Working Towards a Mathematically-Integrated Model of the Universe

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Change the Metaphor – Rewrite the History*

Underpinnings: Given the nature of insight and ideas, high school people with curious minds should be capable of asking difficult questions and proposing new concepts about the nature of space, time, blackholes… We hope that describes us and we don’t waste your time. This is our story. It begins in December 2011 in a high school geometry class. More story

We went inside a tetrahedron. You can, too. Divide the six edges by 2, connect the new vertices; there are smaller tetrahedrons in each of the four corners and an octahedron in the middle. Keep dividing all the edges by 2. In just 45 steps you will be among waves, particles, and quantum fluctuations. Keep dividing by 2; in 67 more steps you will be down within the Planck base units at the limits of space and time. Nothing gets any smaller. Tetrahedral image and octahedrons.

To get back where we began in the classroom, take Max’s Planck Length and multiply-by-2. In just 112 steps, you’ll be back in our classroom. Now, keep multiplying, but watch carefully! In just 90 steps that tetrahedron will be out on the edges of the universe and today’s expansion.

You will have spanned the universe from the smallest to the largest in 202 steps (notations or doublings)! It’s quite a bit more than a base-2 progression of numbers. It is a functional map and it hasn’t been formally studied within our academic and scholarly communities.

In 2011 we created our first, rather colorful base-2 chart of the universe. In 2016 we created a less colorful, but more-detailed, horizontally-scrolled chart. It was all quite challenging; very quickly it appeared to be an alternative to the infinitely-hot start that defined big bang cosmology. Here is our original 2011 Chart, our on-going Working Chart, and a Comparative Chart.

Three short years passed. Still naive, perhaps silly, maybe gutsy, we reflected once more about what we were seeing. More…

We now thought it was more than a STEM tool and more than simple math; those 202 base-2 notations from Planck Time to this day changed our metaphors and began to re-write our history. We tried writing up our STEM analysis for possible publication!

Planck Time (5.39124(60)×10−44 seconds) divided by Planck Length (1.61625(18)×10−35meters) is, of course, 299,792,422.8 meters per second. It is in the numbers, all natural units based on the fundamental constants of nature. We wondered, “What might those four Planck base units look like if the universe, like Lemaître first suggested in 1927, began close to zero degrees Kelvin?” We turned to experts at the National Institute for Science and Technology (NIST) for help.1

Our charts were helpful to us but they were considered idiosyncratic, even poppycock by some, because this unusual look at the universe had not yet been properly vetted within recognized scholarly publications.

By definition, the chart included everything, everywhere, for all time.2 So we asked, “What doesn’t it include?” …infinity? What else? …a path to infinity? Is there any causal efficacy between the finite and infinite? We decided we needed to learn about the many types-and-functions of dimensionless constants like pi. Of course, pi is the most studied, the oldest, the most-used,3 and the most underestimated of all the dimensionless constants. It is not finite. It goes on forever without repeating itself. So, we wondered, “Is pi a bridge between the finite and the infinite?”

Pi gives us a sense of perfection and value. Pi’s never-ending, never-repeating number is a perfection — perfect continuity. Her sphere gives us a perfect symmetry of relations and balance. And, with its many applications of the Fourier Transform, the sphere also gives us another fully dynamic perfection called harmony.

Throughout the first 64 notations, we are looking to see where and how the first imperfections manifest. We have guessed in prior postings that it would manifest as the gap created by five tetrahedrons sharing a common edge. The Gap.

Again, we turned to scholars around the world for insight. More

Our charts opened up at least 64 notations smaller than waves, particles and quantum fluctuations. It wasn’t in any textbooks. What’s down in there? What does it look like? If this was a grid for string theory, does it begin within the first notation? If it is a grid for Langlands programs, what is the bridge to string theory and particles-waves-fluctuations?

We asked a lot of questions; we had no answers. What is going on within that first notation? John Wheeler talked about quantum foam and geons. We tried consulting with Kenneth W. Ford,4 Wheeler’s co-author of Geons, Black Holes, and Quantum Foam: A Life in Physics. We asked, “Might our redefinition of quantum foam and geons work?” Along that path we uncovered many, many attempts to define something more basic than waves, particles and fluctuations.5 There was a logjam. Of all those attempts, might our model-presuppositions-assumptions help to break open this conceptual blockage?

