of it, that calculation was not part of the considerations of Stephen W. Hawking and George F.W. Ellis16 when they made their pronouncements in 1973 within their first and only book together, The Large Scale Structure of Space-Time. A relatively short book, it is rich with mathematical formulas and theory.
Today the best among our scholars are deeply aware of Planck Temperature and have gone through great gyrations to accommodate it into their attempts to recapture the records of the early universe. Often such work today is part of a theory of everything.
The fact is that an infinitely-hot beginning is entirely problematic.
It wasn’t always that way. In 1927 when Georges Lemaître 17 began to theorize a big bang, he postulated a cold start of the universe. Rather mysteriously in 1930s it turned hot and it seems that our best scholars do not know exactly how that happened.
Until our high school geometry class mapped the universe in those 202 notations, the Planck length and Planck time units were just too small to matter. As several possible mechanisms for doublings were discerned, that base-2 progression from those Planck base units increasingly look like a possible alternative waiting to be explored further.
If it all starts near absolute zero, then this cold start would be naturally superconducting. As one watches the mass-and-charge calculations double, it is easy to intuit how basic structures and processes might begin to emerge. There are 67 notations or doublings before the wave-particle duality emerges. That is three notations larger than the classic base-2 expansion introduced and studied as a result of the many stories that evolved from the original chessboard and a grain of wheat story.
It’s huge. It’s significant. Here is an entire domain and a potentially new science waiting to emerge. Plus, there are several mathematical systems without a home that just might find a place on this expanded grid. We have under-estimated the potential within the infinitesimal scale from the Planck Length to particle physics. Although infinitesimal, mathematics and logic still work very well at this scale.
Yet, there appears to be a catch.
Within this model of the universe, everything is building upon the structures before it. Every notation is necessary. Notation-1 is still active and has become a perpetual starting point.18 The first second, which is within Notation-143 is still modulating every new second of our universe.19 The first day is within Notation-160.20 One year is within Notation-168. One million years is within Notation-188. And, none of it is static. Every notation moderates its own self-definition in light of the abutting notations and those synchronized with any other base relation.
Nothing is static. Nothing gets pushed into a static past. Everything is active. With just 202 notations, every notation is not far away. Then, if we add other notational systems along with base-2, everything is ever so much closer.
This model of the universe begins with pi (π). Pi doesn’t exist in the universe, but it begins the process to define the universe. Some scholars claim 30 other dimensionless constants are needed to build this universe. Others have it up to over 300. All these dimensionless constants are by definition part of the infinite and some of them have been crossing the bridge since the first moment in time. These are part of what defines us. They pull us throughout the universe and into the infinite. We have a long way to go to understand it all, but let us all begin again where we left off in grade school. Let us begin to understand pi and all those dimensionless constants anew. Thank you. -BEC
Editor’s Note: There are concepts that are linked but not specially cited with a footnote or endnote. Those concepts that have been well-explored within this website are linked for the advantage of our new readers.
 Getting on track. The prior homepage focuses on four people, Aristotle, Newton, Planck, and Hawking. Often I’ll say, “Let’s go over that one more time. We’re missing something.” Yet, of the four, Aristotle made a very simple-but-important mistake. Obviously he did not have tetrahedrons and octahedrons to build models. He did not grasp simple tilings. That he thought the universe could be tiled and tessellated with just the tetrahedron is a mistake. The 7.35+ degree gap is part of our most fundamental geometries and it necessarily becomes part of our most-fundamental facts of physics.
In our model of the universe where the first object is a sphere and the first dynamic is sphere stacking and then cubic-close packing of equal spheres, the first notations necessarily manifest the perfections being generated from pi and tetrahedral-octahedral tilings and tessellations. Our model caught the spirit of Plato and Aristotle; the first ten notations were allocated to the concept of Forms (and also introduces functions). The next ten notations are allocated to forms becoming Structure then Substance. These earliest doublings (notations, domains) appear to manifest a perfection. The five-tetrahedrons and its gap do not. It seemed that the first gap could not manifest until Systems manifest; and within this model, that would be around Notation-50.
 Geometric gaps. Scholars are not talking about these geometric gaps. That should change. Here is the transition from perfection to imperfection. Here is the beginning of the indeterminate. If we stop to begin to sense the textures from the first notations to this gap, other basic concepts emerge. A perfect tiling and tessellating gives us our sense of (1) continuity and order, (2) symmetry and relations and a sense of balance, and (3) harmony and dynamics. These words are both quantitative and qualitative. Here is a framework for values and the only wiggle room is when that gap finally manifests. Here is the possibility of unique identity. Here opens the possibilities for creativity, and for new concepts and ideas. And here may well be the toxic mix that opens the way for disagreement, for ego, and even for getting angry. One can almost see a pathway where certain types of anger block rationality (continuity and symmetry) and opens a way to violence. Anger, however, can also become creative. To date, in our studies of scholars, no one has reached such idiosyncratic conclusions.
 Analogues in the infinitesimal to the synapse and synaptic functions. Freeman Dyson and I have argued about this point. Are the functions at one of the 202 notations analogically similar to another? I said, “Yes,” and he said, “No.” Though our dear professor has died, one of my goals in all this work is to discern in what ways the synaptic function is similar to, or analogically like, quantum fluctuations, and then how quantum fluctuations are like the very first, infinitesimal fluctuation (geometric gap) well below the notation that includes the particle-wave duality.
 Quantum fluctuations. To say there is confusion within the foundations of physics is understatement. So much is built on hypothetical mathematics built on hypothetical concepts. The study of fluctuations could readily benefit from a bit of simplicity and some analogical reckoning. More to come…
 Newton’s confinement. To break out of Newtonian space-time, we’ll continue our study of the progression of the four Planck base units: length-time and mass-charge. That base-2 progression carries its own logic. The four are necessarily defined by each other which makes them all quite finite and derivative. Yet, these constants are also defined by continuity (order), symmetry (relations) and harmony (dynamics) which obviously gave Newton his inspiration to begin thinking that both are absolute. The logic Newton was sought to define was the very nature of infinity. Space and time are derivative of each other and the infinite.
 New concepts from Langlands and String Theory. When you change your starting points, new concepts emerge. Robert Langlands wanted to start it all with automophic forms. It was a good place to start yet the most simple automorphic form needed some numbers (Planck’s base units) and a naive exploration of how those numbers and a sphere might behave.I was pleased to find an article by Ed Witten struggling to bridge Langlands and string theory.
So dumbfounded by all the new concepts that seemed to be popping up, list after lists were itemized to go over these seemingly new concepts just one more time.
 Insights begin to turn into claims. Sometime in 2012, when it became clear there was no scholarly work done on a base-2 chart of the universe from Planck Time and Planck Length to the current age and size of the universe, we had to decide, “What do you do with it? Is it significant? Is it just a bunch of numbers?” Yet, as a counter argument, one could say, “This chart, Big Board-little universe, started with geometry.” The argument is pushed further, “Just numbers and just geometry. Where’s the connection to reality?”
 The disarming 19th claim: This new model is so radical, most scholars will not accept it. They have too much invested in this universe with an infinitely-hot beginning. Those espousing multiverses, or strings, or Langlands automorphic forms, and at least a dozen other conceptual starting points, most will readily work within this highly-integrated, base-2, mathematical model of the universe. None of these other models are as simple as the Big Board-little universe. No others begin with the Planck base units. None redefine the very nature of time.
It has to be disconcerting to those who have invested their entire life on a platform like COBOL, an early computer programming language that is still being used in many legacy systems, but everyone knows that COBOL will not take us well into the next generation of programming.
 Shortcuts throughout the universe. Discussions about returning to the moon and going on to Mars has been the subject of many speculations online, in print, and with video. One of my favorite scholars wrote the book, Disturbing the Universe, whereby Freeman Dyson set his solar sails to catch the winds of the universe to take him to far away places if only in his mind. Though he rather pooh-poohed the idea that there is a foundational geometry that pervades everything, everywhere, throughout all time, at least he tried to engage. This site will continue to research any scholar’s work that appears to be building on John Wheeler’s wormholes!
This homepage is to explore how a change in our basic perceptions of ourselves and our universe could change everything else. This is a radically different model and it has taken us many years to begin to become comfortable with these conclusions. Unfortunately, there doesn’t appear to be a shortcut to learn how this universe is filled with shortcuts, but as evidenced here, it is very important to try.
Most recently, I’ve begun looking at the QBOL database for bacteria and viruses. There will be analogues everywhere.
First, Natalie Wolchover, an excellent science journalist for Quanta Magazine and the Simon Foundation (ArXiv and so much more) talks about its faster-than-fast expansion. Then, Peter Tyson, back when he was the Editor-in-chief of NOVA Online, declared in his piece, Absolute Hot, “…the Planck temperature, equals about 100 million million million million million degrees, or 1032 Kelvin.”
Now, our high school kids say, “That’s wicked hot.”
Tyson quotes a Columbia University physicist, Arlin Crotts, “It’s ridiculous is what it is. It’s a billion billion times the largest temperature that we have to think about,” referring to gamma-ray bursts and quasars. And though it may have seemed to be a logical place to begin, 1032 K is a most enigmatic concept within which to find answers to questions about the deep nature of our universe.
So, we placed Planck Temperature at the top of the chart within Notation-204 so by the start of the universe, the temperature was close to absolute zero. We are still puzzling Planck Temperature, yet cold start or hot start, it continues to be a difficult subject to begin to comprehend. Of course, we believe it all starts cold. It is just the most simple logic.
 George F. W. Ellis: With his co-author, Stephen Hawking, their 1973 book is simply titled The Large Scale Structure of Space-Time. George F. W. Ellis remembers his postdoc days at Cambridge University. He tell us (private correspondence) that there was no consideration of Planck Temperature in their earliest discussions about the structure of space-time. The Planck base units were not a concern. They started their work, as proclaimed in the opening chapter, “The subject of this book is the structure of space-time on lengthscales from 10-13 cm, the radius of an elementary particle, up to 1028 cm, the radius of the universe.” There is no examination of the range from the Planck base units to the wave-particle duality.
Also, there is within their book, The Large Scale Structure of Space-Time, Chapters 8 &10, consideration of space-time singularities, particularly what they call “the initial singularity of the universe. Of course, this model with its emphasis on an open universe that is constantly exponentially growing, there will, of course, be much more to come!
 Finite-InfiniteThese basic concepts also give us our sense of values and even suggest why it is that we get angry, and how certain types of anger can turn to violence. Yet, our anger can also become creative. Yet, as far as we have come in the studies by our scholars, there are no conclusions that suggest in other than perfunctory ways where our emotions and ideas come from.
In our mathematically-and-geometrically-integrated view of the universe., values manifest as facets of perfected states within space and time. Controversy will follow this model!
Just curious, our high school physics class has discussed this 100+ year search for dark energy and dark matter. Could it be hidden deep down within the infinitesimal scale from the Planck scale to the rather large wave-particle duality with its “gross” quantum fluctuations? It’s a bit counter-intuitive.* BTW, we are trying to think of a way y’all could “measure something too small to measure” (that is, much smaller than the domain of quantum indeterminacy). We are asking ourselves if there might be a way to boomerang back results (vis-a-vis periodic standing wave patterns) that could be measured. Is that a silly concept?
*Of course, at the Planck scale Time and Length are infinitesimal. Mass and Charge are small, yet if all units doubled, then continued to double, by the 50th doubling (notation), Mass is a substantial 2.450532×107 kilograms and Charge is starting to happen at .002111733 Coulombs. By the 64th notation Mass is up to 4.01495×1011 kilograms and charge is 34.59863 Coulombs, yet the Planck Time and Planck Length doublings are still below measurement. Of course, this assumes the universe is exponential, the Planck units are the start, and a doubling mechanism and thrust are identified. -BEC
Replying to @MediaActive @MollyJongFast @AmbJohnBolton @thedailybeast @ProjectLincoln @TheRickWilson @GOP “Will we ever find common ground? I think it is all worldview related; everyone’s just too limited. We need a highly-integrated view of the universe. It just maybe that special place. My early attempts at such an integrated view: http://81018.com are still naive.
The power of 264 is ginormous. Of course, 2202 must be about the whole universe!
Robert Williams in LinkedIn: Robert – http://81018.com is where I hang out virtually. I was with Arthur Loeb from 1970 to 1979 (Philomorphs). Yes, of course, Bucky was part of it all. My rather idiosyncratic viewpoint started with the tetrahedron and sphere: https://81018.com/stacking Would enjoy connecting with you! -Bruce
Introduction. Three thought leaders of our common history were also leaders in their own day. They held their ground when challenged. Throughout the years, their work became sacrosanct. Yet, among all the concepts they each introduced, I believe that they held onto a key conceptual mistake that still blocks us even today.
As a people we long for heroes and leaders; and, these three were ready to accommodate. Headstrong geniuses of their time, once they got into the limelight, they did not easily share it. They were not about encouraging others to discover their own gifts. They were more about imparting their genius to their adoring publics.
But, all three were fundamentally wrong about a foundational concept. They’ve thrown off generations of scholars. They’ve held us all back; and now it’s time to correct their mistakes, forgive them, and get on a path to breakthrough to new levels of insight.
Aristotle (384–322 BC, Athens)1 was wrong about a most-basic geometric fact. Obviously he could not have had perfect tetrahedrons within his toolbox. If he did, he would have known that one cannot perfectly tile and tessellate the universe with just tetrahedrons. He thought it was possible.
There are obvious gaps. Using the very tightest configuration of just five tetrahedrons sharing a simple edge, a most fundamentally important geometric gap is created. Simple logic tells us that it is a relatively early gap in physicality. Aristotle never saw this 7.35+ degree gap; and to his dying day, he promulgated an error as a truth.
