Gauss, one of the great mathematicians of history (an emerging document)

Johann Carl Friedrich Gauss: In 1587 the most efficient stacking of cannonballs was addressed by Thomas Harriot and then in 1611 by Johannes Kepler. It took over 200 years before Johann Carl Friedrich Gauss actually started to prove these conjectures and about another 200 years before the conjectures were more formally proven by Thomas Hales (website) and his people (2014). This question about density had become a key mathematical challenge, deemed by David Hilbert in 1900 to be the eighteenth problem; there appears to be no references to the size of the spheres. For example, I would ask, “Is it possible to have a sphere the size of the Planck Length?” Given the ineffable work of pi, I would argue, “Yes,” and begin sphere stacking at the Planck scale.

Fourier series, transform: The goal here is to bring everyday physics and mathematics to bear to grasp the foundations of our universe so there is nothing esoteric or extra-logical about it. How very satisfying it will be if key mathematicians throughout our history, people like J. Kepler, C.F.Gauss, T.C. Hales (cubic-close packing),  Poincaré-Feigenbam (period doubling bifurcation), and Fourier-Dirac-Strogatz (Fourier transform), are responsible for the concepts that describe and predict the behaviors of our infinitesimal universe.

Cubic close packingOur knowledge of cubic close packing goes back to Thomas Harriot (circa 1587), Johannes Kepler (circa 1611),  and Johann Carl Friedrich Gauss (circa 1801). More recently, through the work of Thomas Hales (1998, 2014), we learned that these scholars were each proven to have calculated a very good approximation of sphere-packed densities . Also, notably, in the 2010 Wikipedia’s summaries of this discipline inspired a programmer to create a simple, but highly-informative simulation of sphere stacking.1

Path integrals and Gaussian fixed point. See Assaf Shomer’s on page 7: “The derivation of the path integral formula in quantum mechanics of a massive particle involves chopping up the quantum evolution into very short time intervals and inserting complete sets of states between them.” Physics 213: Gauss

In the letter, Langlands described a way to extend some of Carl Friedrich Gauss’ pioneering work on prime numbers. Number theorists before Gauss had noticed a hidden relationship among primes: that all the primes that can be formulated as the sum of two squares (for instance, 2^2 + 1^2 = 5 or 3^2+2^2 = 13) have a remainder of 1 when divided by 4, but didn’t know if it held true in all cases Quanta magazine reported. Gauss proved this idea in what’s now known as the quadratic reciprocity law.

Langlands took Gauss’ work and showed that the prime numbers that can be expressed as the sum of numbers raised to the third or fourth power (such as 1^3+2^3+4^3=73) can be tied to the distant mathematical realm of harmonic analysis. (This kind of analysis includes Fourier transforms, a mainstay tool used by scientists and engineers to analyze signals that have a periodic nature, such as sound waves or electromagnetic radiation spectra.)

  1. “As an application of the Law of Large Numbers, let z be a d-dimensional random point whose coordinates are each selected from a zero mean, 1/2π variance Gaussian. We set the variance to 1/2π so the Gaussian probability density equals one at the origin and is bounded below throughout the unit ball by a constant.” Page 13ff
  2. Properties of the Unit Ball
  3. Generating Points Uniformly at Random from a Ball Context-free language:  Hopcroft & Ullman 1979, p. 100, Theorem 4.7.
  4. DFA Minimization
  5. Minimal Supersymmetric Standard Model:
    •  Joseph L. WalshWalsh matrix

November 2018 (update):  “Generate n points at random in d-dimensions where each coordinate is a zero mean, unit variance Gaussian.” from Foundations of Data Science, 2.1 page 12

A précis for this website, our world, and our universe

Left Yellow Arrow
Right Yellow Arrow



Why is there so much trouble in our little world?
Ans: Our little worldviews need the universe.

by Bruce E. Camber

Nations are people; and, all people are in process. No one has all the answers. Surely, there are plenty of very smart people, but we are all still just people. Our leaders are just people, too. Like the rest of us, they have insecurities and gaps of knowledge. It is a recipe for problems!

Throughout our common history, we make many major assumptions. None of these assumptions should ever become sacrosanct. Working on our ideas and concepts never stops. Updating and improving is the essence of life! Notwithstanding, over the years our greatest scholars have made mistakes and sometimes, their students (our scholars) repeat those mistakes for too many years.

Significant Mistake by Aristotle
Ten Full Tetrahedrons Pictured

Perhaps the most egregious mistake by a great scholar is Aristotle’s claim that we can perfectly tile and tessellate the universe with tetrahedrons.1 One might respond, “So what, ho hum. Just a bit technical.” No, it’s a simple but key geometry; one face of the tetrahedrons perfectly covers a surface with no spaces. The flip side (noted by ten triangular peaks demarcated by the red dots within the image on right) cannot be perfectly filled with another layer of tetrahedrons. Therefore, Aristotle’s claim was wrong. It was a mistake and it was promoted for over 1800 years; and today, it is very rarely discussed.2

Today, it seems that we diss simple geometries and we do not grasp basic geometric structures.3

Continuity, Symmetry, and Harmony

Our most important failing: Please think about our oh-so-old, most-ubiquitous, most-studied pi. Mathematics and physics would be dead without it, yet you won’t find it within our concepts about the beginnings of this universe. Today, we tend to fool around with pi.4 If we took it more seriously, we would ask questions about spheres like Lemaître’s primeval atom,5 Wheeler’s geon,6 Pati & Salam’s preon,7 and those defined by the Planck base units.8 We would prioritize the study of the definition of continuity implied by those “never-ending, never-repeating” numbers. We would engage its manifold symmetries and the relations between lines and circles and spheres. We’d be going inside the sphere to study the types of harmony that manifest and the unique dynamics of each.9

These are failings in our time. These are also the failings of our great minds-and-scholars and our leaders; and these failings affect the way we see our universe, our world, each other, and ourselves. The key insight of this article is that we are deeply and profoundly related, connected, and interconnected. If that simple fact was embraced as a fact, we just might be more respectful of each other and be more creative as we engage each other just as we are.

We started working on our chart, a model of this universe on December 19, 2011 in New Orleans … geometry class. There are 202 progressive doublings of the Planck base units to this day and size of our universe. That chart stretches credulity. It is so hard to believe, we’ve asked many great scholars for advice and criticism. We received modest encouragement, no endorsements, and no constructive criticism. Undeterred, we’ve continued on; and, each year the concepts simply become ever so much more rich and encouraging. We even set about critically reviewing “commonsense” worldviews like our understanding of space and time. We had the audacity to question Newton and to support his rival, Leibniz.10 Then, rather naively, we even challenged Stephen Hawking and his understanding of the first moment of space and time. Hawking had an infinitely-hot start. Ours is rather cool!11

A basic idea emerged; an infinitesimal sphere is always the first notation and a simple start.12 And, the universe is populated by infinitesimal spheres with a remarkable interiority.13 A whole new sense of space and time emerged. A whole new physics of interiority began to emerge. The universe was filled with these spheres, quite literally filling all space and time. These infinitesimal spheres are in everything, everywhere, throughout all time. Each is like a little recording device. Everything you say, do, think or feel, is encapsulated, noted, and footnoted!14

Everything matters. Historically, such a construction may naturally give rise to an understanding of the Akasha,15 a concept that is within the beliefs of Buddhists, Confucians, Hindus, Jains, Taoists, and other sects that lift up Wuji philosophies. Yes, within each belief system, there is a record for everything. In other traditions an analogy would be omniscience.

Our model may be simple, yet it is inclusive. It opens a “huge” domain, 64-to-67 doublings that are infinitesimal and can’t be reached by our measuring devices. So small it just might be the basis of a new science for systems theory and a beginning for complexity theory. It is a domain for string theory and has plenty of character to include Langlands programs and even the most oblique and abstract mathematics.16

Yes, everything starts simple. We all do; and to our simplicity we must humbly return if this world is not to collapse with the weight of people who are much too sure of their ideas and actions.


Introduction. Not many people in this world have visited this website. This section is for those who are feeling a bit lost. Our concepts and language always need clarification. We hope that our first-time visitors will find it here.

[1] Aristotle’s Mistake: In 2015, my life changed because I came upon a reference to an article titled, Mysteries in Packing Regular Tetrahedra.” That article amazed me. It took over 1800 years to catch Aristotle’s mistake. Yet, along that way, Averroës (Abu al-WalidMohammad ibn Ahmad al Rushd (1126–1198), Leonardo of Pisa (Fibonacci) (c. 1228), Roger Bacon (c. 1214–1294), and Thomas Aquinas (c. 1225–1274) were among the greats of their time who reinforced his mistake. As a result, none of them would ever know about a most fundamental geometric gap. First, inferred by Johannes Müller von Königsberg (1436–1476), then documented in 1480 by Paul of Middelburg, a professor of astrology in Padua, the discussion was rebirthed by Dirk Struik (MIT) in 1926 while studying in Rome. Most recently, in December 2012, Jeffrey C. Lagarias and Chuanming Zong [also see, May 2020] brought it to life again. Yet, none of these people in their time contemplating that gap ever thought that it just might opened a path to quantum fluctuations, indeterminacy, and imperfection. Such a highly-speculative statement would appear to most physicists today to be uninformed. I do not believe that I would be overplaying my hand to say that this gap makes us all equally human. It is the beginnings of all our imperfections.

[2] Over 1800 Years to Catch a Simple Mistake: We all make mistakes, yet we are not Aristotle, Averroës, Bacon, or Aquinas. Some of our Nobel laureates might come close. Then there are others like Stephen Hawking and Roger Penrose. Nobody is exempt; we all make plenty of mistakes. Admitting our fallibility is the beginning of intellectual integrity, but unfortunately, the need of our cultures throughout time for stars and heroes just does not allow people to be people and the effect on people, culture and scholarship is not attractive.

Arrogance is never attractive.

[3] Basic Geometric Structures: We had an advantage over our scholars; we had boxes and boxes of perfectly clear, plastic tetrahedrons and octahedrons with which to create structures. Most scholars cannot tell you what is perfectly enclosed within a tetrahedron even when you divide the edges by two and connect the new vertices. They haven’t seen the octahedron in the middle. When they look at the octahedron they do not see “half-sized” octahedrons in each of the six corners and eight tetrahedrons, one in each face, with all fourteen objects sharing a common centerpoint. Nor do they see the four hexagonal plates within each octahedron and think about chemical structures. They have not seen how the five tetrahedrons create a most basic gap whereby one can actually make tetrahedrons do the jitterbug. It takes a high school geometry class and playful engagements.

How do these structures come to be? Our antenna were up when we began our studies of cubic-close packing of equal spheres. We thank Phil Davis; he had pushed our faces into the sphere, “…the most basic structure…” And given his expertise, the twinkle in his eye, and his abiding warmth and gracious spirit, we listened. And finally, here we found the work of Harriot, Kepler, Poincare, Gauss, Hales and so many others creating the bridge between spheres and our simple tetrahedron. Bottom line, we could only conclude that the most basic sphere is defined by the Planck base units!”

[4] The Very Nature of Pi: With all the fascination with pi, with the playfulness of Pi Day, with its status as history’s oldest, best-known, most-studied, and most-ubiquitous dimensionless constant, why haven’t we stopped long enough to acknowledge it for our culture, “Here is our deepest key to understanding of everything!” Here is the pathway to understanding continuity, symmetry, and harmony. Here is the transformation nexus between that which is finite and that which is infinite.

[5] Lemaître’s Primeval Atom or Sphere: A sphere is a sphere. How small can it get? How small is the Democritus atom? How small is Lemaître’s 1928 primeval atom? Did it get much bigger with his reintroduction with a very-hot beginning in 1931? Questions abound.

[6] Wheeler’s geon and quantum foam (1955): Perhaps John Archibald Wheeler, one of the great theoretical physicists of our time, had a deeper sense of the sphere when he introduced the geon and quantum foam. Reflecting on their work with him in 1952, Charles Misner, Kip Thorne, and Wojciech Zurek give Wheeler credit for naming Planck Time and Planck Length; they concur that Planck’s 1899 calculations define the most basic units of space and time. However, Wheeler and his biographers were all blinded by an imagined infinitely-hot start, and never asked the question, “What would our universe look like if we take the Planck base units as a description of the very first instant of space and time?” Instead their thinking dropped into an impossible singularity without much clarity. Over 350 dimensionless constants tell us that there is no singularity.

[7] Pati & Salam’s preon: Others made valiant attempts to fill in the blanks and voids when they found them. Jogesh Pati and Abdus Salam were among early attempts at a Grand Unified Theory (GUT). Like the others, it was all top-down and blinded by the big bang. The first 64 notations out of 202 that encapsulate the universe could not be seen.

[8] The Planck base units: In 2011 when we were constructing our very first Big Board-little universe, we asked ourselves, what is smaller than particle physics? What is on each line from Notation-0 to Notation-64? We asked, “Are these Planck units the next big thing?” It would take several years before fully engaging the concept that the Planck base units define an infinitesimal sphere. Then followed the concept that the expansion rate of the universe was one plancksphere per plancksecond, that time is derivative, and the continuity, symmetry, and harmony of the sphere-and-pi were extended to the foundations of the finite and the heart of the infinite.