We were becoming convinced by others. What is the simplest thing in the universe? The sphere. …the most perfect? The sphere. …the most ubiquitous? The sphere. Could a sphere be defined by the Planck base units and some combination of dimensionless constants? It was plausible.6

But as simple as this logic is, just like the simple logic for the unfolding of the 202 notations, it was quickly strained by a closer look at Planck Time.

One Plancksphere per Plancksecond: Calculating the rate of expansion of the universe. That will also amount to the addition of dark matter per second. We adopted the names, Plancksphere and Plancksecond from the internet. Plancksecond (10−44 seconds) is the very first unit of time defined by those Planck base units. The Plancksphere would be the smallest possible object that defines space and time. Though not quite in the spirit of naming by NIST, CODATA, ISO, BIPM1 and others of the yoctosecond (10−24), zeptosecond (10−21), and attosecond (10−18), it seemed appropriate. After all, it was Max Planck who opened up the infinitesimally small universe for us. There is a telltale gap from the yoctosecond (10−24) to the plancksecond (10−44). The academy, including all the scientists and scholars who are responsible for the scientific names of things, has not yet named anything smaller than the yoctosecond. One can imagine that our experts had concluded, “There is nothing to be discovered that is smaller.” So, there are five groups of numbers that await a formal name: 10−27, 10−30, 10−33, 10−36, 10−39, and 10−42 seconds. The 10−42 group includes 10−44 and, of course, there are many who have referred to it as the Plancksecond.

There is nothing easy about Planck Time. It strains credibility. The ultimate time calibrator, simple logic suggest that one plancksphere would be generated each plancksecond. Does that mean that 1044 planckspheres would be generated every second?!? A bit hard to imagine: 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000 planckspheres per second.


As small as our plancksphere is, that is an unfathomable amount of planckspheres! Called a hundred tredecillion, in one second, it would be out in a line 299,792.458 kilometers long and it would fill all available space as it expands, so one can image that lines of spheres would always be equal to the amount of space within that notation. Ostensibly we would be following line 8 within the horizontally-scrolled chart. By that first second, the current chart allocates 1.3937×1042 base-2 vertices. Our math is off but not by too much. We continue working on it. To say the least, it would be a very dense 3-D structure of planckspheres that could be measured in cubic kilometers. We often remind ourselves that is so many factors smaller than neutrinos, these spheres are not in any way measurable by a physical device. That rate of the emergence, is the current expansion of the universe as well as the rate by which the dark matter of the universe increases.

That’s a bit bold so, of course, we know that we have an almost insurmountable amount of conceptual work to do!

Ideas and Concepts. Adopting a new concept is not easy, especially when we have to unlearn old concepts. Along that path we also have to begin to grasp why we didn’t learn other concepts as well as we should have. And then, these new concepts just take time, sometimes years to absorb. Take, for example, “Time is derivative, discrete, quantitative… hardly absolute.” Most of us have looked into the clear night sky and thought that it went on forever. Sir Isaac Newton whispers in our ears from his 1687 book, The Principia (Philosophiæ Naturalis Principia Mathematica). Within Newton’s days there was no science for particles, waves and quantum fluctuations. He didn’t have Max Planck’s infinitesimal numbers. And as sophisticated as our sciences and mathematics seem today, they are still so very young in the grand scheme of things.

Also, we still make a lot of mistakes. Personally, I am only now getting comfortable with our learning curve for time. It’s steep, but the learning curve for space and the instantiation of the first sphere is steeper.

Instantiation? What’s involved? The quick answer is the four Planck base units and at least pi among our dimensionless constants (and we suspect many others), all concresce to make the first sphere(s). Regarding dimensionless constants and our Lambda Cold Dark Matter model (ΛCDM) — remember John Baez has 26 dimensionless constants and Wilczek-Rees-Aguirre-Tegmark have 31. There are many to evaluate. NIST has identified over 350 and mathematician, Simon Plouffe, has 11.3 billion mathematical constants, all through his algorithmic programming.