Aristotle had such stature that this error was repeated by scholars for over 1800 years. Even today, not many people know about the gap. That should change. Our children should see it and begin to appreciate it profoundly.
What is it? I believe this simple gap is the beginning of the geometry of quantum fluctuations. That’s huge, but there is so much more. First, we know this — it is necessarily created by just five tetrahedrons which also outline a face of the dodecahedron, and define the primary faces of the icosahedron and the Pentakis dodecahedron. Aristotle’s mentor, Plato, defined the five basic solids — the tetrahedron, hexahedron (aka cube), octahedron, dodecahedron and icosahedron.
That gap has everything to do with basic structure. It just may also have everything to do with creativity, individuality, consciousness…
Enter Jeffrey C. Lagarias & Chuanming Zong. In 2012 they wrote a most-definitive article about the gap. These two mathematicians provide the background and an introduction to the people in the 1400s who observed and noted Aristotle’s mistake. Then, drawing from the 1926 research of D. J. Struik, they cite Johannes Müller von Königsberg (aka Regiomontanus, 1436–1476) as the first to recognize the error. The first to document it was by Paulus van Middelburg (1445–1534), a professor of astrology in Padua. Even though Aristotle’s error had finally been observed and analyzed, people focused on the fact that it was an 1800-year mistake. They also focused on the concepts within cubic-close packing of tetrahedrons and spheres. Over the years Kepler, Minkowski, Hilbert, and Hales — just to name a few — contributed insights to analyze technical aspects regarding packing densities.
In 2015 Lagarias and Zong were recognized for their work. That is all very interesting, however, we are still looking for the scholars who have asked and answered the question, “What is the net-net effect of that natural gap on our understanding of ourselves and our universe?”
Such questions should never be ignored, so let’s speculate a little.
Projections about the meaning of it all. We turn to our outline of the universe — the 202 base-2 notations from the Planck scale to this current time. Yet, we specially consider the uniqueness of the first 67 infinitesimal notations which are mostly below the thresholds of measurement. Notation-67 is the threshold of wave-particle duality. Notation-84 is the current limit of a measurement of a unit of time. If Notation-0 defines a finite-infinite cusp, these 67 notations are a new field for exploration.
What works survives. Every possible geometric combination that works provides form, function, structure, and then substance, relations, and networks of relations. What works best, survives. The universe, the penultimate opportunist, is creating something big that requires solid foundations. Perhaps somewhere around Notation-50, our universe begins to experiment with those five tetrahedrons with its built-in gap. Out of an abundance of shapes and configurations, the five tetrahedral structure is surrounded by perfectly manifesting forms and structures. Within a moment, that gap comes alive. Perhaps as early as Notation-50, the gap becomes a structural system, and then becomes a systemic fluctuation. Just a guess, the first expression of these systemic fluctuations just might be considered a primitive consciousness. By Notation-67, when it can be measured and “observed”, it will be defined as a quantum fluctuation.
Notation-50 and systemic fluctuations. Here we could postulate the beginning of identity, individuality, creativity, undecidability and unpredictability — a transmogrification from the perfect to the imperfect and indeterminant. Here may well be the birth of life as we experience it firsthand. The perfect is still there, yet it is now beginning to be masked with color, charge, flavors, sounds, and an assortment of other patinas.
A simple mistake by a legendary man has been hiding one of the most substantial mysteries of our time. It is time to absorb it and begin to absorb the new realities that it has been hiding.
It certainly feels true. When you look up into the clear night sky, it goes on forever. Doesn’t it? And, the answer is, “No, it only does as far as the current expansion.”
In 1687 Isaac Newton finished his landmark, three-volume book, Philosophiae Naturalis Principia Mathematica. Better known as just the Principia, it helped to firm up the foundations for what we now know as the scientific method. Though glimpsed by science and mathematics (1) dating back to Babylonian astronomy (c. 1830 BCE) and the Egyptian medical schools (c. 1600 BCE), and then (2) seriously enhanced by Aristotle and the logic within his treatise, The Organon, and then (3) energized with the work of Copernicus, Johannes Kepler, and Galileo, one can say with some confidence that science as we know it today consistently grew out of Newton’s Principia.
Yet, within this landmark writing came his most important contribution to the disinformation of the world’s culture: absolute space and time. These absolutes will not begin to recede as a footnote in our intellectual history until a better orientation is adopted by most people. That is a problem because, to date, alternatives have been non-intuitive. Leibniz came close in 1716 within his indirect dialogues with Newton through Samuel Clarke — Leibniz said space and time are relational, derivative and finite. So we ask, “If not the container for all that is, what is?”
For many that question is about one’s belief in God.
We try not to engage in “God Talk” on this website. One’s personal belief systems are largely a factor of family systems. Our attention is focused on universal systems and their constants.
Too simple for most, Planck had tied Planck Length and Planck Time together: Planck Time is equal to Planck Length divided by the speed of light. Of course, his little formula for Planck Time, can readily be re-written; the speed of light is equal to Planck Length divided by Planck Time.
That formula works! It worked in 1899. Using Planck’s numbers, the value is 299,792,422 meters per second. Without fanfare or celebrations, Max Planck had defined the speed of light using the mathematics of his equations a full 73 years before the National Institutes for Standards and Technology (NIST) accepted a slightly closer estimate, 299,792,456.2 meters per second defined in 1972 by K.M. Evenson and his group within the National Bureau of Standards in Boulder, Colorado.
Planck’s numbers are real; they work with real laboratory measurements. To date, the academy virtually ignores them. Applying base-2 creates a natural progression of those numbers and the first 67 notations have only been marginally explored. Here is an even more logical way to study the earliest universe where space and time are clearly derivative. The question is, “…derivative of what? …light?”
In 1905, Max Planck advised a young Albert Einstein as he began to tie mass and energy together. Yet, at no time has the academy started with Einstein’s sacred formulation, e=mc2, the very first step of the Planck scale.
A simple door with simple logic opens a new path to explore. Another “hiding in plain sight” story, we have been looking at this door since 1899. We seem to have a difficult time opening that door and walking down that extraordinary path on the other side. It follows continuity, symmetry and harmony and puts our unique time within this dimensionality into a whole new light.
“The subject of this book is the structure of space-time on lengthscales from 10-13 cm, the radius of an elementary particle, up to 1028 cm, the radius of the universe.”
They missed the real foundations. They missed the core structures. They missed all the really cool stuff from 10-13 cm down to and including the Planck Length at 10-33 cm. Within our base-2 outline of the universe, that range is from Notations 73-to-75 down to Notation-0.
By 1980 the big bang theory was clearly on the ascendancy. By 1988 with the publication of his book, A Brief History Of Time, especially with its rapid rise to multi-millions of books sold, Hawking was also on the ascendancy as the primary spokesperson for big bang cosmology. In 2016, he rhetorically asked his basic question:
When it comes to theories and mathematics, simple is better than complex.
For most of Hawking’s life, Max Planck’s numbers were considered by the leading scholars of this world to be a curiosity. Dirac had his very-large numbers. Planck had his very-small numbers. Dirac’s were too big to matter and Planck’s were too small to be significant. Again, it wasn’t until 2001 that Wilczek introduced the world to the meaning and value of Planck’s numbers. Slowly, the academy began to test those waters; yet, it was much too late for Hawking to enter. His 1973 co-author, G.F.R. Ellis, on the other hand, was open to explore the failures and deeper problems within the concept of an unfathomably hot beginning.
It doesn’t work, and it’s a conundrum. The big bang theory has been backed up with the humor of a twelve-year television series (2007-to-2019) that is now in endless re-runs. Yet, ever so much more daunting is the mythopoetics of Hawking’s life.
Diffusing the big bang will not be easy, but diffuse it we must.
The Now. In this model, there is no past and no future, only the Now for the entire universe. All of the 202 time periods are still active and everything, everywhere for all time is related to everything, everywhere for all time. It is all constantly encoding and re-encoding the universe.
Every thought-word-and-deed affects the look and feel of the universe.
And, because there are multiple paths throughout the 202 active notations (categories, clusters, containers, domains, doublings, groups, jumps, layers, periods, sets, steps…), in this model, it is not only a small world after all, it is also a small and intimate universe.
Currently there is no way around the naïveté within this three-point charge against three of the foremost scholars of our entire history. I expect each point will be hammered, yet it is only by such hammering can it all be shaped into real possibilities. Thanks. – BEC
Three sections follow: (1) Footnotes & Endnotes, (2) References, Reflections & Resources, and (3)_Miscellaneous Notes including emails and tweets.
Navigation: Please click only on the section number to return go back. This page is a working document and editing continues on the Footnotes & Endnotes, as well as the References & Resources and the Miscellaneous Notes and it will all actively continue to be edited and updated for the next several months. Thank you. – BEC
1b. Dirk J. Struik. If you do not have time to read the “Mysteries in Packing…”, you should know that Lagarias and Zong credit Struik, a Dutch-American and MIT mathematics professor, for reopening these discussions that broke the 1800+ year impasse. The primary reference: D. J. Struik, Het Probleem ‘De impletione loci’ (Dutch), Nieuw Archief voor Wiskunde, Series 2, 15 (1926), no. 3, 121–137
1d. Continuity, symmetry and harmony. Pi has to come into being in some manner. The spheres of the Kepler conjecture have to originate somehow. To answer the question, “Why is there something rather than nothing?” we assume that something is more fundamental than space and time, matter and energy. Here is our attempt to define the concepts that create a finite-infinite relation that gives rise to homogeneity and isotropy. Within this emerging model, the infinite is the qualitative; the finite is the quantitative. Instead of retiring the concept of infinity (Tegmark, 2012), in this model, it is the centerfold but with very specific definitions.
Recognizing how idiosyncratic it is to associate the geometric gap with fluctuations, it is certainly a greater stretch to differentiate types of fluctuations. Yet, that study has begun and eventually we’ll be showing a video of what we call “squishy geometries” and the rather unusual motions created by tetrahedrons.
Systemic fluctuations. Those two words in May 2020 only had 569 references within a Google search. These fluctuations, admittedly a guess, originate with the five tetrahedral structure fully engulfed by perfected systems. With the emergence of particle physics between Notation-64 to Notation-67, they become part of the look-and-feel that define all physical systems. It is a stretch, for sure, however, we will continue to pursue it further.
“All matter originates and exists only by virtue of a force which brings the particle of an atom to vibration and holds this most minute solar system of the atom together. We must assume behind this force the existence of a conscious and intelligent mind. This mind is the matrix of all matter.“
4c. George F. R. Ellis, Hawking’s 1973 co-author, had begun to recognize the deeper problems with conclusions from those early years of explorations at University of Cambridge. In 2012 in his collaboration with Roy Maartens and Malcolm MacCallum, (Relativistic Cosmology [PDF], Cambridge University Press), the big bang model is clearly under close scrutiny and every assumption is on the table: inflation, singularities, the most-recent measurements of the Hubble spacecraft of the cosmic background radiation, fine-tuning…. he is open to explore virtually every issue; yet with close to 50 years of analysis, he can not be absurd to himself. In February 2020, with colleague, A.A. Coley, the topic is, Theoretical Cosmology (PDF), virtually all the same issues are reviewed.
I took the most comfort from an article in 2017, Physics on the Edge, where he names all the key players and essentially shows how confoundingly muddled it all is. The current academy of scholars do not have clear answers.
Please note: The primary links into this section are from the Endnotes & Footnotes from where there is the word, More… Links back to that More… are from the [Numbers].
Aristotle. Our work began in a high school geometry class. We knew it required a tetrahedron and octahedron to tile and tessellate the universe. In 2011 we walked with Zeno deeper and deeper inside each object and learned a lot. You should know that our shapes were all perfectly made according to Plato’s specifications.
Yes, in 1998 we manufactured our own tetrahedrons and octahedrons!
It was hard to believe that neither Aristotle nor 1800 years of scholars (at least 90 generations) did not have their own perfect tetrahedrons in their toolbox. We wondered if geometry had slid from importance or was Aristotle beyond criticism?
The icosahedrons and Pentakis dodecahedrons use the five-tetrahedral configuration; they have gaps, and we dubbed it “squishy” or quantum geometry. By 2011, now with many years of visceral experience, and within our new chart of the infinitesimal scales, we thought that such a pervasive gap had to be significant.
We began thinking of quantum fluctuations and then systemic fluctuations.
[1a] Jeffrey C. Lagarias& Chuanming Zong. In 2011, just about the time we were beginning to explore the infinitesimal universe, Lagarias and Zong had begun writing the best little introduction that I’ve found to this geometric gap. It is a relatively short article (PDF) for the American Mathematical Society (AMS), December 2012. We appreciate that the AMS has made it readily available.
The people of China and the USA — not the governments, but the people — must find common ground. One would think that mathematics and the sciences would give us an abundance of places with which to build ties that are greater than politics. Articles like this encourage us. As important as their personal relation is, these two are also building relations between the University of Michigan and Tianjin Center for Applied Mathematics (TCAM). Zong was initially at Peking National University. I believe that the work of Lagarias and Zong actually changes the quality of life for everyone and for everything within this universe.
So, it is incumbent on all of us to begin to understand this gap (See #19), the first in the universe. It just might teach us all to become more patient with each other, especially with our superficial historic differences.
[1c] Zeno, Aristotle, Planck and Infinite Divisibility. I remember well the puzzled look of our students, when in 2011 I said, “Zeno has bumped into a limit called the Planck Length. We cannot divide-by-2 forever.” Planck gave the universe boundaries and logical conditions for those boundaries. Not entirely satisfied with that perception, a Russian by the name of Sergey Fedosin has taken another step: Infinite Hierarchical Nesting of Matter. I always immediately look to see what their starting points are. Within that document, they do not discuss the Planck base units and so they miss the possibility of defining the domain from the Planck units to the particle physics in a highly textured manner.