[9] Harmony Manifesting Dynamics: First challenged to engage the Fourier Transform through a New Yorker article by Cornell University mathematics professor, Stephen Strogatz, it seemed that Jean-Baptiste Joseph Fourier opened the door to study any and all dynamical relations. Though scale independent, very little work had done to apply the Fourier Transform to the Planck scale. That begins to change here with our initial recognition of five transformations.

[10] Newton and Leibniz: The infamous debate was never completed; Leibniz died so Newton won by default. He shouldn’t have won. Yet, absolute space and time is so ingrained within our beings, most of us will have a bit of trouble breaking it down. His Philosophiae Naturalis Principia Mathematica of 1686 some say is the true beginning of the discipline of physics. Others say that it is possibly the greatest textbook of all of science. One would be judged a fool to take on such standing! Yet, when ones model coheres, we should never be intimidated by an arrogant, unpleasant person. It is entirely possible that he was still wrong about our most fundamental starting points, “What is space? And, what is time?

[11] The Infinitely-Hot Start of Stephen Hawking: As recently as 2016, Hawking was the host of the PBS-special television series titled, “Genius.” Everybody working with those scripts thought the big bang was the only true religion. With such a brief period of human history, perhaps as short as 400,000 years, we do not think it is wise to tout any particular belief systems unless it in some ways reflect the continuity, symmetry, and harmony within pi and the sphere. Here are universals that in some manner reflect the fundamentals of the finite, and the essential nature of the infinite.

[12] An Infinitesimal Sphere, the first notation: Back in 2017 Neil Turok and his colleagues famously proclaimed that the big bang theory was wrong. Most significantly, they added, There is a perpetual state of big bangs. Non-intuitive, the only model to date, where such an assertion makes sense is within 202 base-2 notations whereby every notation is still active and is responsible for the expansion of the universe.

[13] Remarkable Interiority of Infinitesimal Spheres: We have a huge task before us. The references within just the Wikipedia listings for the Fourier transform are voluminous. There is a lifetime of study. We cannot hope to begin to grasp it all. But, we will try! Our web searches began with “Planck scale” + “Fourier Transform” and returned over 50,000 results. We are starting to work through them all. A cursory analysis of a few pages is most encouraging.

[14] Everything, Everywhere, throughout All Time Encapsulated, Noted, and Footnoted: That summary statement just seemed to encapsulate the simple logic that had been guiding us. If Planck Time and the Planck Length represent the smallest possible units of each, is it logically possible for anything to exist outside of those domains. We revisit the question often and will continue on our summary document and our related claims.

[15] Omniscience: Religions jumped ahead of the sciences. To the best of my knowledge, there has not been a scientific-systems theory that would incorporate the concepts found within the belief of an Akasha. That the Buddhists, Confucians, Hindus, Jains, Taoists, and all other sects that lift up Wuji philosophies have somehow seen this from within their own journey will be explored further. The closest Western religions have come to the concept is omniscience.

[16] Simplicity and the Beginning of Complexity Theory. The 64 notations from the Planck base units to the CERN scale of particle physics represents a new domain for science. I have called it hypostatic because it cannot be directly measured with physical devices. Three young scholars have suggested a means to make possible indirect measurements. I believe their work could well open new methodologies for science that go back to 1884 when Lord Kelvin speculated about the nature of dark bodies within our galaxy. The measurement of dark energy and dark matter is a beginning.

We live in a complex universe that starts most simply. Within that simplicity are the key insights about how we can co-exist and more meaningfully engage our days.


Reminders to return to the work of these scholars and to think more deeply about their constructs of reality. THEIR ARTICLES HAVE BEEN PART OF THE CONSTRUCTION THIS ARTICLE.

The last six homepages have been keys to this page:
The Mind, the Self, the Brain and the Human Mystery (December 26, 2020)
The First Instant of the Universe – Pi Circle, Sphere (November 27)
The Expansion of the Universe (November 16)
Change the Metaphor – Rewrite the History (October 16)
Countdown: Minutes, Seconds, Nanoseconds… (October 16)
This World Can Become A Nicer Place (September 23)

EMAILS (1 of 10)

As an article begins to take shape, friends and scholars are engaged to provide initial feedback.

The images at the top of this article are of world leaders, left to right, starting with China’s
Xi Jinping
and going to Germany’s Angela Merkel. Each of those leaders will be sent an email that goes something like this:

“As one of the key leaders of our world today, you are pictured at the top of this page.

“The entire world is in search of peace-and-prosperity, respect-and-dignity, and love-and-integrity. Unfortunately our realities are quite different. It is obvious that our understanding of this world and our universe is incomplete. We need to adjust our understanding of some of the fundamentals through which life takes shape and things develop. I believe our biggest problems relate to our incomplete understanding of one of the most common, oldest, and best-known things in our life. That is pi with her circles and spheres. The three teach us about continuity and order, symmetry and relations, and harmony and her most important dynamics whereby lines, tetrahedrons and octahedrons, become space, time, the finite and infinite.

“More work is here: and

TWEETS (5 of 23)

Eventually to drive traffic to the site and create a little buzz, key words are used to find discussions on the web with people who might be interested.

6:13 PM · Jan 16, 202, Angela Merkel: Our problems require a new vision of who we are and why we are. We must address issues that go right back to how we understand space and time and this world and universe: is a start. Have your best scientists, scholars, and thinkers focus on these issues.

6:18 PM · Jan 16, 2021, Angela Merkel #2: True, but we all believe things that are not true. Old ideas need to be re-examined. What is space? What is time? What is finite? What is infinite? How did it all begin? Some of the giants were only right in their time, not ours. See for more.

10:06 AM · Jan 21, 2021 @davidburkus @mitchjoel @LauraHuangLA @thomaswedell None of us should ever stop growing. Babies are naturally solipsistic; they only have MyView. Most adults have limited WorldViews. Leaders need an integrated UniverseView. Today, Worldviews are too small. We all need the most-inclusive perspective possible. A start:

4:14 PM · Jan 21, 2021 @BillClinton @HillaryClinton @davidnour Did you know that in 1899 Max Planck calculated the smallest possible units of length and time. If you apply base-2 (doublings), there are just 202 notations that define the universe from the beginning of time until now. We all need to be working on an integrated UniverseView:

5:38 PM · Jan 21, 2021 @antonioguterres As SG of UN, lead the world in a new vision of who we are and why. Our little worldviews need to become highly-integrated, mathematical views of the universe. Space-time becomes derivative and finite. Relations become really real. A start:

11:06 AM · Jan 23, 2021 @RamonCruzDiaz Just sent an email; now to follow-up. We will not break free of our narcissism and consumerism until we break free of narrow worldviews. We all need to work on an integrated universe view. Here’s a start: …all simple math but a framework!


The last word, often afterthoughts, about this article and what is happening in our little world.

We so kowtow to our leaders. There is a long, brutal, and largely-forgotten history behind it all. In so doing, we impart a little divinity to them. When we don’t, it may be demanded. The respect that comes with leadership sometimes is not earned, but required.

So, who within our world is the best leader? Who is the wisest? Who is the smartest? Who is the bravest? Who has the most integrity? Who has the most love? Who has the most generosity? We must begin to discern and compare such qualities so all our leaders compete to hold some part of those judgments.

Ten global leaders posted, enough to fit across the page in one row:

Xi Jinping: Peng Liyuan (spouse), Xi Mingze (daughter)
Vladimir Putin:
Kim Jong-un:
Mr. Alejandro Cao de Benos
George Soros:
Pope Francis:
Joe Biden: @POTUS
Queen Elizabeth:
Donald Trump: (still a leader for many people around the world)
Benjamin Netanyahu:
Angela Merkel

Who might undertake such a project?

Key Dates for this document, Precis

  • This document was started on January 13, 2021.
  • First posted for collaborations: January 14, 2021.
  • The URL for this document is
  • The Prior Homepage:
  • First Tagline: A little precis for this website and our universe
  • The last update of this page was on January 24, 2021.

Countdown: Minutes, Seconds, Nanoseconds, Picoseconds… Plancksecond!


A Most-Basic Building Block of the Universe: One Plancksphere per Plancksecond * MADE BY CONTINUITY-SYMMETRY-HARMONY

by Bruce E. Camber Related: World – HistoryExpansionFirst InstantConsciousness

In 1899 Max Planck calculated numbers for Planck Length & Planck Time.1 Max Planck knew he was onto something significant, but he couldn’t quite make it work. One can imagine that he had hoped that the young Einstein could help. Surely Einstein opened new doors to explore, but his were different. And, we have learned over the years that all big ideas and concepts incubate slowly.

Within Max Planck’s equations, space-and-time are necessarily yoked. One is always a face of the other and a primary characteristic of light. As such, both are also yoked to mass-and-charge. Planck and Einstein redefined the very nature of space-and-time, and mass-and-energy. In so doing, they unwittingly redefined the finite and infinite.

A goal of this article will be to examine how this could be so. We start with one of the world’s oldest, best-known, and most-used mathematical constructs, pi, along with her circles and spheres. Most of us have not explored how in some manner of speaking these three are derived from continuity-symmetry-harmony. Another intention of this posting is to see how pi-circles-and-spheres generate space, time, and geometries.

All of this work is slowly evolving out of just over 20 rather basic claims.


Max Planck ignored his own calculations.2 Perhaps he didn’t know what to do with them. It took out-of-the-box thinkers, a bold scholar like C. Alden Mead, to open that door. In 1959 he wrote about the place of the Planck scale. Leading first-principles scholars like John Barrow (1982) and Thanu Padmanabhan (1985) also began wrestling with the Planck units. Frank Wilczek finally broke open the Planck scale to the world in 2001 with a series of articles in Physics Today.

More recently scholars around the world have wrestled with the meaning and value of the Planck Scale. Most top-level posts within this website attempt to be deeper exploration of  those issues.

The four formulas for the Planck base-units

Too small to measure, Planck Length and Planck Time redefine the infinitesimal. It appears nothing is meaningfully smaller, shorter or faster.

When the Planck base units are re-envisioned to create a base-2 chart from the smallest to the largest measurements of space and time, it becomes increasingly evident (1) neither space-nor-time are absolute, (2) time-and-space are Janus-faced, correlated with the Janus-face of mass and energy, and (3) all the notations are, even today, generating one Plancksphere per Plancksecond.

There are around 64 notations (doublings) before waves, particles, and fluctuations. Sixty-four successive doublings of the Planck Length and Planck Time are enough space and time to do a lot of mathematics-and-geometry, but key leaders within the academic-intellectual community think it’s too small for much of anything.

Older, Slower Studies 3

The First Three Minutes.3 In this 1976 book Steven Weinberg begins his study of the universe at about 1/100 of a second after the big bang. That is Notation-138 within our horizontally-scrolled chart. There are 202 base-2 notations or doublings from Planck Time, the very first infinitesimal moment of the universe, to this very day. And, yes, there are just 202 simple doublings.

We all know that life is about doublings. Every living thing doubles in some special way. Yet, Weinberg could not explore from Notation-1 to Notation-137. Though the Planck units have been around since 1899, that progression of Planck doublings did not come out until 2011. To date, we know of no other model of the universe that relies exclusively on the Planck base units.

The First Three Seconds: A Review of Possible Expansion Histories of the Early Universe, (June 2020). Twenty-seven leading scholars from around the world collaborated on a composite article about the first three seconds. The very first second is between Notation-143 and 144. They, too, assumed an infinitely-hot start of the universe so their first three seconds are shrouded in mystery. They didn’t explore the concept of a cold-start model which was first proposed in 1927 by Lemaître when he began to share his earliest thoughts about the beginnings of this universe, so they, too, missed a lot of possible action between Planck Time and that first second.


Planck-Time Transformations ________________5.391 16(13)×10-44 seconds________________

New ideas take time to incubate.4 Max Planck first wrote about Albert Einstein’s new ideas back in 1905. Years later, after Planck died in 1947, Einstein wrote to Planck’s widow about his special memories with Max yet acknowledged how general relativity and quantum theory would, for now, have to stand awkwardly together. These two seemingly irreconcilable pillars of modern physics have continued to stymie the world so much so that the world’s people, especially her leading scholars, have been quite unsure of space and time.

We inherited our commonsense worldview from Isaac Newton.5 So with all due respect and for a very long time, we’ve believed that time and space are absolute.

It seems most people still do. To stray from that bit of so-called commonsense is not easy. If space and time are not absolute, then what is? What holds it all together?

What’s in the heart of every Planck Time transformation? The Planck scale is a different scale but it may not be as strange as so many scholars seem to think today. As a result of our studies of this model since December 2011… though still fuzzy, some clarity is becoming apparent.


Pi’s Continuity-Symmetry-Harmony

A Possible Domain of Perfection.6 Learning a bit more about pi, going over it one more time — even attempting to go inside it — finally my old 1972 definition of a moment of perfection seemed relevant. Inherent within the many definitions of pi is continuity, symmetry and harmony. There is the continuity of the never-ending, never-repeating numbers, the perfect symmetry of the circles and spheres, and special internal and external dynamics, harmonics, introduced to us through the Fourier transform.