We are entirely open to your insights, suggestions, and ideas. What other dimensionless constants might be involved with the production of that constant blast of 1044 spheres?

Though Max Planck’s numbers have been with us since 1899, it has only been since 2001 and 2002 that the scientific community actually began to think about them. It has only been since 2012 that we began to think of these numbers as the first moment of spacetime. It has only been since about 2015 that we’ve begun to think of that the first manifestation of the Planck base units as spheres. Of course, John Wheeler and others have been playing with the idea of primordial spheres since the 1950s, but, they had no container within which they could effectively build a system that would render particles, waves and quantum fluctuations. 

Those first 64 base-2 notations could well be an appropriate start.

If, and only if, pi-circles-spheres are the absolute shape of the first notations, and the container for our universe is continuity-symmetry-harmony, we have a very different model of the universe. It is a working model using cubic-close packing of equal spheres. The simple logic of our notations tells us that each one is always active and constantly building on all prior notations. That includes notations 1-to-201. Notation 202 is uniquely interacting because it includes the current time and current expansion of the universe. Of course, it feels directional.

In 2020 over 20 unique concepts help to support this model.  It is time for each to be critically reviewed by scholars. Let’s review:

  1. What if the universe does start with the Planck base units, what might be the first “thing” created?
  2. What if the first thing created is a sphere defined by those Planck base units?
  3. What if there is an endless stream of spheres and the first functional activity is sphere stacking?
  4. What if sphere stacking opens cubic close packing of equal spheres and tetrahedrons and octahedrons are generated? Doesn’t the rest of Plato follow?
  5. What if the concept of infinity has been so tainted by philosophies, we miss its most simple definition — continuity creating order, symmetry creating relations, and harmony creating dynamics; and then we add, “Please keep all other definitions to yourself. They are not necessary here.”

In the meantime, we’ll try to grasp this model in light of the most challenging problems in mathematics today, such special projects as the Yang–Mills existence and mass gap. Thank you.



Endnotes & Footnotes

* Any link that does not have a footnote is a page often referenced on this site: is Our Story, one of our very first pages. goes to our special image of a tetrahedron. is one of the third key geometries in this universe. is our first and most colorful base-2 chart. is our working horizontally-scrolled chart. begins within the power of metaphors. begins the bold charge into the top leadership.


[1] Experts at NIST. One of our first encounters with NIST was through Prof. Dr. Philip J. Davis. At that time he was an emeritus professor of Applied Mathematics at Brown University and the former Chief of the Numerical Analysis Section at NIST. In 2012 Phil Davis was emphatic and entirely convincing about the sphere as the most basic structure.

Then came interactions with many others, starting with Peter J. Mohr.
• National Institute for Science & Technology (NIST)
• Planck Length (Also: See the definition of uncertainty)
• All things Planck related. 353 Dimensionless constants.
Values of Fundamental Physical Constants

Also, there are the following:.
CODATA, the Committee on Data of the International Science Council. It is part of the International Science Council (ISC).
ISO – International Organization for Standardization

[2] Everything, everywhere, for all time: Many scholars have spent a lifetime in search of a theory of everything (TOE). That is not our goal. We are on an exploration of the Planck base units and the logic of infinitesimal spheres as the first manifestation of space-and-time.

Can we follow the simple math and logic? Not a theory of everything (TOE), but it might be a structure to help understand how a TOE could come to be.

[3] Dimensionless constants: NIST has done an extensive amount of work on dimensionless constants. We will continue our efforts to get some of the NIST scholars to consider parts of the infinitesimal that is currently ignored, defined only by numbers, without nomenclature and formal standing within the ISO. We have approached Bonnie Carroll, the Secretary General of the CODATA group and associated with NIST.
• We will continue to follow related work by John Baez.
We’ll continue our study of the work of the Wilczek, Rees, Aguirre and Tegmark.

[4] Kenneth W. Ford, former director of the American Institute of Physics: From working with the leading theoretical physicists within Princeton’s Project Matterhorn to writing his textbook, Basic Physics (originally published in 1968 by Blaisdell; reissued in 2017 by World Scientific), Ford has soared with the best yet he has been deep in the weeds as well. His book with John Wheeler encourages us to explore the conceptual framework of the geon, blackholes, and quantum foam. We have also been encouraged by Ford’s empathy for ethics. Locate the Geon book near you.