More References, reflections & resources:  Issac Newton did not have the advantage of Leonhard Euler‘s exponentiation. He created the concept after Newton had died. Of all possible manners of notations, base-2, is the most simple, yet it still lacks proper respect. The chessboard stories are told but under-appreciated.
The seemingly simple progression, 264yields a large number, 18,446,744,073,709,551,616. If you were turn turn it into pennies, you could easily retire the world’s debt, all nations and all people… I tried to explain it to my sister-in-law.
2202 is another story. Notationally, 6.42775218×1060 is the raw number. Once there is an amount associated with it, like infinitesimal spheres, it begins to open the imagination.
Newton did not have Planck’s base units. He was arrogantly unsure of himself. This Lucasian Professor (#2) was confident, however, that space and time were absolute. It is profoundly part of our commonsense worldview. Unfortunately, however, it is wrong. Indeed, the approach of Gottfried Leibniz will render a much richer view of our universe.
 Max Ernst Ludwig Planck Within the complex of Max Planck institutes around the world, there have been several attempts to open discussions. In this section, we will look at some of those exchanges more closely.
TRANSLATION: “ARISTOTLE KNOWS THAT CONGRUENT CUBES CAN FILL THE SPACE COMPLETELY, BUT FURTHER CLAIMS THAT THIS CAN ALSO BE DONE WITH TETRAHEDRA. THE AUTHOR FOLLOWS THIS FALSE ASSERTION, WHICH ALSO HAS A CERTAIN MEANING FOR THE TEACHING OF VACUUM, THROUGH THE HISTORY OF MATHEMATICS. THE FIRST TO PROVE THE INCORRECTNESS OF THE SENTENCE IS REGIOMONTANUS. BUT RAMUS AND SNELLIUS FOLLOW ARISTOTLE AGAIN. IT WAS NOT UNTIL THE 16TH CENTURY THAT COMPLETE CLARITY APPEARED (BENEDETTI, BLANCANI, BROSCIUS). (V 3.)” D. J. STRUIK, Het probleem “de impletione loci”. (Dutch) JFM 52.0002.04 Nieuw Archief (2) 15, 121-137 (1926).
“The basic idea is that quantum fluctuations of the inflaton field behave like one-dimensional quantum harmonic oscillators (with time-varying mass). Zero-point fluctuations of a quantum harmonic oscillator induce a non-zero variance of the oscillator amplitude, ⟨xˆ2⟩ = /2ω. Similarly, the inflaton zero-point fluctuations generate a non- zero variance ⟨δφ2⟩. The fluctuation modes (with co-moving wave number k) are stretched from their original small scale (assumed to be above the Planck scale) by the rapid accelerating expansion of the universe, until their wavelength ak−1 exceeds the Hubble scale (when they are assumed to become classical fluctuations).”
• W. Patrick Hooper et al., , 2018, Platonic solids and high genus covers of lattice surfaces We will be proposing one that has been reached by a few well-grounded scientist/scholars. Unfortunately, it still feels a bit more like science fiction, so we’ll come back to it within our final overview and conclusions.
• Alvaro G. López, On an electrodynamic origin of quantum fluctuations, ArXiv, 2020 Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Física, Universidad Rey Juan Carlos, Tulipán s/n, 28933 Móstoles, Madrid, Spain (Dated: January 31, 2020)
Here, space and time appear quite derivative. It appears that he was not ready to challenge absolute space and time. He defines a relation that begins with the Planck units. When we apply base-2, we are looking at the natural unfolding. The two formulas mass-energy equivalence and length-time equivalence are bound by light and appear to be bound to each other.
• Wikipedia: A gauge theory is a type of field theory in which the Lagrangian does not change (is invariant) under local transformations from certain Lie groups. … If the symmetry group is non-commutative, then the gauge theory is referred to as non-abelian gauge theory, the usual example being the Yang–Mills theory.
You might want to ground your people within a very simple model of the universe. All our current models are too big for most of us and those models tend to cause great anxieties. We call our work, The Big Board – little universe. It mathematically connects everything, everywhere for all time within a functional schema that actually seems to be a much better model for cosmology, physics, and mathematics than the models we currently entertain. The universe can be parsed from the Planck base units (it’s our the start)to this current day within 202 base-2 notations (all simple doublings). Once people understand that we live in an exponential universe,this place we live and have our being becomes quite intimate, comforting and secure, plus we realize that we are an important part of the equation and what we do counts. We make a difference. For more, you might start with today’s homepage: http://81018.com
This article was initiated on Wednesday, May 20, 2020. Duped became a homepage or top-level post: Wednesday, June 3, 2020. Last update: Friday, September 25, 2020 The Prior Homepage: https://81018.com/alternative/ The URL for this page: https://81018.com/duped/ The tagline: We reach for the stars, but we’re conceptually blocked…
First, let me congratulate you on your new location. Wonderful. It appears that you are still within 100 miles of Beijing. That’s excellent.
I am still quoting you after all these years (see above). Because the citations were getting so numerous, I created references page for you and Prof. J. Lagarias. My page for you: https://81018.com/2020/05/28/zong/
In these days and times, my most important conclusion is here about all our work, collectively and individually: https://81018.com/duped/#R3-2 Of course, if you would like anything changed, deleted, or added, I will be glad to accommodate your request. Thank you.
Second email: Wednesday, January 8, 2014
Your paper is sensational. It is exactly what I needed to be assured that Frank-Kaspers and many others were not leading us astray.
Your mathematics and analysis are spot on.
Let me share my reasons for my enthusiasm below this note to you. Thanks.
PS. Your work helps us with #2 and #4 below:
1. The universe is mathematically very small. Using base-2 exponential notation from the Planck Length to the Observable Universe, there are somewhere over 202.34 and under 205.11 notations, steps or doublings. NASA’s Joe Kolecki helped us with the first calculation and JP Luminet (Paris Observatory) with the second. Our work began in our high school geometry classes when we started with a tetrahedron and divided the edges by 2 finding the octahedron in the middle and four tetrahedrons in each corner. Then dividing the octahedron we found the eight tetrahedron in each face and the six octahedron in each corner. We kept going inside until we found the Planck Length. We then multiplied by 2 out to the Observable Universe. Then it was easy to standardize the measurements by just multiplying the Planck Length by 2. In 202 notations we go from the smallest to the largest possible measurements of a length.
2. The very small scale universe is an amazingly complex place. Assuming the Planck Length is a singularity of one vertex, we also noted the expansion of vertices. By the 60th notation, of course, there are over a quintillion vertices and at 61st notation well over 3 quintillion more vertices. Yet, it must start most simply and here we believe the work within cellular automaton and the principles of computational equivalence could have a great impact. The mathematics of the most simple is being done. We also believe A.N. Whitehead’s point-free geometries should have applicability. 3. This little universe is readily tiled by the simplest structures. The universe can be simply and readily tiled with the four hexagonal plates within the octahedron and by the tetrahedral-octahedral-tetrahedral chains.
4. And, the universe is delightfully imperfect. In 1959, Frank/Kaspers discerned the 7.38 degree gap with a simple construction of five tetrahedrons (seven vertices) looking a lot like the Chrysler logo. We have several icosahedron models with its 20 tetrahedrons and call squishy geometry. We also call it quantum geometry (in our high school). Perhaps here is the opening to randomness.
5. The Planck Length as the next big thing. Within computational automata we might just find the early rules that generate the infrastructures for things. The fermion and proton do not show up until the 66th notation or doubling.
I could go on, but let’s see if these statements are interesting to you in any sense of the word. -BEC
Summaries. The vastness of space bewilders most of us because our current commonsense understanding of space-and-time is ostensibly Newton’s absolute space and time. It’s challenged by many including our base-2 chart of numbers. We start with Planck Time (and Planck length, mass and charge) and go to the current age and size of the universe in 202 notations or doublings. That horizontally-scrolled chart (34 pages) of just the numbers was completed in 2016. Here space and time are derivative-and-finite. As this data continues to be analyzed, the infinite is necessarily studied and it is also redefined. Within this emerging model, infinity iscontinuity creating a finite order and time, symmetry creating finite relations and space, and harmony creating finite dynamics and space-time moments. No other definition of infinity or the infinite is engaged. Just 202 notations, domains, jumps or steps might seem intimate enough; yet, in the spirit of John Wheeler, the other prime-number bases (base-3, base-5, base-7, etc) are also explored as a series of possible “wormholes” creating another level of intimacy. Within our projected domain for systems theory, between Notation-50 to Notation-60, the Mind (all forms of consciousness), is being explored. By definition, all notations are always active and building on each other, yet particularly within Notation-202, at the current time, the Now, and the current expansion. Considering the diversity of mathematical and physical systems, it is projected that prime-number notations are also how unique mathematical systems develop at or below Notation-64 and how unique physical systems develop above Notation-64. These might be considered matrices within the matrix, or perhaps, subgrids nested within the grid. This picture of the universe is not of a cold-and-hostile place but quite possibly of a warm-and-fuzzy, intimate place. Quick ReviewFinite-Infinite
Could this be a start to construct a warm-and-fuzzy model of this universe? We think the answer is “Yes.”
To that end, our first hypothesis is that Planck Time is the first instance of time and that we are now within the earliest part of Notation-202, right up to this day, this moment-and-instant. That gives us a coherent, little mathematical outline of the universe. And, because it all started in a high school geometry class chasing the tetrahedron-and-octahedron back to the Planck numbers, this outline also begins to demonstrate how numbers correspond to simple geometries.
Now, that should be quite encouraging. We know that geometries and numbers are as much a key part of the foundations of physics as particles and waves, so perhaps we are onto something that could become warm and inviting. We ask many people for feedback.
Pi and Perfections.2 The next hypothesis is that all four Planck base-unit values manifest as an infinitesimal sphere. There is nothing more simple than a sphere. And, the next hypothesis is that there is an endless stream of primordial spheres that follow that first sphere so a natural inflation and simple geometries begin to emerge. More structure, textures, and complexity is observed with all the dimensionless constants that define those Planck units. Eventually even more textures will be added, starting with all the other scientific functions without a necessary length or time dimension. How-why-when-and-where each would manifest is an open question. Notwithstanding, this outline of our universe becomes a working model as more-and-more relations are defined for each notation.
Our hypothesis is that pi (π) is a primary gateway between the finite and the infinite2 and that the qualities of the infinite can be known through the qualities of the sphere. At the Planck base-unit scale, perfect continuity is the never-ending and never-repeating numbers.3 Perfect symmetries can be understood by carefully examining close-cubic packing of equal spheres.4 And, its perfect harmonies are best engaged within the Fourier transform.5 There is something very warm about continuity, symmetry and harmony and here we say, the qualitative expression defines the infinite and the quantitative expression defines the finite.
That is a very different notion of infinity and perfection.
Within this model, however, there are notations that are defined by such perfection and all are prior to the aggregation of the five tetrahedral cluster.6
The Tetrahedron and Imperfections. Aristotle (384 BC – 322 BC) believed the tetrahedron could tile and tessellate the universe.7 It was passed down for at least 1800 years before being debunked. Today we know that an octahedral-tetrahedral couplet is required. Moreover, with just five tetrahedrons sharing a common edge and its two vertices, a natural geometric gap emerges.8 It is proposed that this gap becomes systemic, possibly between Notations 50-60, and opens the first possible systemic fluctuations. Then, around Notation-64 and certainly by Notation-67, systemic fluctuations become measurable and are defined as quantum fluctuations.9
Within this model, quantum indeterminacy now becomes dominant and the universe as we know it continues to unfold.
Our Fuzzy Universe. In 1945 John Wheeler (Princeton) and Richard Feynman (Caltech) proposed quantum field theory or QFT.10 Very well-defined, QFT, more than Gödel’s incompleteness theorem, captures the deep roots for the unpredictable and indeterminate11 within the sciences, mathematics, logic, linguistics, philosophy, and consciousness. Gödel’s constructions using logic are too limited because he never applied that logic to a base-2 model of the universe12 especially considering the perfections within the earliest notations and the dynamics of the finite-infinite relation.
Finite-infinite bridge. This model creates boundary conditions and parameters. The Planck base units and the sphere define our universe with its initial functions and dynamics. There is no singularity per se; all the equations that define each Planck unit and the sphere are all active and define a bridge between the finite and infinite. This bridge is a key to our understanding the very nature and structures of our little universe.
In that spirit let us go over the basics of this model one more time:
Light and the four Planck base units.13 Planck’s natural units, based on the universal constants of G, ħ, c, and kB, are tested within this model by applying Planck’s simple equation for Planck Time adjusted for the speed of light, c. This highly-integrated chart of numbers defines a consistent variable speed of light throughout the model. It is generally within .01% of the laboratory defined speed of light. The next challenge on this path is to understand more deeply G, ħ, and kB.
Simplicity and complexity within the infinitesimal sphere.14 The product of the finite-infinite relation, the qualities of the infinitesimal sphere tell us about the the most basic qualities of the infinite. Going larger, the quantitative is further defined. Going smaller the qualitative is further defined.
Perfected systems are possible up to and around Notation-64. Perfected systems are infinitesimal states of being. It would seem, however, that moments or instants of perfection could spark right through to Notation-202.
QFT, quantum fluctuations, and quantum indeterminacy extend from Notation-64 up to and including Notation-202. The first measurable unit of time (attosecond) is within Notation-84; the first second is between Notation-143 and Notation-144.