Continuity-symmetry-and-harmony are such key concepts, there are links to those three facets of reality at the top of every homepage or top posting within this site.

Where do these concepts reside? Or, could the three be the container for everything, everywhere, for all time? Here the answer is a cautious “Yes.”


Finite & Infinite

Simple logic tells us that the very first spheres are still being pushed forward.7 By going over those progressions of the doublings, over and over again, it slowly became apparent that each notation is always active. Each builds on the prior notations. Time is a face of light, the Janus face of space. So, along with the oldest-most-popular mathematical construct, pi, all of the dimensionless constants became our focus.

If these special equations and relations do not reside within the finite, might we assume that they are somehow aspects of the infinite? Can the infinite be brought into the finite without becoming finite? What connects the finite and infinite? Quite puzzling, it behooves us to ask, “In what ways might David Hilbert have been wrong about his conclusion regarding the finite-infinite relation?”

Simple logic seems to tell us there is an ever-so-active, rather intimate bridge between the finite and infinite. By definition, the finite is the quantitative. Perhaps we should be thinking about the infinite as the qualitative. So, we now propose (and have begun exploring the idea) that this bridge is defined by all the dimensionless constants and the infinite is the qualitative face of reality and beingness.

We’ve all been taught that the infinite is some abstract superlative that is not part of our immediate reality. Here, quite the opposite, it appears that the infinite is not only an intimate part of our experience, it is the experience of the experience that can not be defined by space and time. It is defined by ratios and relations. And, the infinite is experienced as some expression of continuity, symmetry and harmony. That’s a perfection and it appears to extend into the finite.

Yet, recognizing the transitioning from spheres to tetrahedrons and octahedrons, somewhere along that progression of notational doublings, the most simple tetrahedral gap, just five tetrahedrons sharing a common edge, would be among the many combinations that would be tested. My suspicion is such a gap doesn’t become part of fabric of the universe until as late as the first three seconds up within Notation-143, Notation-144 and Notation-145 and then over time, that structure begins working its way back to earlier notations between 48 and 64.

It is a new topic opened for discussions and analyses. Again, in this model, we shall give the infinite everything qualitative. We’ll give the finite everything that is quantitative. And within this model of the universe, quite obviously the quantitative and qualitative co-exist quite well together.

Yet, those age-old questions about good and evil are implied.8 Might we say that all qualities that do not reflect continuity, symmetry and harmony (but do reflect discontinuity, asymmetry, and disharmony) are perspectival and are actively impressed within the very fabric of this dynamic universe? The implications are so far-reaching. More…


Nanosecond to Plancksecond (Planck Time) (10-9 seconds, the nanosecond is just one billionth of a second)

A long, long way from the Nanosecond to the Plancksecond.9 Our mathematically-defined chart of the universe captures the nanosecond within Notation-114 at 1.1197×10-9 seconds. Notation-115 is 2.2395×10-9 seconds, Notation-116 is 4.479×10-9 seconds, and Notation-117 is 8.958×10-9 seconds.

That encompasses the first four groups of nanoseconds of the universe. The related length scale is in the domain in which most of life takes place. Here time is dynamic right back to the first instant.  Each notation defines the look and feel of the universe within that notation.

Wouldn’t you think that our entire universe shares this moment in time? If it is true for the first 116 notations, it may well be true for the next 86 notations.

A nanosecond is equal to 1000 Picoseconds. The Picosecond (10-12) is followed by the Femtosecond (10-15), the Attosecond (10-18) and the Zeptosecond (10-21).

The accuracy of time determination. The greatest accuracy achieved to date, the zeptosecond, was achieved in 2016 by a collaboration of three groups: Max Planck Institute of Quantum Optics (MPQ) in Garching, Technical University of Munich (TUM) and Ludwig Maximilian University (LMU) of Munich. They captured the timescale of photoionization. They were the first to make such a short determination of a unit of time. That followed their earlier work to establish the attosecond under the MPQ leadership of Ferenc Krausz and Vladislav Yakovlev.

The measurement of the Zeptoscond, just one sextillionth of a second — that’s a trillionth of a billionth of a second — is work led by a laser physicist, Martin Schultze. It is truly a measurement by devices, not just a mathematical calculation, and Schultze steps us back into Notation-74 to Notation-77 within our horizontally-scrolled chart.

On to Planck Time. As fast as it is, that zeptoscond is still rather slow when compared to 10−44 seconds given within Planck Time. Next will be the Yoctosecond (10−24), just one septillionth of a second (10−24). Within our chart, the Yoctosecond ranges from Notations 65-to-67.

No Names. The actual words for the next six categories (or groups) down to the Planck scale do not yet exist. Hardly trivial, until each group has a name, they have a limited identity and study of them is more difficult.

The last International System of Units (SI) categories to be added were in 1991. It may well be time to call them back together again. They need to name those next six new groups: 10−27, 10−30, 10−33, 10−36, 10−39, and 10−42 seconds. Planck Time at 5.391 16(13)×10-44 seconds is within the 10−42 seconds’ expansion. It could be named a Plancksecond or PlanckSecond. To date, that combination of words has only been used casually to refer to an extremely short period of time.


Stretching Credulity Even Further

The four base units in lockstep.10 There is literally no room for error within the first groups of notations. It would seem that even with the abundance of scaling vertices, 8.5 billion within Notation-12, and 5.902958×1020 vertices at Notation-24, the thrust of precision would keep everything perfectly aligned. Yet, we know by Notation-64 with its 6.2771017×1057 scaling vertices, quantum fluctuations are emerging. That’s well-established fact.

Consciousness is also a fact.11 We assume it is within the earliest sixty notations. Back in and around 2016, our guess for consciousness was that a primitive consciousness might emerge as early as Notation-48 and that domain could be considered a place for the introduction of a type of fluctuation. We are now researching to see if there may be a better nomenclature already established to distinguish the emergence of various kinds of fluctuations. At Notation-48 there are 2.2300745×1043 scaling vertices. It may well be the area in which we begin our search for the first manifestations of a gap integral to creating a system for the five most primitive perceptions.

Review. These Planck spheres manifest the most complete sense of continuity, symmetry and harmony possible. Sphere stacking would be generating “pure” tetrahedrons and octahedrons. Yet, within each doubling, new dynamics are introduced.

Prime numbers. 12 Rather casually proposed in a few sentences and brief paragraphs in earlier postings is the role of the prime numbers. There are 45 prime notations within the 202 notations that currently encapsulate the universe. There are just nineteen primes within the first 67 notations. The postulation is that each prime supports a new mathematical system that initiates even more diversity and complexity. As within computer programming, there is an order of operations based on the logic of mathematical expression. Each expression gives us clues about our universe and their ordering and we need to pay attention to all of them.

The Hubble Constant.13 Reviewing the work of Wendy L. Freedman, Professor of Astronomy and Astrophysics at the University of Chicago, I learned about their standing enigma — the two values for the Hubble Constant. The editors at PSW Science ask, Is There a Crisis in Cosmology? A New Debate Over the Value of H0. For a November 6, 2020 Zoom webinar with Prof. Dr. Wendy Freedman, PSW Science comments, “If the tension is real, it may signal a new physics beyond the standard model.”

The Hubble constant is the cosmological parameter that sets the absolute scale, size and age of the universe; it is one of the most direct ways we have of quantifying how the universe evolves.” –Wendy Freedman (video), Carnegie-Chicago Hubble Program. VIII. (PDF), ArXiv, July 2019

Astronomers make observations and have an observational framework. We have a strictly mathematical framework.

  1. Can the two work together? Can their observational data work with this simple mathematics?
  2. Can the observational data work if the concept of time is derivative and finite?
  3. Can the observational data tolerate infinity defined as continuity, symmetry, and harmony?
  4. Would the Hubble Constant work with 201 fully-symmetric notations in one manner and with an asymmetric Notation-202 in another manner? That is, might the Hubble measurements that are within Notation-202 be different from any measurement that is within Notations-0-to-201?

Of course, our suspicion is, “Yes, of course,” observation and mathematics most often concur.

We know how idiosyncratic this model of the universe is. We also know how simple and logical it is. We also know there are over twenty assumptions, all departures from the academic norms, that have been made since 2011. With this posting, we’ll add another: Time measurement varies between the 202nd notation and all other notations. Notation-202 feels directional and seems linear; it is asymmetric. Notations 1-201 are all symmetric and each notation has been “filled” with infinitesimal spheres that are defined by the Planck base units.



Fundamental key question, “Is the universe open or closed?”14 In 2017 a most-helpful work by Joseph Silk, Challenges in Cosmology from the Big Bang to Dark Energy, Dark Matter and Galaxy Formation (2016) had come to my attention. In a quick note of thanks and an introduction, I asked for help, “Where did we go wrong?” More recently, I discovered a provocative article that he had written with two others, Planck evidence for a closed Universe and a possible crisis for cosmology (Nov. 2019). With this work I awkwardly engaged his question, “Is the universe an open or closed system?” I was confused; and an email to Joseph Silk, Alessandro Melchiorri, and Eleonora di Valentino (November 2019) certainly reflects that confusion; and rather predictably, that confusion continues to this day.

It seems the answer to their question is “perspectival, yet fundamentally open.” The five dynamics of pi push our faculties to imagine what these small-scale dynamics look like within the largest-possible scale, i.e., the size and age of the universe at this moment in time.

Emergence, Inflation, Acceleration.15 The other dynamic in all these equations is within light. In one of her earliest articles by Eleonora Di Valentino, with her co-authors Alessandro Melchiorri, Valentina Salvatelli, and Alessandra Silvestri, wrote Parametrised modified gravity and the CMB Bispectrum (ArXiv 2012). They conclude, “Cosmic acceleration is one of the major challenges faced by modern cosmology and understanding the very nature of what is sourcing, it is the main focus of up-coming and future cosmological experiments.”

What has happened in the intervening eight years?

Those comments reflect the blinding problems created by big bang cosmology that seem far more approachable within our mathematically-integrated view of the universe. In our simple model, there is a natural inflation, a thrust of the universe, that can all be seen within the numbers of our chart of the universe and these generally approximate the majority of big bang epochs.


Continuity, symmetry and harmony are three facets of reality that define both the finite-and-the-infinite, as well as light, space-and-time and pi-circles-and-spheres.

Old mysteries become evidence as new mysteries are uncovered. Thank you. –BEC



Endnotes and Footnotes

*  One plancksphere per plancksecond, and if Planck Time is equal to 5.391 16(13)×10-44 seconds:

*539,116,000,000,000,000,000,000,000,000,000,000,000,000 planckspheres per second

That number would be the top end. It 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. -BEC


[1] The Speed of Light and the Nature of Light in a Very Different Light: 202 Notations. Light defines (1) each notation, (2) space-time, and (3) mass-energy. Yet, in this model of the universe, each is defined even more fundamentally by continuity, symmetry, and harmony. The speed of light is defined within each notation as well. When we first started exploring the numbers for these Planck Length doublings, we had no idea that we would find a simple correlation between Planck Length and Planck Time. Then, we started thinking about Planck’s formulas, particularly the more simple expression for Planck Time:

Our first reflections began in 2012. By 2014 we began to grasp how well all the numbers worked together. Those formulas worked! We looked for articles by experts but could find no references. Yet, right there on the page, simple mathematics was validating the relation. We began to realize that light is a much broader category than photons, just as photons are a much broader category than visible light.

The results of that simple act of division — Planck Length divided by Planck Time is equal to the speed of light — was nowhere to be found so we began writing it up. Planck’s formula in light of the 202 notations, the instantiation of the sphere as the first expression of space-time, mass-energy, and the building of geometries (cubic close packing), and 64 to 67 notations to quantum fluctuations, particles, and waves, extends the deeper definitions of light as an aether, matrix and grid.

Planck’s calculations render the speed of light more accurately than the 2019 SI number or ISO number (299,792,458 m⋅s). More


[2] Max Planck ignored his calculations from 1899. It would take out-of-the-box thinkers to give meaning to those four Planck base units.

[2a] C. Alden Mead (UMinn) In 1959 he began his struggle to publish his work about the Planck Length. Though finally published in 1964, the article, Possible Connection Between Gravitation and Fundamental Length Phys. Rev. 135, B849 (10 August 1964), was ignored by the scholarly community. Planck Length commanded no respect as a fundamental unit of length.

[2b] John Barrow (1982): With an extraordinary depth and range of scholarship, and a sensitivity to young students, my first letter to John Barrow in 2013 was an earnest request for help, “What do we do with these numbers?” He never commented about my naive attempt to shoehorn everything-everywhere-for all time into 202 notations. Barrow died on September 26, 2020.

[2c] Thanu Padmanabhan: His 1985 article — Physical significance of planck length (PDF) — captured my attention. His nonperturbative approach produced a quantum cosmological model free from singularities and the horizon problem. I was very surprised and gratified to see that his article was published so early in his career. He was just 28 years old (born March 10, 1957). Yet, with guidance from India’s renown astrophysicist, Jayant Vishnu Narlikar, he has been a most prodigious scholar.