[5] What could be more basic than waves, particles and fluctuations? How do we relate algebraic geometries (Grothendieck‘s scheme theory), Euclidean geometries, projective geometry, category theory, Mandelbrot set, Julia set, Möbius transformations, Kleinian group, S-matrix theory, unitarity equations, Hermitian analyticity, Golden ratio (Phi), the Fibonacci sequence, fluctuation theory, ratio analysis, pi, cubic-close packing of equal spheres, ring theory, and lattice generation? Out of that group we settled on pi and cubic close-packing of equal spheres at the Planck scale to begin.

[6] An Infinitesimal Sphere. How could the smallest, most-simple 3-D object be the structure by which the universe is held together, expands, and gives rise to all other shapes and energies? That is the question. And, we shall explore as many answers as we can possibly find.

One Plancksphere per Plancksecond should be equal to the current expansion of our universe. There are so many dynamics within the first few steps… notwithstanding, it is already an impossibly large number to grasp. We should also consider an even larger number by multiplying it by the total number of seconds since the start of the universe. That would give us an approximate total number of Planckspheres within the universe and it would constitute the physical foundations of the universe. It’s a rather novel concept and such a different vision of the old aether. We’ll need to revisit Michelson-Morley and Wilczek’s matrix or grid. Perhaps we should add it to our list of claims or novel concepts.


References & Resources



Thursday, November 3, 2020 at 5:55 PM:
“If I were to write a summary statement about Kenneth Ford, it would begin: At 94 years old, Prof. Dr. Kenneth Ford has done enough. Among hundreds of papers and dozens of books that he has authored, he knows the open questions and the dead ends. With books like The World of Elementary Particles (1963), Basic Physics (1968, 2016), The Quantum World (2004) and 101 Quantum Questions (2012), Ford’s legacy is solid. He doesn’t need to continue to wrestle with all the open questions and tensions within science today.”

“The concepts being put forth within this website are just too much of a quantum leap. From primordial spheres to fluctuations and particles in 64 steps would have been an interesting concept to pursue. But eventually, it is all just too much!

“We extend our great thanks for all Kenneth W. Ford has done! -Bruce”

Tuesday, November 10, 2020 at 8:01 AM:
Dear Steklov Mathematical Institute (RAS):

  • Could the universe start with the Planck base units?
  • What might be the first “thing” created?
  • Could the first thing created be a sphere defined by those Planck base units?
  • Could there be an endless stream of spheres and the first functional activity be sphere stacking?
  • Does sphere stacking open cubic close packing of equal spheres and tetrahedrons and octahedrons are generated? Doesn’t the rest from Plato follow?
  • The concept of infinity has been so tainted by philosophies, we miss its most simple definitions coming from our observations of pi—continuity creating order, symmetry creating relations, and harmony creating dynamics. Perhaps we might consider adding, “Please keep all other definitions to yourself. They are not necessarily useful here.”
  • And so we finally ask, “Is there a glimmer of truth to the observations that give rise to our questions? If so, doesn’t that change our starting point and basic equations a bit?”

Might you have any advice for us? Thank you so much.



A sample of a tweet that went out to many people: “We have all grown up within limited worldviews. We need to include the entire universe to calibrate and understand who we are and why.” A start on a highly-integrated view of the universe:

Here is the same tweet specifically focused for Dr. Jason Johnson who is a Professor within the School of Global Journalism and Communication at Morgantown State University in Baltimore: “Jason, cool it. You know better by now. We are all tied up in knots and so very limited in our perspectives and simple worldviews. Without a complete picture, we always get confused. An integrated view of the universe is better grounding: is only a start.”



Zzzzzz: Back around 2014 we did a science fair project based on this base-2 model of the universe. I wrote up a little blog about becoming comfortable with very small and very large numbers. Exponentiation is a key.

Next: Desmos calculators


The name of this page is History

An artist’s rendering of a blackhole from NASA