The Intimacy of Our Universe
Sphere-stacking and cubic-close packing of equal spheres.15 What started around 1587 with Thomas Harriot, then involved Kepler, Gauss, Poincaré and culminated with the most-recent work of Thomas Hales, continues today under many other labels. Seeing how things fit compactly together, has become today’s work to understand the sub-grid physics modeling and the numerical techniques to validate the predictive results of our numerical simulations.
Within this simulation of sphere stacking, the first black circle is perhaps Notation-0 defined by the Planck base units. Notation-1, the first doubling, gives us the green circle (illustrated just above). Now, although difficult to picture, imagine a highly-dense block of these spheres populating every square inch of the universe with octahedrons surrounded by tetrahedrons creating a blank canvas of dimensionality through connections of the centerpoints of circles. Tiling and tessellating the universe takes on a very new meaning! Imagine if you can that there are literally zillions of these infinitesimal spheres populating every square inch of the universe and this simple tetrahedral-octahedral system, pervasive, is the first level of interconnectivity.
This simple base-2 ordering system quickly becomes complex. Each of the nineteen subsequent prime-number notations — 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, and 61 — introduce even more complex mathematics. The remaining prime numbers — 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 197 and 199 — open physical potentials.
Also, base-2 is just one dynamic of this expansion. This universe appears to be opportunistic so may well use the other prime number bases — base-3, base-5, base-7, base-11, and base-13 (right up on up to base-101 — to introduce yet even more complex functions. Of course, most notations are included within base-3 (there are 67), base-5 (40) and base-7 (28).
One can imagine how any given moment in the universe necessarily involves all 202 notations, yet to experience that moment as a conscious moment, to experience the fullness of a moment within Notation-202, just may actually be limited to specific notations.
We ask questions of many people. We are always asking for insights, comments, suggestions and criticism. Some of the people to whom we have turned are pictured and linked below.
 Planck Base Units. The first moment of time in this universe can be called, Planck Time; however, it did not mark the start of the universe all by itself. Planck Length, Planck Mass, and Planck Charge were all happening at the same moment. It is still a very simple start.Things are simple before they become complex. Doubling by stacking is a very simple process, however, it produces a natural inflation.
Follow the numbers. Our chart of numbers was first inspired by following a simple progression of the simplest geometries starting with the tetrahedron and the octahedron within it. But as simple as these two objects are, there is one object that is technically more simple, the sphere. Our intuitions, very early in 2012, was to say, “It all starts with spheres. And, there is a migration path from spheres to the Platonic solids.” We set out to understand what those statements meant if anything at all.
 The sphere as perfection and a gateway. Too much mystery and hocus-pocus surround the infinite. It was made even more mysterious when in 1687 Cambridge University Lucasian Professor Isaac Newton incorrectly defined space and time as absolutes. When his book, affectionately known as the Principia, became the primary reference for science, his absolutes became forever and universal, preconditions of all that is. And, even though this point of view became the world’s commonsense embrace of space and time, along with mass and charge, all four need to be put back inside the sphere.
Editor’s comment: Time in a bottle.Jim Croce’s 1973 love song, captured the imaginations of most who listened to it, yet time, I believe, is more logically enclosed with space (length), mass and charge with an infinitesimal sphere. Of course, lyrically, it doesn’t inspire one as much as the old seafaring bottle.
There are within the perfect sphere many equations that define each of the Planck units and equations that define a sphere and sphereness. All the equations are reaching across the finite-infinite bridge or gateway and all of them are about continuity (order), symmetries (relations) and harmony (dynamics). It is quite unlike the imagery of one of my favorite physicists, John Wheeler, when he said, “When I became interested in gravitation and general relativity, I found myself forced to invent the idea of quantum foam—made up not merely of particles popping into and out of existence without limit, but of spacetime itself churned into a lather of distorted geometry.”
John Wheeler actually named the two most-basic Planck base units, “Planck Length” and “Planck Time.” Yet, when he thinks about those Planck units, it is in light of the big bang theory. Did he ever consider Planck Length divided by Planck Time is equal to the speed of light? Did he ever consider Planck Time as the first unit of time? Did he ever consider the geometric gap created by five tetrahedrons? To date, we’ve found no records of such inquiries. Yet, he personally knew Gödel and was quoted in a tribute to him by physicists, James E. Peebles & William G. Unruh in Nature, 453, page 50 (2008), “To say Gödel is the greatest logician since Aristotle would be to slight Gödel.” It is wonderful that Wheeler defends Gödel, but it would have been better if they both recognized importance of Aristotle’s mistake and the place and importance of Planck base units.
 Never-ending, never-repeating. Obviously not originating from the finite, but equations that are incommensurable, should not be left out in some never-never land. Fundamentally, I believe any number like pi (π) defines the infinite. Though numbers by their nature are finite, certain very special orders of numbers are infinite; so, I am encouraged to say that this type of ordering is one of the facets of infinity. Of course, people ascribe many other qualities to infinity. We only ascribed continuity (order), symmetry (relations) and harmony (dynamics). Any other quality is solely the choice of an individual and not our concern here. Within the sciences, especially mathematics and physics, continuity is the first principle for order, and here it is. Simple. Simple. Simple. The sphere comes first. What else could there be?
 Close-cubic packing of equal spheres. Had you ever seen this dynamic GIF? It is a key function (so it’s pictured again). Applied to the Planck scale, the first sphere pictured is THE first sphere in the universe. But then look at what happens as the internal dynamics of a sphere “discover” other spheres, geometries (and all of Euclid) begin to emerge. Sphere stacking begins in earnest.
The very nature of the symmetry of a sphere and these first relations tell us how space, time, mass, and charge are each one of the four most-primary facets of light. Gravitation, temperature, motion… are derivative.
 Perfect harmony vis-a-vis the Fourier transform. Can we get inside the sphere and begin to understand all these dynamics? It is so rich with the entire history of the Fourier transform and all its multitude of applications, and here it is within the very first notations. Here is the epitome of fundamentality. It certainly needs much more analysis and discussion, so of course, there is more to come even within this document.
 Five tetrahedral cluster. Something so simple and so geometric, yet so fundamental, I believe it could engender the development of an entirely new science to study the geometry imperfection. Currently there is not even a line or a paragraph about the five tetrahedral cluster within the textbooks of quantum field theory. There are many discussions about tetrahedral configurations (i.e. Quantum Tetrahedra, Mauro Carfora, Annalisa Marzuoli, Mario Rasetti, 2010) and these are now being studied.
Of course, there is so much more to come.
 Tile and tessellate the universe. Cambridge University Lucasian Professor Isaac Newton formalized the study of “science” with his publication of his Principia in 1687. He was sure space and time were absolute, forever and universal, preconditions of all that is. He was only partially right. Although his “absolute point of view” continues as the our commonsense embrace of space and time, it is silly today. First, Einstein locked the two together. Then, he locked mass-and-charge together and his mentor, Max Planck, gave us the formulas that tied all four to the speed of light.
We ignore those equations to our peril. Yet, as a result of ignoring Aristotle’s mistake, we fail to see the obvious. Though it took 1800 years to stop repeating his mistake, it still lives on within our refusal to engage the fullness of geometry and its first-and-most-simple imperfection. It takes a tetrahedron and an octahedron to tile and tessellate the universe. We all need to understand this role of that tetrahedral-octahedral cluster as deeply as Newton wanted space and time to be absolute. Geometries and numbers connect the universe. Here, Aristotle’s 1800 year mistake confronts Newton’s 300 year mistake. It took an Einstein to break that grip. We will know that we are getting successful when Hawking’s infinitely-hot mistake goes cool and becomes an historical footnote.
This outline, a nascent model and construction project, just might help to change things.
The first three epochs of big bang cosmology — The Planck Epoch, the Grand Unification Epoch, and the Inflationary epoch — will eventually be understood notation-by-notation. For example, from 10-44 seconds to 10-41 seconds, Notation-1 to Notation-10 maybe known as the The Initial Processes of Forms. Notation-11 to Notation-20, or 10-40 to 10-38 seconds, might be known as The Initial Processes of Structures. Notation-21 to Notation-30, or from 10-37 to 10-35 seconds might be known as The Initial Processes of Substances. Notation-31 to Notation-40, or from 10-34 to 10-32 seconds, might be known as The Initial Processes of Qualities. Notation-41 to Notation-50, or from 10-31 to 10-29 seconds, might be known as The Initial Processes of Relations. Notation-51 to Notation-60, or from 10-28 to 10-26 seconds, might be known as The Initial Processes of Relations. One might say that these are the domains of perfection or the pure, there is no room or time for the imperfect. Yet, with the initiation of real relations, one could intuit how processes within Notation-202 might effect one of the notations in the 40s and 50s.
Big bang cosmology has barely gotten out of the Planck Epoch, into the Grand Unification Epoch and off to the Inflationary Epoch! The big bang cosmology is built on wishful thinking and not on numbers, mathematical functions, and logic. More to come…
Now, how many notations before the five tetrahedral cluster with its natural geometric gap would find a functional place? It may be a question of logic that the Langlands programs or string people have answered.
Our research is ongoing.
In this model, it appears that the tetrahedral-octahedral cluster could perhaps be manifest by the fourth notation. It is hypothesized that it becomes the basic building structure for the universe, a core of physical beingness. So, we now ask, “What might happen to create the five tetrahedrons (which are carried into the icosahedron and the Pentakis dodecahedron)?” When might such a combination manifest in the physical universe? It is proposed that this gap becomes systemic, possibly between Notation-50 and Notation-60, and opens the first possible systemic fluctuations, which between Notation-64 and Notation-67, become measurable and are defined as quantum fluctuations per se.
Somewhere near here quantum indeterminacy becomes dominant. More to come….
I believe we’ve been blinded by basics that were not basic enough; and as a result, we’ve additionally adopted blinders by adopting the mistakes of the luminous and superluminous such as those of Aristotle, Newton, and, yes, even Hawking. Even the best among us make mistakes. And, we all still have much more to learn.
Although Felix Klein (1849-1925) said “Physics is geometry” and John Wheeler (from about 1952 to 1972) also made the statement, “Physics is geometry,” no scholar has brought that geometry down to the initial gap that creates quantum fluctuations, or cubic-close packing of equal spheres for the generation of simple geometries, and then to a primordial sphere itself.
 Quantum field theory (QFT). Within this model, we are now exploring a range of notations when-and-where quantum field theory can actually manifest within this universe. We are exploring whether the perfected notations are a variable and could actually be dependent on the thoughts and activities within Notation-202. One of our little paths within this model just might open QFT to consciousness and the how the qualities of a conscious moment become translated to quantitative entities.
 From the Physics to Logic of the Indeterminate. This convergence of three footnotes, 10, 11, and 12, is a direct result of a challenge by the scholarly group, FQXi. They challenge thinkers to engage Gödel, Turing, and QFT, to discern in what ways the most basic principles of all three impact our understanding of undecidability, uncomputability, and unpredictability. For further work within this area, we refer you to this first draft, a response to their challenge.
 Gödel, base-2 model and Planck’s units. Gödel didn’t know there could be a mathematical progression from the Planck units in such a manner that unites space, time, matter and energy. Although special relativity pushes the absolutes out of the picture, Gödel gives Newton’s absolute time a place within General Relativity. Given Einstein’s special relations with Max Planck, it is of some interest to note that neither Einstein nor Gödel truly engaged the Planck base units. You would think it might have come up during Gödel’s time as a teacher-professor (1940-1978) at the Institute for Advanced Studies which included those long walks with Einstein. Even with his work on numbering and base-2, Gödel did not clearly demarcate a beginning of the universe but like Einstein, assumes the big bang.
 Revisit: Light and the Planck units. Aristotle failed to understand the tetrahedron. Newton failed to understand space and time. And, Hawking failed to understand the Planck base units. Their failures would not be important if they didn’t throw generations of scholars off their search for the truth, for better explanations, and to understand more deeply the way things really relate. I believe we would be so much further along the path of self-understanding had we had a better understanding of the universal constants of G, ħ, c, and kB. Planck’s simple equation for Planck Time gives us the speed of light, c, in 1899, yet we ignored it.
What are we to do with this highly-integrated chart of numbers as it defines a consistent variable speed of light throughout the model? It is generally within .01% of the laboratory defined speed of light.
Yes, quite obviously, our next challenge is go further with c, and to take on G, ħ, and kB.
Perfected systems on the same grid as quantum indeterminacy is hard to fathom.
Let me re-iterate: QFT, quantum fluctuations, and quantum indeterminacy extend from Notation-64 up to and including Notation-202. The first measurable unit of time is within Notation-84; the first second is between Notation-143 and Notation-144.
 Revisit: The basics. The three key transitions of our thinking begin with 1) spheres, 2).sphere-stacking, and 3) cubic-close packing of equal spheres (ccp). The key people in this emergence span four centuries starting with Thomas Harriot (1587), Kepler (1611), Gauss (1801), Poincaré (conjecture – 1904) and Thomas Hales (1998). Sub-grid physics modeling is one of our continuing research projects: the numerical techniques to validate the predictive results of our numerical simulations.
Working References & Resources:
Please note: Always a work-in-progress, the following references and resources are still very rough, not even a first draft. -BEC (May 8, 2020)
3. The sphere. It did not result in a theory of everything, but with mathematics it necessarily encapsulated everything, everywhere for all time. It then became mathematics in search of a theory (theoria).
7. Aristotle’s 1800 year mistake: • We are redefining space, time, mass and energy. All notations are active and interdependent, all connected through base-2, possibly 67 notations interconnected through base-3, and forty through base-5. Once we get a pathway opened to some of the string theory and Langlands people, it will be fascinating to begin getting their inputs.