[2d] Frank Wilczek (2001) became a Nobel Laureate in 2004, yet he continued his wide-eye, open and enthusiastic approach to the unknowns within life. He was one of the first of those within his caliber who encouraged our explorations. His three articles about Planck units truly opened the door for the rest of us.


[3] From Three Minutes down to Three Seconds. It seems that most within the general population and many of our best scholars hold the positions spelled out in 1976 within the book, The First Three Minutes, by Steven Weinberg. A Nobel laureate and one of our most cited scholars, Weinberg still missed over two-thirds of the most fundamental structures of the universe.

In June 2020, twenty-seven leading scholars from around the world argued most convincingly about the early structures of the universe. A tribute to Weinberg, their article was titled, The First Three Seconds: A Review of Possible Expansion Histories of the Early Universe, (June 2020). Forty-four years after Weinberg’s popular work, they acknowledge the abundance of mystery within the first three seconds. More

Those are the magical three seconds that open up to an entirely new and profoundly simple universe.


[4] Yes, New Ideas Do Take Time To Incubate, Sometimes Centuries. Those “seemingly irreconcilable pillars” have been screaming at us for over one hundred years, “It’s incomplete. You’re missing key parts.” So instead of going in circles with the same parameters trying to do a different thing, break the circle so it becomes a spiral and find those missing parameters! Our simple guess is the first 67 notations.

Brandon Brown wrote the book, Planck: Driven by Vision, Broken by War, OUP, 2015. He is Prof. Dr. Brandon Brown on the University of San Francisco campus and he has become a foremost Max Planck scholar with his comments and analysis from his 2015 book published by Oxford University Press. The Planck-Einstein relation was one of the most formative personal relations in history and one of the reasons their silos stand today is because physics has not fully embraced mathematics and mathematics hasn’t fully embraced physics. Arrogance and turf wars keep them apart. The first 64-to-67 notations are grounds for reconciliation; and with it, I predict will come the birth of entirely new science.

Though it appears that Planck was unable to break out of Newton’s commonsense worldview, Einstein made some progress. Yet, he was wrapped up in his own vision. With the help of Max Planck Institutes and people like Brandon Brown, an actual dialogue between Planck and Einstein about the Planck base numbers may yet be uncovered. To date, there is no record of it. If the two of them really focused on those Planck numbers, they just might have discovered those 64-to-67 notations prior to particles, waves, and fluctuations.


[5] “Commonsense is not so common.” It took the intellect of Sir Isaac Newton to define perhaps the most widely-held “commonsense” worldview which today makes very little sense at all. That is, of course, Newton’s claim that space and time are absolute. To this day, it is still pushed forward by respectable scientists and philosophers among us. For me it stands as one of history’s most egregious intellectual mistakes that has created walls and silos within our current picture of the universe. The expression has been credited to several — Voltaire (1764), poet and political writer, Nicholas Amhurst (1726), and a Roman poet, Decimus Iunius Iuvenalis (aka Juvenal, Book III of Satires, circa AD 181). It has become so true, today it’s a truism!

We have missed something most fundamental.

Tensions are real and there are many reasons why these tensions are signals for what people have in so many different ways characterized as a “new physics beyond the standard model.” That expression has been repeated so often, it now has its own acronym, BSM.

More to come: breaking though commonsense and the BSM


[6] Perfection and possible degrees of perfection. We have all experienced a moment that was profoundly moving. Possibly it felt transcendent. Some might call it a moment of perfection. In 1971, confronted with such, I attempted to describe them in the most general mathematical and scientific terms that caught the spirit of such an experience — continuity-symmetry-harmony. I quickly learned that all experiences are within space and time and Newton’s container universe was the generally-accepted, commonsense description of such. It wasn’t enough for me, so I took on those conditional expressions — continuity-symmetry-harmony — to evaluate all new information about the structures of reality. So within quantum physics, I gravitated toward quantum chaos theory and the EPR paradox and what has become known as quantum entanglement.


[7] The very first spheres are still being pushed forward. Hard to believe, of course, but so much about this sphere generation is hard to believe. There is no name for 1044 spheres. One could say, “Somewhere just under a trillion-trillion-trillion-trillion.” It doesn’t compute easily. Then, if we are to imagine that amount per second, it becomes rather unfathomable. Nevertheless, very large and very small numbers are the next steps for all of us to begin to grasp the boundaries of this universe. More


[8] Qualitative, values, ethics, morals, aesthetics. Continuity, symmetry and harmony became the basis to make valuations: What is good? What is better? And, what is the best? These valuations are perspectival. They slide around plancksecond-by-plancksecond, second-by-second, day-by-day, and year-to-year. So even though seemingly arbitrary, there are aggregations of value which also become a basis for judgement and evaluations such that a quantitative value may be assigned to an experience, an artform, or performance, and especially to things. More


[9] Scale of Infinitesimal Measurements: From Nanosecond to Plancksecond. The six groups of numbers between the Yoctosecond and the Planck Time need formal names that everybody within the global scientific community recognizes.

The group responsible for such standards is CODATA, the Committee on Data of the International Science Council (ISC). They are charged to improve the availability and usability of data within all areas of research. Having a name for these infinitesimal measurements is the first step in having the reality of these dimensions become adopted science. I have asked many of the members of their Executive Committee to take up that cause. More


[10] Four Planck base units in a base-2 notational lockstep: At this point, we can only use our intuition and logic to construct the very nature of simplicity and perfection. Langlands programs and string theory may help yet those disciplines will have to incorporate, then evolve out of the functions of pi, circles, and sphere. They’ll have to find places into which they can pick up and integrate all their earlier work, accommodating the four Planck units working together. In this model, each of these initial 64-to-67 notations represent unique opportunities, relations, and functions. More

As we continue this analysis, it will be coming increasingly important that the six groups of scales of the universe with no names, be named officially-and-formally by the ISO and CODATA and all their consultative entities like NISTBIPM, and standards groups within 141 other countries. Prior to quantum fluctuations, there are these clusters of measurements that have no name: 10−27, 10−30, 10−33, 10−36, 10−39, and 10−42 seconds. As we observed earlier, Planck Time at 5.391 16(13)×10-44 seconds is within the 10−42. More


[11] Consciousness: We know where the brain is located, but we do not know how where the mind is located nor are we profoundly sure how the brain/mind relation works. The first experts to whom I turned for insights about these issues was John Eccles. His report with Karl Popper is The Self and Its Brain (Springer, 1977). It set the stage for me. The next was Roger Penrose, author of The Emperor’s New Mind (Oxford, 1989) and Shadows of the Mind (Oxford, 1994). Their genius is clear and their concepts robust, yet neither Eccles nor Penrose were able to define a grid within which the mind and brain were in concert.

Enter Computational Neuroscience. While an undergraduate in-and-around 1957, Stephen Grossberg posited nonlinear differential equations for neural networks. One could say that it was a stroke of genius as well as a deep-seated intuition.  Grossberg was among the first to begin to define computational neuroscience. Yet, the scientific community has been limited without a simple grid that includes everything, everywhere, for all time, including consciousness. The first grid by Kees Boeke used base-10; and though digital, it encouraged analogical thinking with no causal efficacy. Here within a base-2 grid, causality is driving the exploration.


[12] Prime numbers: Prime numbers have a key role within encryption technologies today, but one might still ask, “Is that all there is? Could primes be playing a more fundamental and pivotal role within the structure of things?” We think so. To that end, as an exploration, we would assign the first 19 primes — 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, and 67 — to absorb the expansion of structures and functions within Mathematical Systems Theory  (Also, see the text of that title by Diederich Hinrichsen and Anthony J. Pritchard, Vol. 48, 2005).  Prime elements and irreducible elements, prime ideas, group theory There will be more to come.


[13] Two values for the Hubble constant: In lectures and papers, Wendy Freedman is an expert and leading scholar regarding all things related to the Hubble Constant. She says, “Our value of the Hubble constant, Ho = 69.8, with statistical and systematic uncertainties of 0.8 and 1.7 km/sec/Mpc, respectively, falls midway between the value obtained from the Planck Cosmic Microwave Background analysis, and that obtained using Cepheids.” Freedman has not seen our base-2 scale from the Planck Time, assumed to be the first moment of time, to our current time. Nor has she seen the comparative analysis with the big bang epochs. She is also unaware of the difference between Notation-202 and the first 201 notations. Using the simple logic of this simple model of the universe, the age of the area in the universe being measured will render one result for all images within 2.82+ billion years and another Hubble Constant for all images within areas older than 2.82+ billion years (perhaps as high as 3.2 billion). More to come


[14] Open or closed: Joseph Silk is highly-regarded within the Astrophysics and Cosmology community. His overview gave me a chance to go over the basics just one more time, and ask myself, “What am I missing? Why is our idiosyncratic theory so wrong and the big bang theory so right?”

So, I empathize with those people who started within this industry back in the 1970s. Now fifty years later, those people (like me who are now over 70 years old) remember when there was a certain respect for Sir Fred Hoyle’s steady-state theory as an alternative, competitive theory to the big bang. Since that time investments in the big bang theory skyrocketed and reached an all-time high under the leadership of Stephen Hawking. With its many unresolved problems exacerbated by time and with the Planck base units finally becoming recognized, the value of big bang cosmology is falling. Those who have invested heavily in it cannot be absurd to themselves. Yet, we all know that which is more-true-than-false prevails.

Of course, these kinds of transitions are not easy.

I was glad to discover that second article by Joseph Silk with his two younger colleagues. There is an honesty to their question and we need to explore these kinds of questions. For me, it helped to shape the finite-infinite analysis, especially our focus on dimensionless constants: finite, infinite or in that hyphen between the two. This time, however, it shaped that quantitative-qualitative analysis; and so far, it feels good and seems to work.


[15] Inflation, expansion: A young, Italian astrophysicist, Eleonora diValentino, was part of the “Closed or Open?” discussion with Joseph Silk (just above). In 2014 she was asking key question about  cosmic acceleration, particularly “…the very nature of what is sourcing it…” She appears to be the next generation of the Wendy-Freedman-types who are in search of fundamental truth; these are  people who expect that their quest will never ever finish. As an academic exercise, we are going to focus on a joint article,  Parametrised modified gravity and the CMB Bispectrum (ArXiv 2012) where Eleonora Di Valentino and her co-authors Alessandro Melchiorri, Valentina Salvatelli, and Alessandra Silvestri arrived at that conclusion. The study now is to answer the question, “How can you believe in big bang cosmology when you have done this analysis and made these conclusions?”  So, yes, there is more to come


References & Resources

  1. Planck scale: A special page of references and resources will be dedicated to exploring the work of people like Serge Timashev of the Karpov Institute of Physical Chemistry (Moscow) with his ArXiv article, The Planck numbers and the essence of gravitation: phenomenology (2017),  and Ronald J. Adler of  Hansen Experimental Physics Laboratory (Gravity Probe B Mission) of Stanford University with his ArXiv article, Six easy roads to the Planck scale (2010).
  2. Wendy Freedman, Professor of Astronomy & Astrophysics, Univ. Chicago, November 2020: Is There a Crisis in Cosmology? A New Debate Over the Value of HAlso see: Measuring and Understanding the Universe, 
  3. HubblesiteGalaxies Used To Calibrate The Hubble Constant
  4. April 11, 2019, Marclay’s Clock: 24-hour installation highlights a modern obsession with time, Jean-Michel Johnston, University of Oxford Also in Nexus NewFeed
  5. In 1925, the great mathematician, David Hilbert wrote, “We have already seen that the infinite is nowhere to be found in reality, no matter what experiences, observations, and knowledge are appealed to.” Even today, many scholars would agree, but perhaps Hilbert and those scholars are mistaken.
  6. Research: “Primordial adiabatic and Gaussian perturbations
  7. Is a time symmetric interpretation of quantum theory possible without retrocausality? Matthew S. Leifer and Matthew F. Pusey
  8. Not all mathematical advances relating to π were aimed at increasing the accuracy of approximations. When Euler solved the Basel problem in 1735, finding the exact value of the sum of the reciprocal squares, he established a connection between π and the prime numbers that later contributed to the development and study of the Riemann zeta function.  Fore more: • Complex numbers and Euler’s identityNumber Theory and RZF
  9. October 24, 2020 at 12:22 pm · Reply to “Infinity Is Not The Problem” Does the qualitative reside within the finite? Could the perfection of the sphere be an example? Is pi an example? I’ve been playing with it within an examination of the Planck scale:   Might all the dimensionless constants be the bridge between the finite and infinite
  10. Experimental result cannot be explained by the Standard Model (SM): Non-zero masses for the neutrinos (elementary particles traveling close to light speed, electrically neutral, and weakly interacting). The SM assumes that they are massless. Therefore, particle physics explores a new physics beyond the SM.
  11. The Standard Model is not a complete description of Nature: it does not account for dark matter, dark energy, gravity, or neutrino masses and mixings. There are also remain many features of the Standard Model itself which are not understood, and which may find their answers in speculative ideas beyond the Standard Model such as supersymmetry, large extra dimensions, and/or extended Higgs sectors.
  12. Conspiracy of BSM physics and cosmology, Maxim Yu. Khlopov. Nov 2019: The only experimentally proven evidence for new physics is the effect of neutrino oscillations, but the physical nature of neutrino mass is still unknown. “… the conspiracy of Beyond the Standard model (BSM) Cosmology [1] is puzzling taking into account the plethora of nontrivial cosmological consequences of BSM particle models. “
  13. Yakov (YaB) Zeldovich: ”…though the probability for existence of these phenomena seems low, the expectation value of their discovery can be hardly overestimated” (ArXiv, Conspiracy of BSM physics and cosmology, Maxim Yu. Khlopov, Nov. 2019
  14. The Crisis in the Foundations of Mathematics, José Ferreirós, Universidad de Sevilla 2008, 2011
  15. Proyecto Hephacos, 2014:
  16. Is The Universe Finite? PBS, sponsored by Brilliant, 2019


Emails (just a few)

1. Marios Christodoulou, Andrea Di Biagio, Pierre Martin-Dussaud, An experiment to test the discreteness of time

2. Wendy Freedman, Kavli Institute for Cosmological Physics, Astronomy & Astrophysics University of Chicago

3. Andreas Battehberg, Entering the field of zeptosecond measurement, Nov-2016

4. Bonnie Carroll, Secretary General, CODATA. Calculator


Tweets (just a few)

@Pontifex What you are saying is actually backed up by a mathematical-and-scientific model of the universe. There are just 202 base-2 notations from the Planck units to the current-time-and-size of the universe: Also: @lori_deschene You’ve got good spirit, a good heart… now we have to break free of our little worldviews and get a highly-integrated, mathematical view of the universe (all within 202 base-2 notations). Everything we say and do affects the universe.


Zzzzs (afterthoughts)

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.

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Key Dates for this document, The-Three

Image by Mikhail Leonov, April 11, 2019, Marclay’s Clock and what could be called, “Fibonacci Time.”

[17] Fuzzy Universe

Foundational Questions Institute: Undecidability, Uncomputability, and Unpredictability Essay Contest (2019-2020) Support materials for the submission from Bruce Camber in April 2020.
[1] Decidability
[2] Computability
[3] Predictability
[4] Undecidability
[5] Uncomputability
[6] Unpredictability
[7] First units
[8] Grand reductionism
[9] Triangulation
[10] Fourier
[11] Lorentz
[12] Poincaré spheres
[13] Planckspheres
One second: 299,792± km
[14] Automorphic forms
[16] Aristotle’s Mistake
[17] Fuzzy Universe
[18] Scholars
Background: FQXi called for papers. It encouraged people to focus on the raw power of logic and mathematics to anticipate the structure of real realities. If there is logical and mathematical cohesion, there is probably a real physical reality that it describes. Matching them up and learning where and how such a unit of cohesion fits within the larger frameworks is the challenge. More...

[17] Our Fuzzy Universe.  The work of John Wheeler and Richard Feynman are central to this work. With the help of Gödel, Einstein, Planck, Poincaré, Gauss, Euler, and Leibniz, a new path will be found.  The age-old questions about fluctuations and dark energy and dark matter will be answered.

An Alternative: The Universe As An Intimate Place


Warm & Fuzzy or Cold-and-Hostile

By Bruce Camber Key dates   FQXi: In conjunction with this article Today’s work

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 is continuity 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 Review  Finite-Infinite

Pythagoras said that all things come to be in accordance with number on the grounds that order in the primary sense is in number and it is by participation in order that a first and a second and the rest sequentially are assigned to things which are counted. – Theano

Introduction. For as long as I can remember we were taught the universe is a vast, empty space. More recently, we learned the universe has over two-trillion galaxies (see Chris Conselice, 2016) with many-more trillions of stars. Either way, it is too big for intimacy. Coming to our rescue is Theano’s Pythagoras who gives numbers a role and stature. In our hyper-networked model* where everything is connected to everything, we hope to open paths that point to a very special intimacy within our universe.

Max Planck’s four base units.1 When we backed into a simple mathematical outline of the universe by applying base-2 to the Planck units, there are just 202 doublings. It all seems quite simple and the 202 notations seem entirely manageable.

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:

Planck 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.[3][4][5] 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 to tetrahedron-octahedron couplet

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.

Tetrahedrons enclose an octahedron.

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.

Stephon Alexander Alexander
Massimo Amelino Amelino
Mariah_Carey Carey
Gates Gates
Magueijo Magueijo
Michela Missimi Missimi
Dean Rickles Rickles
Leonard Susskind Susskind
Maryna Viazovska Viazovska

Conclusion. All these concepts are being explored. And, with just this introduction, we open paths for further exploration; there is so much more to be explored. -BEC


Endnotes and Footnotes

Please note: Most of these endnotes and footnotes are working drafts; a few are still quite rough. Work and rework will continue for awhile longer.

* Hyper-connected construction: The universe like the human brain or the internet, is discussed in many documents throughout this site: This Shifting Paradigm Changes Our Perception Of Everything (October 2014), Everything Starts Most Simply. Therefore, Might It Follow That The Planck Length Becomes The Next Big Thing? (May 2014), On More Fully Recognizing The Infinite (November 2017),  Four Key Missing Pieces from Our Puzzle (December 2019), and On Asking Nobel Laureates Key Questions  (September 2019).

[1] 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.

[2] 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.

[3] 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?

Sphere to tetrahedron-octahedron couplet

[4] 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.

[5] 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.


[6] 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.

[7] 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

[8] Natural geometric gap. Spheres and spherical geometries are the gateway to Euclidean geometry and the most simple tetrahedron.  As we’ve seen with the dynamic image of cubic-close packing of equal spheres, the octahedron manifests immediately.


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….

[9] Quantum fluctuations and quantum indeterminacy. If this mathematical outline and construction is on the right path, quantum fluctuations are redefined. There is a causal efficacy.

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.

[10] 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.

[11] 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.

[12] 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.

[13] 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.

[14] Revist: Infinite-finite bridge, simplicity-complexity within the infinitesimal sphere. If the most simple sphere is the product of the finite-infinite relation, and if continuity, symmetry, and harmony define the essential qualities of infinite and quantitatively give us the essential quantities of the finite, we have a very different intellectual starting point. So, none of this materials will be easy. Perhaps Planck was right about the absorption rates within the scholarly community. I hope not.

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.

[15] 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)

1. Simplicity and our historic mistakes: This is the Aristocracy of the Elite.
Aristotle’s 1800+ year mistake. Tiling/tessellating with the tetrahedron
Newton’s 300+ year mistake. Absolute space and time.
Hawking’s 40+ year mistake. An initially infinitely-hot start is certainly a hell-of-a-start.

2. Finite-infinite. From the beginning of time to the Now.
The beginning of time is now included within the Now.
Gödel finite-infinite relation
• spheres-and-pi
• the limitations of the Planck base units for time, length, mass and charge
Trialogue on the number of fundamental constants, Michael J. Duff,  Lev B. Okun, Gabriele Veneziano

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).

4. Close-cubic packing of equal sphere:
The power of structural modeling of sub-grid scales …
Planck-scale models of the Universe, Fotini Markopoulou
Visualizing a Hypothetical Planck Scale Substructure of realityDugan Hammock

5. Fourier:
•  Visualizing a Hypothetical Planck Scale Substructure of realityDugan Hammock
•  Six easy roads to the Planck scale

6. Five tetrahedral cluster

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.

8. Newton’s absolute time and space in general relativity, American Journal of Physics 68, 350 (2000);, Ronald Gautreau

In 1998/99 Merab Gogberashvili published on arXiv a number of articles on a very similar theme. [1] [2] [3] 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.”

10. QFT as uncomputable and undecidable. Within this model, the universe is an inclusive instance of all things, everywhere, for all time. The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an effective procedure (i.e., an algorithm) is capable of proving all truths about the arithmetic of natural numbers. For any such consistent formal system, there will always be statements about natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem, an extension of the first, shows that the system cannot demonstrate its own consistency.

On Patrick Suppes’ studies –

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?

Jim Peebles, Michel Mayor and Didier Queloz, 2019 Nobel Prize in Physics

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.”

John Wheeler, relativity, and quantum information, C.W. Misner, K.S. Thorne, and W.H. Zurek,  Physics Today, page 40, April 2009

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 complete recursively 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

As families of formal languages mostly have a decidable emptiness problem, the non-emptiness of the intersection of a formal language and a regular language is decidable.” FROM DECIDABILITY TO UNDECIDABILITY BY CONSIDERING REGULAR SETS OF INSTANCES, Petra Wolf arXiv:1906.08027v1, 19 Jun 2019

12. Gödel, base-2 model and Planck’s units.

Time and Causation in Gödel’s Universe (PDF), John Lane Bell, Transcendent Philosophy 3, 2002 gives time its primary status such that time travel becomes a rational conjecture.

Alonzo Church (June 14, 1903 – August 11, 1995) :  “He is best known for the lambda calculus, Church–Turing thesis, proving the undecidability of the Entscheidungsproblem, Frege–Church ontology, and the Church–Rosser theorem.” – Wikipedia

“His proof that the Entscheidungsproblem, which asks for a decision procedure to determine the truth of arbitrary propositions in a first-order mathematical theory, is undecidable. This is known as Church’s theorem.[8]  – Wikipedia

Lambda calculus (also written as λ-calculus) is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution. It is a universal model of computation that can be used to simulate any Turing machine. It was introduced by the mathematician Alonzo Church in the 1930s as part of his research into the foundations of mathematics.

When Einstein Walked with Gödel, a video by Robert Wright with Jim Holt, author of the book by the same title, Farrar, Straus and Giroux (May 2018).

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 Scholar  M. Planck, 1900 Ann. d. Phys. 1 69  Crossref   Google Scholar  reprinted in Max Planck, Physikalische Abhandlungen und Vorträge, Band I. Friedr. Vieweg. 1958 pp. 560-600, 614-667  Google Scholar

14.  Revist: From Simplicity to complexity within the infinitesimal spheres.

15.  Revisit: Sphere-stacking and cubic-close packing of equal spheres.
•  Sub-grid scale model classification and blending through deep learning (December 2018)
The power of structural modeling of sub-grid scales – application to astrophysical plasmas, Georgiev Vlaykov, Dimitar, Grete, Philipp (August 2015)

  • Modification + divergence-structure + perturbation theory +  spacetime foam
  • Exponential factors + finite.
  • Dynamics generated + the Hamiltonian of general relativity + energy + bounded + stable theory

Please send along your comments or questions.

Header image:

Credit goes to Pablo Carlos Budassi of South Africa where his Facebook, Twitter and YouTube projects are all linked. A high-resolution image is here on Wikipedia under “Universe” (just page up from “See Also“).

Key Dates

This page was initiated on April 5, 2020
Posted online for collaboration: Tuesday, April 7, 2020
Homepage date: April 23, 2020 @ 11:11 PM
Last edit: Wednesday, May 19, 2020

This page is:
Related pages:
We’ve been duped!

[12] Poincaré spheres

Foundational Questions Institute: Undecidability, Uncomputability, and Unpredictability Essay Contest (2019-2020) Support materials for the submission from Bruce Camber in April 2020.

Determinant becomes unpredictable, uncomputable, and undecidable (PDF)

[1] Decidability
[2] Computability
[3] Predictability
[4] Undecidability
[5] Uncomputability
[6] Unpredictability
[7] First units
[8] Grand reductionism
[9] Triangulation
[10] Fourier
[11] Lorentz
[12] Poincaré spheres
[13] Planckspheres
One second: 299,792± km
[14] Automorphic forms
[16] Aristotle’s Mistake
[17] Fuzzy Universe
[18] Scholars
Background: This FQXi challenge brings into focus the role of sphere, particularly the Poincaré sphere. The high school students who interpret the role of spheres and transformations to their entire family will also include the Poincaré spheres.

[12] Poincaré Spheres: Leibniz-Euler-Gauss helped to give us Henri Poincaré. The Poincaré sphere with the Fourier transform and Lorentz symmetries and transformations will continued to be studied and the progress indexed to relevant pages on this website and the most relevant articles throughout the web.

Grasp A Mathematically-Integrated Structure of Universe

No Big Bang  — Focus On the Planck Units

“What difference would it make? Even if this model of the universe is right, so what?’

  1. Planck Time is a real number generated by fundamental physical constants. In this model it is assumed to be the very first moment of time, the very start of this universe. That unit of time is 5.391 247(60)×10-44 seconds.
  2. Planck Length and Planck Time define each other, a necessary tension and a necessary relation, expressed by a simple formula: Planck Length divided by Planck Time is equal to the speed of light.
  3. The work of pi (π) within the formulas for these physical constants shape the first manifestation of beingness, necessarily a sphere, and with the sphere come all the sphere dynamics uncovered by Euler, Gauss, Poincare, Fourier and others.
  4. The universe is a system that builds logically and systematically on itself and everything that happens, right from that very first moment, structurally continues within this instant.