In 1998/99 Merab Gogberashvili published on arXiv a number of articles on a very similar theme.  He showed that if the Universe is considered as a thin shell (a mathematical synonym for “brane”) expanding in 5-dimensional space, then there is a possibility to obtain one scale for particle theory corresponding to the 5-dimensional cosmological constant and Universe thickness, and thus to solve the hierarchy problem. It was also shown that four-dimensionality of the Universe is the result of stability requirement, since the extra component of the Einstein field equations giving the localized solution for matter fields coincides with the one of the conditions of stability.
9. Felix Klein (1849-1925) said “Physics is geometry.” John Wheeler from about 1952 to 1972 also made the statement, “Physics is geometry.”
Usually a quantum critical point is a point in the phase diagram of a material where a continuous phase transition takes place at absolute zero. In this model, there are 202 notations and it would appear that each has many “quantum” critical points, some superconducting cold and others superconducting hot. “Quantum” is in quotes because it has not been fully defined. If the packet of energy is not measurable and will never be directly measurable with an instrument, is it systemic or quantum?
“Eudoxus, arrived at an answer that, in one form or another, would survive for two thousand years. For mathematical purposes he imagined the heavens as a series of nesting, concentric, transparent spheres… Aristotle, amended this system. He assumed the spheres were not just mathematical constructs but physical realities; to accommodate the mechanics of an interlocking system, he added counter turning spheres.”
Lincoln Kinnear Barnett, editor and author, Life Magazine, author The Universe and Doctor Einstein, Harper & Brothers, 1948, and The World We Live In, published by Life magazine, 1952-1954. He said, “The gateway to universal knowledge may be opened by the unified field theory upon which Einstein has been at work for a quarter century. Today the outer limits of man’s knowledge are defined by relativity, the inner limits by the quantum theory. Relativity has shaped all our concepts of space, time, gravitation, and the realities that are too remote and too vast to be perceived. Quantum theory has shaped all our concepts of the atom, the basic units of matter and energy, and the realities that are too elusive and too small to be perceived. Yet these two great scientific systems rest on entirely different and unrelated theoretical foundations. The purpose of Einstein’s unified field theory is to construct a bridge between them. Believing in the harmony and uniformity of nature, Einstein hopes to evolve a single edifice of physical laws that will encompass both the phenomena of the atom and the phenomena of outer space. Just as relativity reduced gravitational force to a geometrical peculiarity of the spacetime continuum, the unified field theory will reduce electromagnetic force—the other great universal force—to equivalent status.”
11. Unpredictable and indeterminate. “A theory is a set of formulas, often assumed to be closed under logical consequence. Decidability for a theory concerns whether there is an effective procedure that decides whether the formula is a member of the theory or not, given an arbitrary formula in the signature of the theory. The problem of decidability arises naturally when a theory is defined as the set of logical consequences of a fixed set of axioms.”
“There are several basic results about decidability of theories. Every inconsistent theory is decidable, as every formula in the signature of the theory will be a logical consequence of, and thus a member of, the theory. Every completerecursively enumerable first-order theory is decidable. An extension of a decidable theory may not be decidable. For example, there are undecidable theories in propositional logic, although the set of validities (the smallest theory) is decidable.” – Wikipedia
13. Revisit: Light and the Planck units. In the first notations, there is such a thrust of light and Planck Charge, continuity-symmetry-and-harmony define everything. The building blocks are spheres and whatever new mathematics can be injected into the emerging forms. A long, long way from particles and waves, dynamic forms (aka automorphic forms)..: • M. Planck, Über irrevesible Strahlungsvorgänge, 1899 S.-B. Preuss Akad. Wiss. 440-480 Google ScholarM. Planck, 1900 Ann. d. Phys.1 69 CrossrefGoogle Scholar reprinted in Max Planck, Physikalische Abhandlungen und Vorträge, Band I. Friedr. Vieweg. 1958 pp. 560-600, 614-667 Google Scholar
And, because I have made so many references, I have started pages for you and Prof. Dr. Chuanming Zong. Your page is here: https://81018.com/2020/03/28/lagarias/ Of course, if you would like anything changed, deleted, or added, I will be glad to accommodate. Thank you.
Second email: 27 March 2020
Dear Prof. Dr. Jeffrey Lagarias:
I thank you again and again for your scholarly work. I endorse your work! Yet, given our work is so idiosyncratic, you probably would prefer that I didn’t.
11Aristotle’s failure is our failure. Perhaps the gravity and nature of this error is only now beginning to be understood. We all make mistakes. When we are challenged, we defend our concepts as best we can, and then adapt. We change or our associates change.
Some people become larger than life within their own time. Three examples are Aristotle, Newton and Hawking. All three were wrong about one key impression about the nature of life, yet their egos and their position and their person were so illuminated, it became increasingly difficult to challenge their assumptions.
First email: Saturday, 31 August 2013 at 8:19:21 PM
Jeffrey C. Lagarias, Professor of Mathematics, University Chuanming Zong, Professor of Mathematics, Peking University
Just a terrific job. A wonderful read. Thank you.
Coming up on two years now, we still do not know what to do with a simple little construct: https://81018.com/planck-length-time/ I have a hunch that that object made of five tetrahedrons plays a key role.
Your work gives me a wider and deeper perspective.
Bruce ************* Bruce E. Camber
PS. Long ago I studied with David Bohm, Phil Morrison, and so many others like them, but to make a living, I became a television producer! We had the longest-running television series on PBS stations in the USA and the Voice of America around the world about best business practices. http://smallbusinessschool.org/page18.html
Here are some key points within my current thinking:
1. The universe is mathematically very small. Using base-2 exponential notation from the Planck Length to the Observable Universe, there are somewhere over 202.34 notations, steps or doublings. NASA’s Joe Kolecki helped us with the first calculation and JP Luminet (Paris Observatory) with the second. Our work began in our high school geometry classes when we started with a tetrahedron and divided the edges by 2 finding the octahedron in the middle and four tetrahedrons in each corner. Then dividing the octahedron we found the eight tetrahedron in each face and the six octahedron in each corner. We kept going inside until we found the Planck Length. We then multiplied by 2 out to the Observable Universe. Then it was easy to standardize the measurements by just multiplying the Planck Length by 2. In somewhere around 202 notations we go from the smallest to the largest possible measurements of a length.
2. The very small scale universe is an amazingly complex place. Assuming the Planck Length is a singularity of one vertex, we also noted the expansion of vertices. By the 60th notation, of course, there are over a quintillion vertices and at 61st notation well over 3 quintillion more vertices. Yet, it must start most simply and here we believe the work within cellular automaton and the principles of computational equivalence could have a great impact. The mathematics of the most simple is being done. We also believe A.N. Whitehead’s point-free geometries should have applicability.
3. This little universe is readily tiled by the simplest structures. The universe can be simply and readily tiled with the four hexagonal plates within the octahedron and by the tetrahedral-octahedral-tetrahedral chains.
4. And, the universe is delightfully imperfect. In 1959, Frank/Kaspers discerned the 7.38 degree gap with a simple construction of five tetrahedrons (seven vertices) looking a lot like the Chrysler logo. We have several icosahedron models with its 20 tetrahedrons and call squishy geometry. We also call it quantum geometry (in our high school). Perhaps here is the opening to randomness.
5. The Planck Length as the next big thing. Within computational automata we might just find the early rules that generate the infrastructures for things. The fermion and proton do not show up until the 66th notation or doubling.
I could go on, but let’s see if these statements are interesting to you in any sense of the word. -BEC
Observations. The 202 steps from one end of the universe to the other constitute an actual working, mathematical model.6 Those numbers logically pull the universe together. Certainly, 202 steps doesn’t sound like much. And as small as that seems, our universe could be smaller!
We are now beginning to see possible shortcuts throughout the 202. Historically, scholars (like John Wheeler) called such shortcuts, wormholes.7 Today, the universe is currently unfolding and expanding uniquely within Notation-202.8 All other notations are fully symmetrical, yet continue to evolve; yes, every notation is changing, adapting, and responding to Notation-202 and to each other. In this model the butterfly effect9a applies to the universe. Yes, this paradigm shift opens as many questions as it seems to address.9b
There is one thing we do know with some assurance from this model: this universe is a totally-connected, integrated, and a rather intimate place. Also, it would appear that the first thirty-to-fifty notations manifest perfections of continuity, symmetry and harmony. Somewhere, perhaps around the 50th notation, a geometric gap10 becomes part of the landscape.
First, Aristotle failed to grasp it and made incorrect assumptions about it.11 Then scholars repeated his mistake, over and over again, for about 1800 years. Within a simple configuration of five tetrahedrons sharing a common edge, there is a gap and then it is extended within both the icosahedral and the Pentakis dodecahedral structures. By Notations 64-to-67, that gap becomes systemic and quantum indeterminacy becomes dominant and the universe as we know it continues to unfold.
Conclusion. Our mathematical outline of the universe is slowly becoming a simple, logical, and mathematical model of the universe that fundamentally begins to redefine space-and-time as well as infinity.12
1Sir Isaac Newton. Absolute space and time is still the commonsense view of the world’s population. Even our best scholars defer to it although they know it is not right. But until people have an alternative that works, we all resort to what we know. Going back to Newton’s time, back to 1716, there are the makings of an alternative; Newton was arguing (through his associate, Samuel Clarke) with Gottfried Wilhelm Leibniz. That Leibnizian point of view could be an actual opening to an alternative understanding of space and time. More…
3Redefining Space and Time. In 1971, as a result of studying the Einstein-Podolsky-Rosen (EPR) thought formula of 1935, I asked, “What could be more fundamental than space and time?” Out of college in 1969 and part of an invention and problem-solving think tank in Cambridge (Massachusetts), it seemed that continuity was the basis for order which is experienced as time. Symmetries fundamentally defined relations and are experienced as space. And, then harmonies were the essence of dynamics, or interacting symmetries, and that is experienced as space-time moments. So, that was it — continuity (order-time), symmetry (relations-space) and harmony (dynamics-interacting symmetries) defined (and were) the infinite. Sir Isaac Newton was wrong. Space and time were not absolute, but derivative.
4 An Outline of the Universe. In 2011 within a high school geometry class, we divided the edges of a tetrahedron in half, connected the new vertices, and like Zeno, we kept on going until we hit Max Planck’s wall. There were just 112 steps. When we multiplied our classroom tetrahedron by 2, in just 90 steps we were out to the edges and age of the universe. Much like the Sierpinski Triangle (base on the equilateral) and Sierpinski space, it was disconcertingly simple and the numbers told wild and crazy stories. Yet, at no time has our little naive outline of the universe been discussed by the scholarly community. I’ve tried (and will continue to do so). And, things could change. FQXi, a scholarly group, invited papers for March 2020 around some of these issues and I’ve submitted one. It is still quite rough, so I continue to update it within these homepages. Given this work started in a high school, our construct is entirely too simple for most. More…
7Shortcuts throughout the universe. Base-2 is magical because it is so simple. Doublings, as we’ve examined sphere stacking, generate the simplest geometries. There is cohesion and there are structures within structures. We could see how the universe literally was growing together. Then that vision began started checking out. At one second, between the 143rd and 144th notations, the number is within .01% of the laboratory-defined speed of light.
There will be many more self-validations within this model. Those 202 notations are simple math and reality should follow. And, if base-2 works, perhaps base-3 and every other prime-base between Notation-1 and Notation-202 will as well. Of course, the prime numbers have been called “the atoms of mathematics.” Those primes up to Notation-199 are here-on-in considered the most fundamental among all primes.
Observations: If base-2 works, then base-3 should be scrutinized closely. Base-3 goes from Notation-3 to Notation-201 in 67 jumps. It is fully symmetrical. Each of its 67 notations are “completely filled” with planckspheres as it works right alongside base-2 and Notation-202 which is being populated now as the current expansion, literally filling up with planckspheres.
Note: The count of the jumps includes from Notation-0, the Planck base units, to the prime number base.
If base-2 and base-3 work, then base-5 should work. It would jump to Notation-200 in 40 steps. Of course, all the other possible bases will also require analysis. Base-7 jumps goes to Notation-196 in 28 steps. We might think of these primes to be like an express train to its highest notation closest to 202, then you would go to a transfer station to the local trains, where a “local” base would then bring that base into the present moment within Notation-202.
Base-11 jumps to Notation 198 in 18 steps. Base-13 jumps to Notation 195 in 15 steps. Base-17 jumps to 188 in 11, base-19 jumps to 190 in 10, base-23 to 184 in 8, and base-29 to 174 in 6. Base-31 goes to 186 in six jumps. Base-37 goes to Notation 175 in five jumps.
Just what is this passageway? Right now, that’s anybody’s guess. I’d say, mathematical possibilities are actively being explored with the addition of every new plancksphere.
We can readily calculate that base-41, base-43 and base-47 reduce the jumps to 4. Base-53, base-59, base-61, and base-67 reduce the jumps to 2. The following primes, 71, 73, 79, 83, 89, 97, and 101, reduce the jumps to 1 and, of course, it would always be working with base-2.
The nineteen primes thereafter — 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 197, and 199 — are themselves an especially unique place within the universe. Each new notation initially replicates all prior notations, yet each evolves with its unique functionality.
Wheeler’s wormholes is an open study. For more see:
8Notation-202 and the symmetry of the universe. The current notation is still filling with infinitesimal spheres exactly like the first one to emerge and all 200 that have since emerged. All are still currently active. Notation-202 is directional and will continue to be directional until it is filled with infinitesimal spheres and the next notation begins. Yes, our current notation is Notation-202; it has a duration of 10.98+ billion years. Only about 2.82 billion years has transpired. That creates directional, asymmetric moments. Symmetry is needed for our Standard Models, our physics and mathematics. Enter sleep. Consider sleep to be like an overnight recompile of a computer. All the programs are reset and re-calibrated so everything, everywhere, for all time works in conjunction with each other. Symmetries are momentarily restored. Of course, there is more to come.