This page was initiated in February 2020.   See Ethan Siegel.

12 Key Concepts: How To Integrate The Entire Universe In Just 202 Steps


A Simple Mathematical Model of the Universe

by Bruce Camber  The arrow pointing left  just above or the title link to a related homepage.                                   

Abstract: The long-standing “infinitely hot” big bang cosmology blocked the view of a simple mathematical construct that encapsulates the entire universe within ordered relations that provide an outline, possibly a new foundation, that starts very cold with the Planck base units. By applying base-2, the universe is parsed and parameterized within 202 notations (doublings, groups and sets). Here is a different approach to space-and-time, mass-and-charge, and the finite-infinite relation, all concepts needed for the sciences, mathematics, and commonsense. Here we focus on twelve basic concepts to transition from Newton’s absolute space-time, big bang cosmology, and the place of particles-and-waves to an infinitesimal, sphere-based model that gives rise to a finite, discrete space-time, and then to quantum fluctuations and indeterminacy, and then to our universe as we know it.

Background. Closely-associated with other homepages, this article will continue to be updated. Twelve Concepts (this page) is associated with Transformation (Aug. 2019), Bottom-up (Sept. 2019), Map the Universe (Oct. 2019), Finite-Infinite Bridge (Nov. 2019), Our young, cosmological model (Dec. 2019), A Simple View (Jan. 2020), Claims (Feb. 2020) and Perfection (Feb. 2020)

The twelve foundational concepts of this model:

#1 Grasp the boundaries.
The Approximate Age of the Universe in seconds:
436,117,076,640,000,000± seconds1

“436 quadrillion, 117 trillion, 76 billion, 640 million seconds.” A review of this very big number resulting from a rather simple math exercise begins to give us a sense that the universe is finite. It has a starting point and the current time is always defined by the current expansion.

Within this model there is no need for absolute space and time. We surely envision a space and time that goes on forever but that doesn’t ascribe a primordial status to it. It is our hope.

Now, if you will, think what happens when we all know and can say, “The universe is just over 436 quadrillion seconds old!?!” If we all used a working Universe Clock (a little like the US Debt Clock), it just might help us all feel some commonality with each other within this world, with our little Solar System, then within the Milky Way and even our Local Group.*

Yes, just think what might happen when we all envision everything we do and think in context with the entire universe.1a

A Dilemma: In 2011 when we first engaged the question, the universe was estimated to be between 13.772-and-13.82 billion years, with an uncertainty of just 59 million years. That estimate continues to be more refined. Some now put the number closer to 14.4 billion years. 

Yet, throughout the universe, how long is a year? We are on a Solar-Time System based on our Sun. The rest of the universe is not. Our year has 365 days except for Leap Year when it has 366 days. A simple question is, “For every thousand years, should we add 250 days?” We can all readily do the easy calculations — the number of seconds in a minute (60), an hour (3600), a day (86,400), and then a 365-day year (31,536,000 seconds). There are 31,556,952 seconds in a 365.2425-day year. And, 31,557,600 are in a 365.25-day year.

We chose 365.2425 days/year, 31,556,952 seconds/year or 31,556,952,000,000,000 per billion years (an aeon). Multiplied by 13.82 is 436,117,076,640,000,000 seconds.

At some point in the near future, we’ll be increasing that number. First, we’ll probably be multiplying by 14.4. Then we will need to add the approximate amount of time that will have passed from the moment the experts make their calculation. In 2011 we used the numbers from NASA’s Wilkinson Microwave Anisotropy Probe (WMAP).1b There is also ongoing work with the Hubble Space Telescope,1c the ESA Planck satellite,1d and NASA’s Spitzer Space Telescope.1e

Just a second: The value of a second was established by a consortium of government agencies around the world. In 1841 they unofficially began cooperating and sharing insights. In 1947 they formed the International Organization for Standardization1f (ISO) to share “official standards” but those standards were more fluid than most expected.

How does one define one second? The ancient approximation was 1/86,400 of the time that it takes the Earth to rotate once on its axis. In the 1700s it was determined by the pendulum swing of a carefully-calibrated grandfather clock, and then more recently by the very stable calibrations within the cesium 133 atom.

We’ve been advocating that the second be defined by an exact multiple of Planck Time. At Notation 143, it is .60116 seconds. Why not add that fraction that brings it as close as one needs to the current one second mark? We proposed such a concept back in 2012 when we advocated that a standard length be based between the multiples of the Planck Length.  In 2012 a retired NASA scientist1g thought it was a worthwhile proposal.

Reviews: (1) Our 2017 analysis for Huntsville’s NASA SpaceApp:,  (2).earlier calculations of an actual Planck time line, the 2016 horizontally-scrolled chart,  and (3).the first calculations for time in 2014,  a vertically-scrolled chart!



#2  Capture everything, everywhere, for all time.
Apply base-2 to the Planck units:
202 notations from the first moment of time to the Now.2

2202  By definition, the Planck base units are key initial quantities of physical reality. Pi (π) and other dimensionless constants are part of the dynamics. Taken as a given, if these base units are doubled, then doubled again and again, in just 202 doublings, the current time (the Age of the Universe right now) and the current size of the universe (right now) are inscribed. There are well over 1000 numbers that are generated within this chart (horizontally-scrolled). It is entirely predictive; each notation necessarily builds on the prior notation; and, all notations are never ending  and never repeating.

The very first instant of the universe is Planck Time: 5.391 16(13)×10-44 seconds. It is so infinitesimally small, the first second of this universe only emerges between notations 143 and 144. The first year emerges within notation 169. The first thousand years, a millennium, emerge between notations 178 and 179. The first million years are between notations 188 and 189; and, the first billion, an aeon, is between notations 198 and 199. Notation 200 has 2.744 billion years. Notation 201 has 5.4908 billion years. All of human history and most of the history of our Planet Earth are within Notation 202 (10.9816 billion years in duration). Given these notations appear to be cumulative, we would be in the earliest part of Notation 202. Here we assume the universe has been expanding for 13.81 billion less the cumulative 10.9816 billion years.

For earlier perspectives, go to:


#3 Eliminate some basic mysteries.
Dark energy and dark matter
Study those notations that cannot be physically measured.

264  The first 64 notations.3  We take as a given that all 202 base-2 notations defined within the horizontally-scrolled chart are real. Logic tells us. Mathematics tells us. Then there are repeated confirmations between the systems of mathematics and our physically-measured realities. The first 64 notations are foundational and require much more study. The first few notations are initially displayed when one opens the chart online. If one were to scroll through the chart (yes, horizontally scrolled), especially observe how those first 64 notations begin to yield an impressive mass and charge. At just the 64th notation, the Planck Mass multiple is now 4.01495×1011 kilograms and Planck Charge multiple is 34.5986 Coulombs. Yet, the Planck Length and Planck Time multiple are still infinitesimally small. The first actual “measurements” of a multiple of the Planck Length appear to be between Notations 64 and 67 and currently the smallest measurement of a fractional unit of time is within the 84th notation.

A rather naive-and-startling conclusion for many, yet seemingly quite obvious, is that  the first 60± notations define dark energy and dark matter. It is simple, all-natural, and logically follows.

The Process: Again, to derive the 202 steps or notations from the Planck base units to the approximate age and size of the Universe today, multiply the four Planck base units by 2 (and the results by 2, over and over again). Perhaps the simplest line to follow is the Planck Time as it doubles up to and beyond the current age of the universe within Notation 202.

The challenge: Now, particularly watch lines 5 and 6 within our large horizontally-scrolled chart:  We start with Planck Mass (2.176.470×10-8 kilograms) and Planck Charge (1.875×10-18 Coulombs). And it bears repeating — at the 64th doubling or notation, the time and length measurements are still below our abilities to measure directly, yet Planck Mass has increased to 4.01495×1011 kilograms (400 billion kilograms) and Planck Charge has increased to 34.59863 Coulombs (perhaps the inverse of a neutron star yet of similar densities).  It is assumed that our infinitesimal universe is now heavily-laden with what we call planckspheres, a primordial sphere constantly being generated from Notation #1.

That’s stretching credulity, but then it goes extreme.

Just beyond the first second of the universe at 1.2023 seconds at the 144th notation, the other values have exponentially increased to 4.8537×1034 kilograms and 4.1827×1025 coulombs. Today we are within the 202nd notation and the Planck time doubling now approximates the age of the universe and the Planck Length doubling approximates the size of the universe. Planck Mass approximates the total mass of the universe and Planck Charge approximates the total coulombs value of the universe. Some percentage of these totals are below our thresholds for measurement, and so, yes, one might conclude, “Here is our ever-so-illusive dark energy and dark matter.”

To review our very first article about this conclusion, go to:  


#4 Start with the most simple and build. Complexity comes quickly. Redefine the infinite and the infinitesimal: Start with the equation for π.

{\displaystyle \pi ={\frac {C}{d}}}
C is the circumference of a circle, d is the diameter.

The finite-infinite relation.4 We will continue studying the role of  dimensionless concepts like π (pi). We hypothesize that the qualities that define fundamental physical constants and the dimensionless constants (ratios) will inform a perspective that will help to define the infinite (infinity) as well as the infinitesimal. This definition will not satisfy those scholars engaged in the current debates about the definition and use of infinity within our cultures, yet the hope is that it opens new possibilities for new discussions.

The most-simple, most-ubiquitous, never-ending, never-repeating ratio, pi, is a starting point. By definition, it (1) defines and redefines the infinite, (2) it opens a definition of a finite-infinite bridge, and (3) it opens a range of quantities (and qualities) that capture the infinitesimal across as many as 64 notations.

The abiding concepts that flow within the finite-infinite relation and throughout pi are continuity (order), symmetry (relations) and harmony (dynamics). Yet, these three concepts are also the backbone of the finite, particularly our sciences, th e finite-infinite-bridge, and the infinitesimal. Yes,  our starting pointis — it all started with π (pi) and then her sphere.4  

For an earlier analysis, go to:
And also, see:π/ 


#5 Follow the math and the simplest logic.
Mathematically confirm the speed of light:
Always follow the numbers.

Light.5 Based on just the accuracy of the determination of the values of Planck Time and Planck Length, the speed of light can be confirmed mathematically. At one second, the Planck Time value is 1 and Planck Length is the distance light travels in one second.  The experimentally-defined speed of light is 299,792,458 m/s in a vacuum. Outside the vacuum, it is a variable. Mathematically, it is also a variable. Our current calculation is: 299,792,437.99 meters/second.


Of course, it is not at all surprising that the Planck Time, Planck Length, and the speed of light correlate throughout the chart given that both Planck Time and Planck Length are defined by the speed of light.


What is surprising is that this simple formula begins to corroborate the basic integrity of the chart and base-2 exponentiation, and it all begs for a much deeper analysis.

For earlier analyses, go to

Yes, the speed of light is approximated at every notation. It generally ranges around ± .1% of the laboratory defined speed, 299,792,458 meters/second.  Just .1% of that value is 299,792 m/s; our current calculations create a range of approximately 213,648 m/s.  With our current figures, the highest calculation is 299,982,157 (Notation #16) and the lowest (#3) is 299,768,509.931 (a range of 213,648 m/s). So, Planck’s little equation for Planck Time, Planck Length (lP) divided by light equals Planck Time (tP), seems to be telling us an important story throughout all 202 notations. Of course, more analysis is required.

Work on the chart began in 2011. To review, go to 


#6 Force fit what you can.
Compare their intellectual expansion to our mathematical expansion: The numbers are the numbers; logic is logic.

Big Bang Subsumed Within Quiet Expansion.6 There is a concresence between the events of the current big bang theory and our mathematically-defined Quiet Expansion. This analysis opens many questions based on the fact that observational data from the intellectual definition of the Big Bang actually works within the mathematically-defined  inflation of our “Quiet Expansion.” 

Editor’s note regarding the Quiet Expansion: 20-to-20,000 Hz is the generally accepted range of audible frequencies for human hearing; these convert to wavelengths of 1.7 centimeters to around 17 meters or from the 109th to the 120th notation.

For a direct comparison and earlier review, go to


#7 Analyze the logic.
Examine six samples from across the 202 notations or doublings: Caution! This is even more of a stretch.

The Logic of Numbers.7 To grasp each parameter that defines this chart is not trivial. It requires transitioning beyond our commonsense boundaries. Thinking of the universe as exponential at its core is difficult. Thinking about time as an interval without a past but as encoded as a necessary effect within the entire universe even stretches our sense of relationality. I suspect it will do the same for you. The net-net of studying the simple doubling formula is an entirely different orientation to our little universe of just 202 notations.

Studying the logic of these six samplings does not make it much easier. 

For review, go to: and Also, if you have not looked at the chart of numbers, please do:


#8 The first sphere.
Sphere stacking is at the heart of all doublings:
Study this most pivotal image.

Sphere to tetrahedron-octahedron couplet

Stacking/Doubling.8 This simple dynamic image has no less than four key stories. The first story is the emergence of that very first sphere. Sphere stacking is the next story.

The Planck scale, if compared to the atom, is like the atom compared to our solar system. Out of the 202 doublings, 1 to 80, goes from the Planck scale to the atom, and 81 to 160 goes from the atom to the solar system. Surmised is that these planckspheres which have a very small mass and charge, and the smallest length and smallest duration, literally fill the universe (See #3).

That first sphere. It is, by definition, the most simple dynamic of the universe. Yet, there is nothing simple or singular about it. The next sphere comes, and then the next. The centerpoints all connect. That’s an active equation. Stacking beings. More centerpoints connect. The first sphere interacts with all other spheres, centerpoint to centerpoints to centerpoints.

The story is being demonstrated with the dynamic image (above right) whereby geometries are created. Vertices or nodes, lines, triangles, tetrahedrons and octahedrons literally tile and tessellate the universe as it emerges. 

Aether. Many physicists continue to work to justify the concept of an aether. Johannes Kepler, with his 1611 work publication of Harmonices Mundi, had a primitive aether using basic geometries. Kepler then gave us clues with his study of the packing of cannonballs; and with this image just above, both projective and Euclidean geometries are opened. A case could be made for the classic Michelson-Morely 1887 work which looked for a carrier of light waves. When brought down to the Planck scale, the dynamics of the first infinitesimal spheres tell a new story and a very different concept of the aether is introduced. 

For an earlier review, go to


#9  Open up period doubling bifurcation.
Get to know the Feigenbaum constant:
δ = 4.669 201 609 102 990 671 853 203 821 578

Period doubling Bifurcation.9  In 1975 mathematician, Mitchell Feigenbaum, discovered a limiting ratio for each bifurcation interval. It was a constant. The plain vanilla version of period doubling was guided by such constants, yet, from my initial studies, nobody could discern why. Nobody was looking at the 202 notations, so certainly nobody was looking below that 64th notation.

There is nothing simple about Henri Poincaré and his work. Yet, when seen from the Planck scale and through the most simple-but-dynamic sphere, an aether that Poincaré may have well envisioned (p.7, The Foundations of Science: Science and Hypothesis…, Science Press, 1913 by H. Poincare, authorized translation by George Bruce Halsted) appears to manifest.

Jules Henri Poincaré brings so much to this exploration — our current studies include applying his conceptual framework for the Poincaré sphere so to open discussions about polarization and its applicability within the dynamics of the Fourier transform. Ostensibly all these dynamics, including Fourier A and Fourier B (below) are part of Notation #1.

That first infinitesimal sphere brings with it a huge agenda, but a deeper analysis of period doubling has been blocked by big bang cosmology.

Here we can begin to discern the mathematics and geometries that are the basis of period doublings bifurcations. Of course, we’ll continue looking at sphere stacking. We’re learning how to dance in this sphere of influence. Still entirely clumsy, there is a lot to learn.

In a recent article, I said, “…now included are Mandelbrot’s work on fractals, the Santa Fe Institute and their work on complexity and chaos theory, and Stephen Wolfram on computational irreducibility. In 2006 Ari Lehto refocused his work to explore period doubling at the Planck scale and in 2014 Charles Tresser added insights regarding its universality.”

For review, go to: and


#10 Test the Fourier series.
Learn about the Fourier transform dynamics at the Planck scale:

Fourier A

Fourier Transform.10 There are so many equations within the Fourier transform, it rather quickly spins one’s head with calculations. Where period doubling captures the moment when things become two, here we discover how two things are dynamically related to all other things. The Fourier transform has such a diversity of applications, it touches every part of our life. Yet, to the best of our knowledge, at no time has its inherent power been explored at the Planck scale. It, too, has been hidden by big bang cosmology.

File:Circle cos sin.gif
Fourier B

It appears that the Fourier transform at the Planck scale could open a new discussion of the very nature of “gravitational weak” and “electromagnetic and strong.”  Just observe the two graphics on the right. Click on each image to go further. 

More coming from equations of motion, structural dynamics…

For review, go to:


#11 This is where it is all going.
We’re living in an exponential universe and among all things infinite:
We’ll all continue studying Euler’s equations.

Euler.11 Our chart and all these formulations are inherent within an application of Euler’s exponentiation. It assumes a Planck-scale sphere, herein called a plancksphere, which has been populating the universe from the first instance of space and time, and, it creates a basis for a mathematical physics, a new foundational science that gives rise to particles and waves.

Euler’s formula, named after Leonhard Euler, is a mathematical formula that establishes the fundamental relationship between our most basic and our more complex mathematics.

If the universe is fundamentally exponential, it changes the paradigm. It is a very different foundation. If space and time are derivative and finite, this model becomes a major transition. There are basic concepts to be re-instantiated. And, over the past 40 years, there are many very confident articles, books, videos and movies that will need to be updated and rewritten, and a whole new universe needs to be explored. 

For review, go to:


#12 Geometries of probabilities and fluctuations.
Study the geometries of gaps:
Indeterminacy is geometrical.

Logic is logic. A gap is a gap.12 This model of the universe brings pure mathematics into the picture within the first notations. Everything is used and needed — algebra, calculus, geometry and topology, combinatorics, logic, and number theory. Then, by the 64th notation much of the applied mathematics is required, especially dynamical systems and differential equations, mathematical physics, information theory (and, yes, even signal processing).


Within this particular article, we have not discussed the application of probabilities-and-statistics and game theory. These studies are also put to the test. At some notation (or notations) within the scale from 1-to-64, a simple geometry of probabilities is introduced. That geometry has been discussed in many documents on this website. Even before we had our first Big Board, we were discussing squishy geometry in our high school.  It all begins with just five tetrahedrons (seven nodes), but includes the icosahedron, and then the Pentakis dodecahedron. That gap is real and ubiquitous within basic geometries.


This article is just a start. Yes, it’s part of an eight-year start! The work is still in the earliest possible stage of exploration. Yet, I think we have brought it far enough that you can now either rationally slap it down (and tell us why it is not so), or be a little bold, and “like” what you see as an exploratory introduction. That would give us a little encouragement to continue on.  -BEC

Yes, a little like an old-fashioned 12-step program: Here for those who need a little help to break free of big bang cosmology (it has been so profoundly ingrained and is totally addictive), yet, be cautioned, this is an early first draft!


Endnotes and Footnotes

1  What is a second?  Technically within the International Organization for Standardization (ISO standards) the second is one of seven SI base units. They define a second as “9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom.”

There is nothing intuitive or simple about it and we believe there is a more intuitive way.

In light of the entire universe. By using the chart of 202 base-2 notations, we actually start at the Planck scale and go to the current time. Here a second is easily defined in light of the entire universe. Based on a a multiple of Planck Time, one second is between Notations 143 (just .60116 seconds) and Notation 144 (where the doubling is now 1.2023 seconds). This second is defined by dimensionless constants that are universal, not by physical measurements done by people and machines.

Could these Planck units provide a basis for standards? To attempt to open up that question for dialogue, we will engage many of the people throughout the International Organization for Standardization (ISO). We have asked Dr. Ian Robinson of the UK’s National Physical Laboratory. We have also asked scientist/scholars at the National Institute for Standards & Technology here in the USA. Can we get beyond the “local measurements” of the instruments like those done by the Wilkinson Microwave Anisotropy Probe (WMAP), the Hubble Space Telescope, ESA Planck satellite, and NASA’s Spitzer Space Telescope?  Perhaps all forms of measurement with devices, instruments, and tools should be to confirm not to determine a standard.

When the second is defined in light of Planck Time, the entire chart will come alive for many more people. In 2012 we first suggested a concept like it, “Use the Planck Length to define a meter.” A NASA scientist who was helping us with some of our calculations thought it was a worthy idea.  Defining the number of seconds in a year should be trivial and not quite so tortured!

The dilemma.  Almost everyone in this world believes that space and time are absolute. That concept is part of the fabric of our commonsense. To think of time as the Now, where the universe is constantly changing and evolving in part based on what is happening Now, flies in the face of that commonsense. That the universe is always the total sum of all its relations, an actual composite of all relations everywhere for all time, is just too big for most of us. A little like Kermit, a frog singing his song, “It’s not that easy being green,” it is also not easy to change our orientation.


2 202 doublings of the Planck base unitsThe Planck units have had a slow ascendancy. In December 2011 when we backed into our model of 202 base-2 notations, we were chasing simple embedded Euclidean geometries (tetrahedron and octahedrons). It had structure, logic, simplicity, and energy (light and coulombs). It was comprehensive and predictive, but it flew in the face of (1) Isaac Newton’s absolute space and time and (2) big bang cosmology. Both have had sharp and well-informed critics over time, so it seemed a natural coalition to build on their arguments. Also, it appeared that the cost-benefit analyses weighed in our favor. It was therefore “just” a matter of making the case for it, building a consensus, and then a group of advocates with as many possible people and within as many disciplines as possible. 

May the circle be unbroken. In September 2019, the the television producer, Ken Burns, released a block of shows about the history of Country Music. The sixth episode featured an iconic, highly-symbolic country song about unbroken circles; our focus turned to pi and the infinite.

In the original 1907 Ada R. Habershon version, the question was asked, “Will the circle be unbroken?” The 1935 Carter Family version asks, “Can the circle be unbroken?”  We proclaim, “May the circle be unbroken.” It is not a question. Human circles are always broken, physical circles tend to be imperfect. Yet at the deepest levels (Notation #1), conceptually these circles are never broken and give rise to the laws of conservation of energy and matter. 

The mystery of pi. A ratio, never-ending, never-repeating, always a relation, it stretches well beyond us, and it is ubiquitous. It doesn’t have space-time coordinates but is part of all space-time coordinates. While dancing around a new definition of the infinite, I say, “Let pi be our initial definition of the infinite and infinity. Forget all other definitions.”  If we start with pi, we can let pi inform any and all more complex definitions. So, within this context, we considered potential infinite qualities. Today, these are continuity/order, symmetry/relations, harmony/dynamics. There is also the randomness and uniqueness within the “never-repeating” aspect of the ratio. What else? I could not grasp more so turned to the scholars and scientists who have made pi their primary study.

So, there should be more to come, yet those concepts of professional scholars and scientists could also be spread out in the endnotes, footnotes, references and resources…


3 Dark Energy & Dark Matter.  This work is not theoretical. It is mathematical. It is practical. It is logical. It really is a new order of commonsense that asks everyone to reconsider the very nature of their own logic. It is a simple chart that easily defines dark energy and dark matter. Yet, it is also non-intuitive. Time has a new face. Infinity has a key role. Base-2 exponentiation, a doubling mechanism, burst out of the most simple geometries which burst out of a stacking of spheres. The whole chart of 202 notations became a working model of the universe because it is so very simple. And, there is no extralogic to carry it forward. And, yes, although the Planck units were overlooked for many years, that is not the case today.

 Possibly more to come…


4 Redefine the infinite and the infinitesimal. Throughout history, the finite-infinite debate is fraught with emotion with those who “believe” and those who do not. Those debates are ignored. We attempt to start with a clean slate — tabula rasa — by starting with most simple concepts that defy a finite label to the most advanced, the current work within renormalization and regularization whereby infinities are carefully analyzed so  to control that influence. Here are perhaps are the most advance concepts about infinity and infinitesimals. Also, the highly-technical descriptions of infinity and the infinitesimals by out most erudite scholars have yielded concepts with highly-specialized language. That work will be engaged, but always with a goal of discovering new relations that could be used to influence the general perceptions of the world’s people.

To start simply and modestly, the qualities of pi were analyzed.

Within classic studies of the electron, the electron is a point particle with a point charge and no spatial extent. The radius formula is:


Within our base-2 chart, this “point” particle falls within Notation 60. There are 60 doublings of the Planck scale prior to reaching the size of a “point” particle. Denying this infinitesimal range seems a bit too sure that particles and waves are our fundamental building blocks with which to construct our universe. As we shall see within the remaining key concepts of this article, the simple sphere, once it begins stacking, is simple no longer.

More to come…  Question: What is an “infinity moment”?


5 Light: Time, Length, Mass and Charge  Two of the most simple-and-basic formulas open a key  discussion. Reviewing line 10 of the horizontally-scrolled chart is the simple division of each Planck Length multiple by its associated Planck Time multiple for all 202 notations. We discover that the result is very close — within .1 % of the laboratory-defined speed of light in a vacuum: 299,792,458 meters per second. Space and time are the first-order equivalence of light and they are a Janus face. Einstein’s oft-quoted formulation is a second-order equivalence whereby, with the speed of light squared, mass and energy become a Janus face.

Line 10 of the chart is the first simple confirmation that this model has some cogency.

There should be more to come...  


6 Big Bang easily swallowed by a Quiet Expansion. The big bang’s isolation is absorbed within the logic of our natural inflation. Guth, Linde  et al  have been hit long and hard enough about the extraordinary measures taken to hone that old theory. The 1999 University of Cambridge, Newton Institute conference, Structure Formation in the Universe (the book),  may have set the foundation for a major revision, yet something as simple as the base-2 chart, perhaps Wheeler’s Dream, is apparently just too simple for many.