11Aristotle’s failure is our failure. Perhaps the gravity and nature of this error is only now beginning to be understood. We all make mistakes. When we are challenged, we defend our concepts as best we can, and then adapt. We change or our associates change.
Some people become larger than life within their own time. Three examples are Aristotle, Newton and Hawking. All three were wrong about one key impression about the nature of life, yet their egos and their position and their person were so illuminated, it became increasingly difficult to challenge their assumptions.
Aristotle’s geometric gap, Newton’s absolute space and time, and Hawking’s infinitely-hot big bang have each misled scholarship and then, most of us lost the scent and direction of the chase with its resultant discovery and creativity. We’ve been blinded by basics that were not basic enough. Throughout our ever-so-youthful human history, such people can readily continue to mislead us. We have to be vigilant to review and re-review all the concepts we hold near-and-dear and begin to adjust them appropriately.
12 New definitions of space-and-time and infinity. A rather uncomfortable view of space-and-time, certainly non-intuitive, this model pushes us to consider the four Planck base units are so tightly wrapped together — always and forever an active, bundled expression — the five faces of the speed of light are what we first experience and observe: light, space, time, matter and energy. Those five faces are each uniquely defined as a Planck unit and then collectively-and-uniquely defined in relation to the entire universe. Yes, though unique, each is ever-changing in relation to the entire universe. This finite universe has wholeness and thingness. Yet, it is balanced within an infinite continuity, symmetry and harmony.
Please note: In the course of compiling data-information-and-insights, writing about them, editing and re-engaging new concepts, there is much that no longer fits, however, it is still worthy and perhaps should be preserved as a reference, resource, or reflections about the roots for this article.
1. Questions about groups and sets: Does each notation or step or doubling complete a set? Can those sets can be grouped? In quite a different light, there are three groups of scholars to which we re-engaged for this rather brief article. These are: FQXi, the Physics of Quantum Electronics (PQE) group, and the 1999 Structure Formation attendees.
2. Poetry about the small universe? Consider the intimate part of our universe. The simple part. Perhaps once we learn something about the very small (infinitesimal) universe, the big-complicated universe, larger-than-life and rather overwhelming, will get a new handle and the universe will come home.
What seems so far away, “…back at the very beginning,” begins to become accessible. So, let us start with that infinitesimal circle that becomes a sphere, and one sphere follows with another and another. Here is the youngest universe; forever young, ebullient, reaching out to you like a baby. Yes, here is our evolving universe, always beginning, even today (See Turok).
3. Working models. Do you see that very small stack of spheres (just below)? No sphere is empty. There’s always a centerpoint, an active radius; there’s dancing. There’s music, too. There’s particularity; there’s continuity. There are relations and symmetries. The infinite (qualitative) has become the finite (quantitative). And, the harmonies of the universe begin.
6. Spacetime. I began reading the work of Stephen Wolfram. He watched his incredibly simple algorithms generate incredibly complex results; but, to do so. it requires a computer to be in the mix. Even quantum computing doesn’t get simple enough.
Abstract.A simple mathematical model of the universe,a unceremoniously developed in 2011 within a New Orleans high school,b has 202 base-2 notations which start at the first moment of time, Planck Time, and goes to the current age of the universe, today or the Now. Discussed in our prior homepage,c the first notations out of the 202, instantiate the perfections of continuity (numbers and order), symmetry (geometries and relations), and harmony (dynamics). These are the qualitative perfections of infinity and the quantitative perfections of a primordial, infinitesimal sphere. This sphere is the first thing to manifest in space-and-time.d As sphere dynamics extend those perfections, a finite-infinite relation becomes emergent and is always active. Yet, a simple geometry of five tetrahedrons sharing a common edge, sometimes called a pentastar,e has a gap and therein is a potential that gives rise to imperfection. At some early notation, that potential becomes actual, a dynamic geometry manifests as quantum indeterminacy. By Notation-67 it becomes a fundamental face of all subsequent notations. It masks moments of perfection or perfected states within space-and-time. The deepest underlayment for these primordial, infinitesimal spheres is base-2 exponentiation through which all notations or domains organize. The ever-so-limited domains of perfection with no quantum fluctuations are well-hidden within the much larger domain of imperfections where quantum fluctuations have become dominant.
Introduction: On backing into a base-2 exponential model of the universe. In our high school geometry classes we were dividing the edges of a tetrahedron by 2 and connecting those new vertices; there are four smaller tetrahedrons, one in each corner, and the octahedron in the middle. Dividing the edges of the octahedron by 2, there is a smaller octahedron in each of the six corners and eight tetrahedrons, one in each face. All 14 objects share a common centerpoint. Also, all create plates of triangles, squares, and hexagonals.
From our classroom models, going further and further within, in just 45 steps, we were down in the domain of particle physics. Within another 67 steps we were within the Planck scale. We then used the Planck length for our edge and multiplied by 2 to return to our classroom model in just 112 steps or doublings. We continued multiplying the edges by 2 until we were out to the age and size of the universe in just another 90 steps. A total of 202 notations mapped our universe. Notation-202 is 10.9816 billion years in duration. The cumulative aggregate from Notation-1 to Notation-201 is also 10.9816 billion years so just 2.82+ billion years of the 202nd notation have evolved (if we take as a given our universe is approximately 13.81 billion years from its start).
Planck Base Units. We learned about Max Planck’s 1899 definition of his four base units of time, length, mass and charge. We took their face values as a given, and asked the question, “What would the universe look like if the very first moment begins with the instantiation of those four values?” Of course, these values are the result of equations with additional values used by Max Planck to render his basic numbers.
Consider the four equations and their numbers for space (length), time, mass and charge:
We asked, “If the Planck Length and Planck Time are the smallest possible units of length and time, does it follow that these are also the first units of length and time?  Does it follow that these equations, with all their dimensionless constants, come together to become the very first moment of physicality?” Unwittingly we had opened the “CDM of the universe” and soon wondered if Steven Weinberg would consider our model a “grand reductionism.” 
Our postulate is that the Planck’s units are really-real physical entities, not zero-dimensional point particles, but an actual entity defined by the Planck base units. So our next question was, “What would that entity look like?”
Every equation is in part defined by the speed of light, pi (π), and the Planck Constant.
Because our students were studying basic geometric structures, they had a few answers. Yet, after some discussion, the students of pi, circles, and spheres prevailed. We then assumed not one sphere, but an impossibly-fast, steady stream of spheres emerge. We then wondered what the next dynamic could be.
We decided to follow Kepler (1611) and his sphere-stacking exercise of that year. Analogically, a little like Kepler, we now have this infinitesimal, raw stacking of primordial spheres. It seemed that there could be many ways to count each notation. When does it become Notation-2? …Notation-3? If the first notation is defined by a sphere, what defines the next? Does it progress 2, 4, 8, 16, 32, 64, 128, 256, 512 and 1024?
We thought, “If there is no absolute space and time, the spheres can’t roll. There is nowhere to go.” So if the spheres fill space perfectly, what does that look like?
There already exists a rather deep science and mathematics of sphere stacking. There is another science of cubic-close packing of equal spheres. With our naive, rather cursory overview of both, we left our questions open. Within Notation-10 we guessed there could be as few as 256 spheres and as many as 67,108,864. We were analyzing the base numbers within our Chart, column 10, lines 8 & 9. Within that range is the advice of Freeman Dyson, “Since space has three dimensions, the number of points goes up by a factor eight, not two, when you double the scale.” We had to learn about scaling vertices and dimensional analysis and that gave us the high-end of our range.
It is confusing; so we simply concluded, “Nothing is easy,” and went on to the next question, “What happens next?”
Our students had a quick answer, “The spheres come alive.”
First, there are those dynamics within cubic-close packing of equal spheres. The radii “discover” radii and triangulation begins (aka, triangulated coordination shells . The discovery process continues and a tetrahedral layering begins enclosing octahedral cavities. There are structures within structures:
Indeed, the entire structure of spheres and tetrahedrons-and-octahedrons comes alive.
We concluded that any-and-every known, scientifically-defined dynamic that does not have a time or length dimension just might be applied within these primordial on-going events.
There is so much to learn here and we’ve only just begun. The key point is that everything fits perfectly. Pure geometry meets physics 101.
Spherical perfections. Creating perfect continuity, perfect symmetry, and perfect harmony, this infinitesimal universe begins taking shape and the ubiquitous Planck Sphere dominates . It is all a quiet emergence within a simple perfection. Infinitesimal and way-too-small-to-measure, here are domains reached only by logic and mathematics. Actual physical measurements of a length do not begin until around the 67th base-2 notation; and, a unit of time, the attosecond, is not measured until the 84th base-2 notation. That is an extraordinarily large amount of intellectual space to tie logic, numbers and geometries together. Of course, it will be a challenge, but it just may be relatively straightforward for some of the scholars who have focused on the Langlands program or string theory throughout their professional careers.
Of those three facets of pi coming out of those never-ending, never-repeating numbers, the first is the face of continuity. It is a perfect ordering system that creates new sets of numbers and flow. Also, within those circles and spheres is a deep and abiding symmetry that gives rise to tetrahedrons and octahedrons — see Illustration 2 (above) — which is also a perfection. Those symmetries begin to discover symmetries and there is a simple harmony. Kepler’s music comes alive well before there is any range for human hearing!
It is a tangible perfection. And, it creates our homogeneous and isotropic universe. It is a study of perfected states in space-and-time. You will notice on every top-level page of this website are the words, CONTINUITY•SYMMETRY•HARMONY. Here is our definition of infinity and the foundations of perfection.
The expansion is now geometric, arithmetic, and exponential. And, we do project that Langlands scholars and string theorists have done a major amount of work to define this space and its automorphic forms . Here is a discovery process whereby every equation within Langlands programs can be tested within a highly-structured environment. We need help with simple questions: “Might the radius of our infinitesimal sphere be a string?” We need help with more difficult questions, i.e. “Does this model reopen Witten’s equations of state within the Seiberg–Witten invariants?”
Complexity and shortcuts: This simple base-2 ordering system quickly becomes complex. Each of the nineteen subsequent prime-number notations — 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, and 61 — could be used to test and instantiate more complex mathematics. The remaining prime numbers — 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 197 and 199 — also open new potentials.
Our little universe gets smaller. Base-2 is just one dimension of this expansion. This universe appears to be opportunistic so may well figure out how to use base-3, base-5, base-7, base-11 and base-13. What might be the linear components? How would linearity relate to the dynamics of base-2? Might there be simple coupling that introduces yet even more complex functions for this universe?  How would John Wheeler’s infamous wormholes work? How do we constructively open our imaginations and its science fiction?
Within this model all notations are active all the time. They build off of each other.
Finite-infinite. This system is slowly evolving with its own rules and axioms that are grounded in what will be a problematic statement for many people — the origins of these perfections are not finite. Opening the finite-infinite relation is an age-old enterprise fraught with tensions so please allow me to close that door rather quickly by emphasizing our simple definition of the infinite: “The infinite is the qualitative expression of continuity, symmetry and harmony whereby continuity begets order, symmetry begets relations, and harmony begets dynamics.” The finite is the quantitative expression. That’s it. Nothing more. Anything else anybody wants to impute is their business; it is probably not relevant here.
Within a little over one second, the base-2 expansion is out to the 144th notation. Planck Length is 360,424.632 kilometers. Planck mass is a hefty 4.8537×1034 kilograms. Just to put that in context, the mean average distance between the earth and the moon is 384,402 km (238,856 mi). It is 356,500 km (221,500 mi) at the perigee and 406,700 km (252,700 mi) at the apogee. The sun’s mass is around 1.989 × 1030 kg so at this notation, the density of the universe has analogies to a neutron star. The universe as we know it begins to take shape between Notation-196 and 197. At 10,829,559,004,640,000 seconds, Notation-196, the emergence is at 343.15 million year mark.
Just within these 202 notations, here are a few highlights of this base-2 model:
The first quantum fluctuations. With this discussion, we push that gap down into our infinitesimal and primordial work with Planck spheres. We postulate that there are many potential dynamics to begin the first quantum fluctuations as it moves up those notations. At some point, the pentastar becomes part of a system and begins to move. It is dynamic. It can start and stop. It can move around and up and down notations.
We do know that by Notation-67 it has become part of the fabric of the notations and so begins the measurements that are characterized as undecidability (subjects), uncomputability (relations), and unpredictability (objects).
We are in search of answers to the question, “When and where do these fluctuations manifest?” We are begging for help. These are all new studies for us. Our simple history begins in 2011. Our more critical history didn’t really begin until 2016. Notwithstanding, we are speculative people and believe the fluctuations actually fluctuate, first between notations, then between sets of notations, and then within groups of notations.
Yes, since 2011 we have been asking scholars for advice about it all. It is a very different, entirely idiosyncratic model based on eighteen divergent points of view. Notwithstanding, there is much more to explore, and so we will continue as best we can.
Please note that this is still a rough first draft. It becomes a bit rougher below this line. -BEC
 Although there are links throughout this website to the December 19, 2011 story of our high school geometry students and their teachers chasing tetrahedrons and octahedrons down into the Planck scale and then out to the age and size of the universe, here are other key links to tell a bit more of that story:
Because of its systematic ordering, this project was initially considered a STEM tool.
When we could find no place within our grid for Plato’s Eidos and forms, Aristotle’s Ousia, binary operations, pointfree geometries, Langlands programs, string theory or loop quantum gravity (please see line 11 of our horizontal chart), we decided, “That’ll all be within the first 67 notations.” We knew then there would be an endless amount of work to do within this model.
We started as everything does, simple. We take little steps and ask simple questions. We try to respect all prior scholarship. When we become confused, we step back to something more simple. So, it was with deep respect that we engaged the CDM approach to the universe and read that the most-distinguished Steven Weinberg (book, Facing Up) might call this model, “a grand reductionism.” We continue wrestling with his work and with these other scholars:
 Our initial studies of the work of F. C. Frank (H. H. Wills Physics Laboratory, University of Bristol, England) and J. S. Kasper (General Electric Company’s Research Laboratory, Schenectady, N.Y.) opened the concepts within cubic-close packing of equal spheres, the triangulated coordination shells, and the emergence of the tetrahedron from just four spheres. That all opened the way to engage The Physics of Quasicrystals edited by P J Steinhardt and S Ostlund. We struggle to grasp the work of scholars within this area:
 Our first introduction to the Fourier transform was through Steven Strogatz on Pi day in 2015. His article for The New Yorker Magazine resonated at that time and it still does today. Now we are attempting to really dig into the Fourier work. Of course, we have a long way to go. Here are some of the scholars to whom we are currently turning for help:
 Linear transformations are part of the dynamics within a notation. Yet, there is a homogeneity with all the contiguous planckspheres so geometries may readily extend within notations and across notations. It seems that the dynamics of all geometric models of the universe may hold insightful keys. With that mindset, we are open to all studies of space and time symmetry:
 The word, Planck sphere, is a key, core concept and we will continue to research it until we find the best possible resources that go back as early as possible. To date, we start with John Wheeler’s work with quantum foam; it could hold a key. Here are others:
 Who are the scholars to whom we can turn to learn about automorphic forms? Of course, there are the scholars within the Langlands programs and string theory. They have done sustained work since the 1970s and they have done a major amount of work to define its automorphic forms:
 One of the world’s leading scholars within string theory is Ed Witten. He is also a gentleman. Because the majority of his career has been in the shadow of big bang cosmology, his work has had an impossible starting point with which to contend. There is no easy migration to a theory that pushes time-space-and-light together at the Planck scale, and then again with mass-and-charge at the next level (c2). It will be fascinating to see if they will do better within a cold start that redefines the historic æther, and gives their discipline the radius of the plancksphere within Notation-1 and every plancksphere through Notation-67, and at least the first 67 notations of each subsequent notation.
 Of course, base-2 is the first exponential expansion of this model such that no point within the universe, right from the first instant, is more than 202 notational steps away. Yet, I believe our opportunistic universe will also test base-3. If necessarily held in check by base-2, there may step by three to Notation-6, then Notation-9 and so on, such that there is an aggregate of 67-steps through to Notation-201. Base-5, if necessarily held in check by base-2, might be 40 steps through to Notation-200.
How do these other bases manifest? What is their relation to the prime number and base-2?
Take base-7. If necessarily held in check by base-2, would it still provide 28-steps to Notation-196? Take base-11; if necessarily held in check by base-2, might there be just 18 steps to Notation-198? Of course, we have more questions than answers. and base-13 just fifteen steps through to Notation-195. These clusters of notations possibly may well introduce even more complex functions.
The largest square of a prime, 1313 is 169, obviously under 202; and, 1717 (the next prime) is over (289). So, although included within the base-2 progression, are Notation-13 and Notation-169 fully-relational with base-2? What does it look like?
 Imperfection. One of history’s greatest thinkers made a most fundamental mistake that was repeated for about 1800 years. That is a mistake of epic proportions. We are all taught to have such great respect for scholars, yet sometimes it holds us back. Aristotle (384–322 BCE), one of the greatest Greek philosophers and a polymath, obviously had imperfect models of the tetrahedron, otherwise he would have seen and felt a geometric gap. Five tetrahedrons all sharing one common edge opens a gap. My first discussion about it was in 2016.
1800 years later. The greats that followed Aristotle repeated his mistake and we failed to grasp a most-essential quality of simple geometry. One of our primary source articles is “Mysteries in Packing Regular Tetrahedra (PDF)” by Jeffrey C. Lagarias and Chuanming Zong. They relied heavily on the Dutch article by D. J. Struik, Het Probleem ‘De impletione loci’, Nieuw Archief voor Wiskunde, Series 2, 15 (1925–1928), no. 3, 121–137. We’ve all got to reflect on this mistake.
We are undoubtedly among a very few who claim that this gap is the basis for quantum indeterminacy, imperfections, free will, unpredictability, undecidability and uncomputability, so yes, they is much more to come!
Notes. Throughout the process, many new resources are uncovered. It takes time to do a critical review. Much is discounted and deleted. Some things are not. Though new to us, the statements seem to capture a truth, so for a few weeks they are left down here below the footnotes. Some of these comments will be used with the editing process for this article. Some of them may become part of the next article. -BEC
clusters, groups, and sets can readily be parsed giving us a way of checking the equations of state, the physical numbers, and potential relations.
sets are dynamic while groups are not
Time symmetry in modern physics, Andrew Holster, New Zealand
Discrete Calculations of Charge and Gravity with Planck Spinning Spheres and Kaluza Spinning Spheres, Michael John Sarnowski
What? Be careful here. “The fundamental ratio of surface to volume quantizations is what yields mass. The proton has 1040 on surface, 1060 in volume, it’s ratio is 10-20 * planck mass = proton mass.” Nassim Haramein and physicists at the Resonance Science Foundation and the Hawaii Institute for Unified Physics. (http://hiup.org)
As for As for Gödel’s undecidability and incompleteness theorems, within the first 64 notations, extending the mythopoetics of the Chessboard and the Wheat stories, there are nineteen prime numbers to initiate new mathematics within that base-2 foundation.
_______________________________________ Determinant becomes undecidable, uncomputable and unpredictable _______________________________________
Bruce E. Camber
Big Board-little universe Project: http://81018.com
500 East Fourth Street #484, Austin, TX 78701
April 22, 2020
Apply base-2 exponentiation to the Planck base units and the universe is parsed within 202 notations or doublings. These initial Planck units are derivative and finite. All the values by which each is defined opens questions about the nature of the finite-infinite relation. Within this emerging model, infinity is: 1) continuity creating a finite order and time, 2) symmetry creating finite relations and space, and 3) harmony creating finite dynamics and a space-time moment. No other definition of infinity or the infinite is engaged. Within this construction there is a small range of notations, the dynamics of which are determinant and are also understood to inculcate the following: decidability, computability, and predictability. Then comes a range of domains, the dynamics of which transition to the indeterminate and a transmogrification to undecidability, uncomputability, and unpredictability. There is a domain of perfection with no quantum fluctuations and a much larger domain of imperfection where quantum fluctuations have become dominant.
The Focus. We are projecting that within the first 64-notations, just below thresholds of physical measurements, there is a dynamic range of perfectly-defined domains, and then, an even more dynamic range that transitions between determinacy and indeterminacy. These first 64 notations, we believe, are the grounds for the decidable , the computable , and the predictable . An irony is that although logically determinant, these values are too small to be measured. Once measurable, the measurements become indeterminate. It is a fundamental transmogrification to undecidability , uncomputability , and unpredictability .
Planck base units. We begin with the Planck base units of time, length, mass and charge. We take these face values as a given and ask the question, “What would the universe look like if the very first moment begins with the instantiation of those four values?” Of course, these values are the result of equations with additional values used by Max Planck to render his four basic numbers.
Consider the four equations and their numbers for space (length), time, mass and charge (Illustration 1).
Background. In our high school geometry classes the question was raised, “If the Planck Length and Planck Time are the smallest possible units of length and time, does it follow that these are also the very first units of length and time?  Does it follow that these equations, with all their dimensionless constants, come together to become the very first moment of physicality?” We were unwittingly opening the “CDM of the universe” and wondered if Steven Weinberg would call it a “grand reductionism” .
Our postulate is that these Planck’s units are really-real physical entities, not zero-dimensional point particles, but an actual entity defined by those base unit values. So our next question was, “What would this entity look like?”
Every equation is in part defined by the speed of light, pi (π) and the Planck Constant.
Because our students were studying basic geometric structures, they had a few answers. Yet, after some discussion, the students of pi, circles, and spheres prevailed. We then assumed not one sphere, but an impossibly-fast, steady stream of primordial spheres emerge. We then wondered what the next dynamic could be.
We decided to invoke Kepler (1611) and his sphere-stacking exercise of that year. (Illustration 2). So analogically, like Kepler, we now have this infinitesimal, raw stacking of spheres. We consider the first ten notations. Within Notation-10 there could be as many as 256 spheres. However, if we follow the advice of Freeman Dyson regarding scaling vertices and dimensional analysis, there could be as many as 67,108,864. Go to the (Chart, column 10, lines 8 & 9). We decided that at some point we would learn a deeper logic and we would be able to decide.
The dynamics still beg the question, “What happens next?” Our students had a quick answer, “The spheres come alive.”
First, there are the dynamics within cubic-close packing of equal spheres. The radii “discover” radii (see Illustration 3), and triangulation begins aka, triangulated coordination shells . The discovery process continues and a tetrahedral layering begins enclosing octahedral cavities.
Spherical perfections. Within this thrust to create perfect continuity, perfect symmetry, and perfect harmony, this infinitesimal universe takes shape. The plancksphere  dominates. In theory, there is nothing that is undecided, uncomputed, and unpredictable. It is all a quiet emergence within a simple perfection. It’s creating an isotropic and homogeneous universe. Infinitesimal and way too small to measure, these are domains reached only by logic and mathematics. Actual physical measurements of a length begin around the 67th base-2 notation; and, the first unit of time, the attosecond, is not measured until the 84th notation. Thus, there is an extraordinary amount of intellectual space to tie logic, numbers and geometries together. Of course, it will be a challenge, but it just may be relatively straightforward for some scholars.
It is a study of perfected states in space-and-time.
The Expansion: Geometric, arithmetic, and exponential. Scholars within Langlands programs and string theory have done major work to define this space and its automorphic forms . Here is a discovery process whereby every equation within Langlands programs has a place within a highly-structured environment. Every radius of each sphere (a string) opens Witten’s equations of state and the Seiberg–Witten invariants .
This simple base-2 ordering system quickly becomes complex. Each of the nineteen subsequent prime-number notations — 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, and 61 — introduce even more complex mathematics. The remaining prime numbers — 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 197 and 199 — open new physical potentials.
Also, base-2 is just one dynamic of this expansion. This universe appears to be opportunistic so may well use the other prime number bases — base-3, base-5, base-7, base-11, and base-13 (right up on up to base-101 — to introduce yet even more complex functions . Of course, a majority of notations are included within base-3 (67), base-5 (40) and base-7 (28).
Within this model all notations are always active. They build off of each other.
Finite-infinite. This system is its own self-enclosed system, a working finite model theory, with its own rules and axioms that are grounded in a problematic statement — the origins of this perfection are not finite. Opening the finite-infinite relation is an age-old enterprise so we close that door rather quickly with this simple definition of the infinite: It is the qualitative expression of continuity, symmetry and harmony whereby continuity begets order, symmetry begets relations, and harmony begets dynamics. The finite is the quantitative expression. That’s it. Nothing more.
The Results. Within a little over one second, the base-2 expansion is out to the 144th notation. Planck Length is 360,424.632 kilometers. Planck mass is a hefty 4.8537×1034 kilograms. Just to put that in context, the mean average distance between the earth and the moon is 384,402 km (238,856 mi). It is 356,500 km at the perigee and 406,700 km at the apogee. The sun’s weight is around 1.989 × 1030 kg so at this notation, the density of the universe is like a neutron star.
The universe as we know it begins to take shape between Notation-196 and 197. Here the universe is at 10,829,559,004,640,000 seconds or about 343.15 million years.
Just within these 202 notations are a few highlights of this base-2 model:
The first 1000 years is between Notation-178 and 179.
The first million years is between Notations-188 and 189.
And, the first billion years is between Notations-198 and 199.
This model is primarily about the very early universe. Within the process, while it is being filled with planckspheres and with the emergence of geometries from the simplest to the increasingly complex, a five-tetrahedral cluster will manifest.
Although Aristotle thought it was a perfect configuration , and that understanding had been perpetrated by scholars for over 1800 years, the truth became apparent in the 1500s. Yet, even at that point, the real realities remained under-reported and less understood. As recently as the 1950s, chemists who recognized Aristotle’s mistake, calculated that size of that gap. It is an important gap.
Pentagonal, icosahedral and Pentakis-dodecahedral structures have such a gap or the surfaces are stretched and the internal angles are not exactly 60 degrees. In the 1960s the first concepts around aperiodic tilings were introduced. In 1976 Roger Penrose introduced his unique tilings and Alan Mackay followed up experimentally to show how a two-dimensional Fourier transform (with rather sharp Dirac delta peaks) manifests a fivefold symmetry. In 1982 Daniel Shechtman began his public-struggle to open the exploration of quasicrystals. This struggle to understand these geometries are current and on-going.
Pentagonal faces introduce new dynamics. The most fundamental dynamic would be the beginning of quantum fluctuations and its aftermath, undecidability (subject), uncomputability (relation), and unpredictability (object).
We are in search of answers to the question, “When and where do these fluctuations manifest?” We’ve beg for help. These are all new studies for us. Our simple history begins in 2011. Our critical history didn’t really begin until 2016. Notwithstanding, we are speculative people and believe the fluctuations actually and measurably begin to fluctuate, first between notations somewhere above Notation-64, and then between sets of notations, and within groups of notations.
Our Fuzzy Universe. In 1945 John Wheeler (Princeton) and Richard Feynman (Caltech) proposed quantum field theory or QFT, and, it has increasingly become a bedrock of physics . Very well-defined, it can be argued that QFT has the deep roots to the unpredictable and indeterminate within the sciences, mathematics, and even logic, linguistics, philosophy, and consciousness. However, those within QFT studies have not yet considered the 202 base-2 notations and the implications of Planck base units, an exponential universe, a dynamic finite-infinite relation, and the dynamics of a structural evolution from the sphere and cubic-close packing of equal spheres. Within this model, the old epochs of big bang cosmology get readily absorbed by an all-natural inflation. There are at least eighteen very special claims that provide a possible foundation for QFT that deepens its roots and broadens its reach.
Gödel’s constructions. Gödel gives Newton’s absolute time a place within General Relativity. And, given Einstein’s special relations with Max Planck, we are still in search of any references where Einstein or Gödel engage the Planck base units. Though Gödel was a teacher-professor (1940-1978) at the Institute for Advanced Studies and Einstein and Gödel had long walks, the infinitely-hot big bang probably got in the way. Even with Gödel work on numbering and base-2, he did not clearly demarcate a beginning of the universe.
Conclusion. We will continue asking scholars about our simple configuration as we learn more about those topics required to make our universe work. Thank you. –BEC
Endnotes, Footnotes, References, and Resources
 Decidability (Subject). A key to this theory and construct (Wikipedia) is the coherence of its logic. Are Planck Time and Planck Length the smallest possible units of space and time? Some scholars say it is. By starting at those Planck base units, does that base-2 progression necessarily include everything, everywhere, for all time? Max Planck never applied base-2, so he was not able to declare the base-2 progression of doublings to be a logical system. We do. It is totally predictive. One validation point is between the 143rd and 144th notation; it logically confirms the distance light travels in one second. That number is 299,792 km and it is within .1% of the speed of light confirmed within the laboratory. That requires a certain coherence of our four most-basic natural units, those Planck units, and their dimensionless constants. That would include c with special relativity, G with general relativity, and ħ with quantum mechanics. Also, there is ε0 (vacuum permittivity) with electromagnetism and kB (Boltzmann constant) with temperature/energy). More…
 Computability (Relation). The question is asked, “Is it possible that computable functions, including Turing degrees, are not necessarily set within just Notation-202? In this model where space and time are derivative, computability theory logically begins within Notations 0-to-1 and builds logically to include Notation-202. The algorithms of computational logic, like dimensionless constants, do not necessarily reside within machines. Questions about the mind and consciousness are stretching us to start a theory of computation that logically starts within Notations 0-1 and grows to Notation-202. More…
 Predictability (Object). Because the only notation that has a past and future is Notation-202 —all others are complete, yet fully dynamic and fully symmetric — there is still change. That thrust for change is a dynamic that effects every notation. To begin to enter this intellectual space, some of the more recent work regarding predictability is being engaged:
Within this model, time is finite and derivative. More…
Transmogrification: Starting with Notation-1 and going as high as Notation-67, logic prevails, yet our universe remains opportunistic. There is a thrust for more continuity, symmetry and harmony, reaching for a higher perfection. The first ten notations have been generally associated with Plato’s forms which today are associated with the automorphic forms of Langlands programs and string theories. Essentially this is the first-order of the planckspheres. We hypothesize that the next set of notations are a second-order for structures and a third order of planckspheres is for substances in the spirit of Aristotle’s Ousia. The emergent face of forms, structure and substance is qualities, the fourth face of the planckspheres. Perhaps as early as Notation-40 the first group of five-tetrahedrons, twenty tetrahedrons (icosahedron), or sixty tetrahedrons (Pentakis-dodecahedron) are manifest. When there is an ordering within the system, that gap, a squishy geometry, creates the indeterminant. We are projecting that transitions between Notation-50 and Notation-67. That’s a guess. More…
 Uncomputability (Relation). The historic brain-mind discussion will be part of these discussions. The Mind, that is, all minds, are currently projected to emerge within Notations 50 to 60 while consciousness as we understand it today would always be within the current time within Notation-202. To begin to grasp the boundaries conditions between computability and uncomputability we are engaging many documents including:
 Unpredictability (Object). In 1970 I became enamored with the work of John Bell at CERN labs in Geneva. He had a new way of looking at the EPR Paradox. In time I went to meet him stopping along the way to visit with David Bohm and Carl Friedrich von Weizsäcker. My goal was to better understand how the relations could be the primarily real within the subject-relation-object formula. It was only by making space and time derivative did the simple formulation of continuity-symmetry-harmony, a moment of perfection begin to open up. We wanted a container universe but did not discover the container until 2011. Now, all these articles are beginning to make some sense. Unpredictability is built into the very essence of geometries of the universe, yet the universe itself can be profoundly knowable. Freedom, creativity and moments of perfections are all built into the structure of unpredictability. Though it is not the fundamental structure, it intersects with the most fundamental. So, to help better understand this nascent model, we continue to explore a few related articles:
These are new studies for us and we have “miles to go” before it all coheres. More…
 The story behind this story. There are links throughout this website to the December 19, 2011 story of our high school geometry students and their teachers who chased tetrahedrons and octahedrons down into the Planck scale and then out to the age and size of the universe. Here are other key links to tell a bit more of that story:
When we could find no place within our grid for Plato’s Eidos and forms, Aristotle’s Ousia, binary operations, axiomatic set theory, pointfree geometries, Langlands programs and string theory (please see line 11 of our horizontal chart), we decided, “That’ll all be within the first 67 notations.” We knew then there would be an endless amount of work to do within this model.
Naïveté is often a curse; but in this instance, it allowed our most simple concepts to emerge. More…
 Grand Reductionism. We started as everything does — simple. We take very little steps and ask simple questions. We try to respect the scholarship that has gone on before us. When we become confused, we step back to something more simple. So, it was with deep respect that we engaged the CDM approach to the universe and Steven Weinberg and his book, Facing Up. Yet, we could not imagine a larger “grand reductionism” so, we did wonder!
We continue wrestling with his work and with these other scholars:
Beyond the Dynamical Universe: Unifying Block Universe Physics and Time as Experienced, by Michael Silberstein, W. M. Stuckey, Timothy McDevitt, Oxford (2018) Also: https://www.relationalblockworld.com/
In 1979 I first met Steven Weinberg at his office in Jefferson Laboratory at Harvard. He had not received his Nobel prize, but The First Three Minutes was out. Everybody fights for their legacy.
 Triangulations. Our initial studies of the work of F. C. Frank (H. H. Wills Physics Laboratory, University of Bristol, England) and J. S. Kasper (General Electric Research Laboratory, Schenectady, N.Y.) opened the concepts within cubic-close packing of equal spheres, the triangulated coordination shells, and the emergence of the tetrahedron from just four spheres. That all opened the way to engage The Physics of Quasicrystals, World Scientific, 1987 edited by P J Steinhardt and S Ostlund. We struggle to grasp the work of scholars within this area:
Objections to set theory as a foundation for mathematics
Jonathan P. K. Doye and his group are very helpful
 Fourier. Our first introduction to the Fourier transform was through Steven Strogatz on Pi day in 2015. His article for The New Yorker Magazine resonated at that time and it still does today. Now we are attempting to really dig into the Fourier work. Of course, we have a long way to go. Here are some of the scholars to who we are currently turning for help:
A huge study, we have barely scratched the surface. More…
 Lorentz. Linear transformations are part of the dynamics within a notation. Yet, there is a homogeneity with all the contiguous planckspheres so geometries may readily extend within notations and across notations. It seems that the dynamics of all geometric models of the universe may hold insightful keys. With that mindset, we are open to all studies of space and time symmetry:
Rovelli: Reconcile Planck-scale discreteness and the Lorentz-Fitzgerald contraction
Planck scale space time fluctuations on Lorentz invariance at extreme speeds
 Poincaré. In 1980 I worked with Jean-Pierre Vigier and Olivier Costa de Beauregard at the Institut Henri Poincaré. Our focus was solely on the EPR paradox, Bell’s theorem, and the experimental work of Alain Aspect at the SupOptique or “IOGS” in d’Orsay (just outside of Paris). Never did we look back at the work of Henri Poincaré. Today, a focus is on the Poincaré sphere and its underlying Lorentzian symmetry as a geometrical representation of Lorentz transformations. More work with these scholars:
The Poincare Conjecture: In Search of the Shape of the Universe, Donal O’Shea (2007)
Sphere in Various Branches of Physics, Tiberiu Tudor (February 2018)
Harmonic Analysis on Symmetric Spaces—Euclidean Space, the Sphere, and the
 The Infinitesimal Sphere. Also called the Planck sphere, it is a key, core concept and we will continue to research it until we find the best possible resources that go back as early as possible. To date, we start with John Wheeler’s work with quantum foam believing that it could hold a key. Others are:
Discrete Model of Electron, April 2019, DOI: 10.13140/RG.2.2.28408.49920, Discrete Universe Project, Jose Garrigues-Baixauli, Universitat Politècnica de València, Spain PDF
Physical Significance of Planck Length, Thanu Padmanabhan, Ann. Phys. 1985 165(1) 38-58
We continue wrestling with the work of scholars within this domain, so there will always be more…
 Automorphic forms. We turned to the scholars within the Langlands programs and string theory to learn about automorphic forms. We are learning about Loop Quantum Gravity n . They have done sustained work since the 1970s and have done a major amount of work to define its automorphic forms:
Automorphic forms (Wikipedia) “One of Poincaré’s first discoveries in mathematics, dating to the 1880s, was automorphic forms.”
Langlands program (Wikipedia)
Is there an analytic theory of automorphic functions for complex algebraic curves?, Edward Frenkel (ArXiv – December 2018)
We continue wrestling with the work of scholars within this domain, so there will always be more…
 String theory. One of the world’s leading scholars within string theory is Ed Witten. He is also a gentleman. Because the majority of his career has been in the shadow of big bang cosmology, his work has had an impossible starting point with which to contend. There is no easy migration to a theory that pushes time-space-and-light together at the Planck scale, and then with mass-and-charge at the next level (c2). It will be fascinating to see if they will do better within a cold start that redefines the historic æther, and gives their discipline the radius of the plancksphere within Notation-1 and every plancksphere through Notation-67, and at least the first 67 notations of each subsequent notation.
The logarithmic equation of state for superconducting cosmic strings, Betti Hartmann, Brandon Carter, November 2008 arXiv:0803.0266
We continue wrestling with the work of scholars within this domain, so there will always be more…
Of course, base-2 is the first exponential expansion of this model such that no point within the universe, right from the first instant, is more than 202 notational steps away. Yet, I believe our opportunistic universe will also test base-3 which would aggregate a 67-step shortcut through to Notation-201. Base-5 would provide a 40-step shortcut through to Notation-200. Base-7 would provide a 28-steps to Notation-196, base-11 just 18 steps to Notation-198, and base-13 just fifteen steps through to Notation-195. These clusters of notations possibly can introduce even more complex functions.
The largest square of a prime, 1313 is 169, obviously under 202; and, 1717 is over (289). More…
 Geometric gap. One of history’s greatest thinkers made a most fundamental mistake that was repeated for about 1800 years. That is a tragedy of epic proportions. We are all taught to have such great respect for scholars, sometimes it holds us back. Aristotle (384–322 BCE), one of the greatest Greek philosophers and a polymath obviously had imperfect models of the tetrahedron, otherwise he would have seen and felt this geometric gap. Five tetrahedrons all sharing one common edge opens that gap.
1800 years. The greats that followed him repeated his mistake and we failed to grasp a most-essential quality of simple geometry. One of our primary source article is “Mysteries in Packing Regular Tetrahedra (PDF)” by Jeffrey C. Lagarias and Chuanming Zong. They relied heavily on the Dutch article by D. J. Struik, Het Probleem ‘De impletione loci’, Nieuw Archief voor Wiskunde, Series 2, 15 (1925–1928), no. 3, 121–137. Two chemists, F.C. Frank and J.S. Kasper with their article, Complex Alloy Structures Regarded as Sphere Packings, took it further and calculated that gap. An key part of it all is the work of Daniel Shechtman, I. Blech, D. Gratias, and J. W. Cahn, Phys. Rev. Lett. 53, 1951 (1984). https://doi.org/10.1103/PhysRevLett.53.1951, Google Scholar Crossref
We are among a very few who claim that this gap is the basis for quantum indeterminacy, imperfections, free will, unpredictability, undecidability and uncomputability, so there will be much more to come…
 Fuzzy Universe. The concept of a warm and fuzzy universe certainly flies in the face of current cosmology and even with the physics that has grown out of the work of people like John Wheeler and Richard Feynman. Yet, perhaps we have the makings for a mathematics of a hyperconnected universe. Let’s open it up for discussion. Let’s see if our universe is as hyperconnected as the internet and our brain. More…
Aristotle’s mistake was so ingrained and so pivotal, quantum fluctuations have remained a mystery to this day. Here the belief that physics is geometry bears out in a most simple way to explain a most difficult concept.
Newton’s absolute space and time held back scholars with a deep understanding of continuity, symmetry, harmony, and yes, even beauty. The infinite became a place where only fools dared to tread.
Hawking was trapped by his disease and by becoming a Lucasian professor. He couldn’t deny Newton,. Had he, perhaps he would have been moved to redefine the infinite as continuity (order), symmetry (relations) and harmony (dynamics) whereby space and time would have become derivative and relational.
Even with his work on numbering and base-2, Gödel never applied his logic to a base-2 model of the universe from Planck’s units to the current time. It is our loss. More…