Maybe more to come…


7 Logic 101: The chart has too much information to analyze quickly, but the simple logic tells us that the universe is exponential. It tells us that there is an infinitesimal universe that is ordered, has a geometry, and has at least 60 domains, 10-34 meters to 10-18 and 10-44 to 10-26 seconds, each increased by doubling (base- 2) each step of the way.

Of the many people with whom I have spoken about this work, a biology professor, George Fox, asked for some clarity. Dr. Fox is a leading biologist —  he first conceived of the archaeon as a graduate student and had firsthand experience introducing new concepts. He asked me to examine the logic of samplings across the 202 column grid and this 2016 article resulted.

Examining that logic was challenging. Yet, because we believed we were engaging a new model of the universe, a new concept of time-and-space and charge-and-mass, we have been patient with the challenge and with our own gaps in knowledge. What we did learn was that although it stretched our understanding of logic, it was not illogical or extralogical.

Notations #1 to #64: The challenge now is to build a mathematical foundation for the universe that starts with spheres, the deepest dynamics inherent within all spheres at the Planck scale, and to date, we are examining the structuring within cubic close packing, period doubling bifurcation, and the Fourier transform. Eventually, we will attempt to bring in the more classic studies of holonomy and conformal field theory as well attempt to build a mathematical-and-natural progression to particles and waves.

Along this path whenever possible, dimensionless physical constants and  fundamental physical constants will be applied. Dimensional analysis and Buckingham π theorem will be employed. The concepts of nondimensionalization and natural units will be further analyzed until this new system of measurement begins to make sense to others.

Planck units use the gravitational constant G  (a constant of proportionality) measured in a laboratory to just four digits. The modern notation of Newton’s law involving G was introduced in the 1890s by C. V. Boys. The first implicit measurement with an accuracy within about 1% is attributed to Henry Cavendish in a 1798 experiment.

Still rough and more to come…


8 Sphere Stacking: A Fundamental Dynamic Of The Universe. Three necessary facets of physicality:

  • One plancksphere created the initial bridge.
  • Endless planckspheres are being generated from Notation #1; this is the expansion.
  • Sphere stacking (doublings) define all subsequent notations.
  • Nodes or vertices, lines, triangles, tetrahedrons, and octahedrons begin their definitions within the first few notations, and quickly become projective and Euclidean geometries.

What began as a centuries old problem to maximize one’s use of space on the deck of a ship was then applied to crystal and atomic structure. Now it goes ever so much smaller into the Planck scale.

Cubic-close packing of equal spheres began around Notation 114; it was then applied chemical structure and went down into the notations around 90. Then it was applied to atomic structure and went down into notations around 80. With this next application, we bring it all the way down into Notation #1.


9 Period doubling bifurcation. We are trying to discern if anybody has defined a mechanism that could explain a period doubling hierarchy. Though studied as early as 1887 by Lord Rayleigh, it appears to be somewhat of a mystery. I’ll check with Ari Lehto, Charles Tresser and some of the other scholar-experts to find out where the edge of that research is today.

So, of course, there is much more to come…


10 Fourier series, transform: The goal here is to bring everyday physics and mathematics to bear to grasp the foundations of our universe so there is nothing esoteric or extra-logical about it. How very satisfying it will be if key mathematicians throughout our history, people like J. Kepler-C.F.Gauss-T.C. Hales (cubic-close packing),; Poincaré-Feigenbam (period doubling bifurcation), and Fourier-Dirac-Strogatz (Fourier transform), are responsible for the concepts that describe and predict the behaviors of our infinitesimal universe.

To that end, we are going over these details just one more time. I know we are missing  a lot.

So, of course, there is much more to come…


11 Euler’s Equations: The most beautiful equation, Euler’s identity, will have a key role in this model. I am not sure how and who yet, but some mathematical genius, somebody like Po-Shen Loh (video), will come along and tell us. Exponential notations, 22, 23 and 264 and 2202 are all key steps for our students to begin to grasp the “singularity” and harmony of our universe.

Again, of course, there is much more to come…


12 The place and power of simple logic: Where there is continuity, there is discontinuity. Where there is symmetry, there is also asymmetry. And, where there is harmony, there is dissonance-and-the discordant.  If the infinite is ever to be meaningful for science, the historic definitions of infinity from philosophical and religious thinkers should be ignored. Defining the infinite as continuity, symmetry, and harmony allows infinity to penetrate the finite and we need not be concerned with Hilbert’s consternation about the exclusivity of each from each other. There is a transformation between the two and it is believed that a finite-infinite bridge can be defined much more clearly.

So, of course, there is much more to come…

Research, references and resources:

Planck Length compared to the atom is like the atom compared to the Solar System. We have a lot to learn about the infinitesimal!

Key Dates for World Initiated in private on September 5, 2019
Protected posting: Thursday, September 12, 2019
First-draft homepage: Monday, September 30, 2019
Most active editing: September 9 to October 5, 2019
Re-initiated as a homepage: 9 February 2020

Let’s redefine the historic æther.

Aether or Æther or Ether

” The modern concept of the vacuum of space, confirmed every day by experiment,
is a relativistic ether. But we do not call it this because it is taboo.” –RB Laughlin
By Bruce E. Camber  20 January 2020

Introduction.  Prior to December 2011 the scientific-and-scholarly communities had not seen a base-2 chart of the universe from the Planck base units to the current time and size of the universe. It has 202 notations. Notation-199 holds the first billion years.  Notation-188 holds the first million years. The first second comes within Notation-143.  The first 64 notations are below the thresholds of measuring devices so basic logic, the formulas that define the Planck base units, all the necessary dimensionless constants, and the actual Planck-derived numerical units at each of the first 64 notations, should guide us.

Finite, Physical, Quantitative. Here are foundations to redefine the aether (or æther or ether). This æther is finite and physical yet beyond the reach of any possible measuring device. Quantum indeterminacy makes measuring impossible, yet here are the foundations of all measurements. Here are the foundations of all things physical. Here things are quantiative, yet we get the feelings of the qualitative.

We will consider the history of Aether theories, the luminiferous aether, the timeline of luminiferous aether, and the Lorentz ether theory (LET) .

We will find as many facts as possible, then within the structure of the first 64 notations, begin to make some guesses and postulations about the steps from simplicity to complexity. Because these numbers all originate from with the perfections of continuity, symmetry, and harmony, our most encompassing definition of the infinite, the first notations ostensibly define the finite-infinite transformation.

The formulas that define the Planck base units. First, we have the four formulas for the Planck base units. Don’t let those formulas confuse you. We’ll be taking them apart step by step. First, get visually familiar with each.

  • Planck Length: PlanckLength

  • Planck Time: PlanckTime

  • Planck Mass: PlanckMass

  • Planck Charge: PlanckCharge

  • Planck’s constant:  6.62607015×10−34 J⋅s
  • Reduced Planck constant (H- bar): ħ = h/2π

These Planck base units are defined by dimensionless constants. Although every constant will eventually be studied until each becomes a friend, we start with the two constants knowingly shared by us all, π (pi) and c (the speed of light). The most simple dimensionless constant to shape space is pi, first, as a circle, then as spheres, then as sphere stacking, and then as the simple geometries generated from cubic-close packing of equal spheres.

A Primary Assumption.  There are finite and infinite systems that : (1).can best be described as continuity (order), symmetry (relations), and harmony (dynamics).

What else can we assume? What about the Poincaré sphere? Could the polarization happen in the first doubling? When and how might the Fourier transform apply, even at these infinitesimal frequencies? What about Maxwell’s equations?

The first doubling could readily involve even more of the greats of mathematics, everyone from Leibniz (calculus, relational space and time), Euler (exponentiation), Gauss (beams, kernels,  and Spherical Gauss-Laguerre Basis Function and Gauss‐Legendre algorithm), Langlands (automorphics),  Laplace (operator on the 2‐sphere), Lorentz transformations, Witten (string/M-theory), Dijkgraaf (topological string partition function)… (yes, to be review, researched, and continued)

Also, every hypothetical elementary particle (like the axion, for example, postulated within the 1977 Peccei–Quinn theory to address the CP problem in QCD) has a place within these 64+ notations. We’ll eventually be wrestling with all of them.

So we start with the sphere. We ask, “Is the simplest sphere the first manifestation of the physical?” Can the universe be populated by what we call the plancksphere?  Is it the current expansion of the universe, the forward thrust of the earliest notations?”

My quick answer is, “Yes. Yes, and Yes.”

Within this model the æther is this forever changing domain, always expanding, the most-infinitesimal spheres from the first moment of time, plus a never-ending generation of planckspheres always filling the current notation.

Quite roughly, there are eight billion more years within this current notation. It is still “filling up” causing a sense of the directionality of time. And, “Yes” that conclusion flies in the face of many, possibly most — and probably all — current theories held by the many within our scholarly and scientific communities.

That steady stream of planckspheres blasting into the universe is a penultimate mystery. Mathematical logic is the only way we can visit there. My simple guess today is that there is a progression from circle to sphere,  an infinitesimal circle with the qualities defined Planck base units and those currently understood qualities of spheres such as the Fourier transform.

In a figure of speech, “It’s a long way to Tipperary!”   Much more to come….


Various notes (resources and references  below).

A Few References and Resrouces:

Odds & ends:

Robert B. Laughlin, Nobel Laureate in Physics, endowed chair in physics, Stanford University, had this to say about ether in contemporary theoretical physics:

It is ironic that Einstein’s most creative work, the general theory of relativity, should boil down to conceptualizing space as a medium when his original premise [in special relativity] was that no such medium existed. The word ‘ether’ has extremely negative connotations in theoretical physics because of its past association with opposition to relativity. This is unfortunate because, stripped of these connotations, it rather nicely captures the way most physicists actually think about the vacuum. Relativity actually says nothing about the existence or nonexistence of matter pervading the universe, only that any such matter must have relativistic symmetry.  It turns out that such matter exists. About the time relativity was becoming accepted, studies of radioactivity began showing that the empty vacuum of space had spectroscopic structure similar to that of ordinary quantum solids and fluids. Subsequent studies with large particle accelerators have now led us to understand that space is more like a piece of window glass than ideal Newtonian emptiness. It is filled with ‘stuff’ that is normally transparent but can be made visible by hitting it sufficiently hard to knock out a part. The modern concept of the vacuum of space, confirmed every day by experiment, is a relativistic ether. But we do not call it this because it is taboo.[6]

Isaac Newton suggests the existence of an aether in the Third Book of Opticks (1st ed. 1704; 2nd ed. 1718): “Doth not this aethereal medium in passing out of water, glass, crystal, and other compact and dense bodies in empty spaces, grow denser and denser by degrees, and by that means refract the rays of light not in a point, but by bending them gradually in curve lines? …Is not this medium much rarer within the dense bodies of the Sun, stars, planets and comets, than in the empty celestial space between them? And in passing from them to great distances, doth it not grow denser and denser perpetually, and thereby cause the gravity of those great bodies towards one another, and of their parts towards the bodies; every body endeavouring to go from the denser parts of the medium towards the rarer?”[9]

In the 19th century, luminiferous aether (or ether), meaning light-bearing aether, was a theorized medium for the propagation of light (electromagnetic radiation). However, a series of increasingly complex experiments had been carried out in the late 1800s like the Michelson-Morley experiment in an attempt to detect the motion of Earth through the aether, and had failed to do so. A range of proposed aether-dragging theories could explain the null result but these were more complex, and tended to use arbitrary-looking coefficients and physical assumptions. Joseph Larmor discussed the aether in terms of a moving magnetic field caused by the acceleration of electrons.

James Clerk Maxwell said of the aether, “In several parts of this treatise an attempt has been made to explain electromagnetic phenomena by means of mechanical action transmitted from one body to another by means of a medium occupying the space between them. The undulatory theory of light also assumes the existence of a medium. We have now to show that the properties of the electromagnetic medium are identical with those of the luminiferous medium.”[10]

Hendrik Lorentz and George Francis FitzGerald offered within the framework of Lorentz ether theory a more elegant solution to how the motion of an absolute aether could be undetectable (length contraction), but if their equations were correct, Albert Einstein‘s 1905 special theory of relativity could generate the same mathematics without referring to an aether at all. This led most physicists to conclude that this early modern notion of a luminiferous aether was not a useful concept. Einstein however stated that this consideration was too radical and too anticipatory and that his theory of relativity still needed the presence of a medium with certain properties.

Gundelfinger quartic and quartic surface and projective space and infinity

Octet truss:


4:46 PM · Jan 22, 2020: We’re certainly not given much time on this earth. We’ve got to come to terms with that fact and when we know our time is close, settle down, make the rounds to say good-bye, “I love you” and “Thank you” and engage those last days with peace and joy.

Robert B. Laughlin, Nobel Laureate in Physics, Stanford University, A Different Universe: Reinventing Physics from the Bottom Down. NY, NY: Basic Books. (2005).pp. 120–121. ISBN 978-0-465-03828-2.


Our cosmological model is young but comprehensive. With a natural inflation, it cleans up key problems and opens new domains to explore.


From Planck Time to today, the Now.

by Bruce Camber   A working draft, very-much in process. Initiated: December 21, 2019

ABSTRACT.  Max Planck’s 1899 calculations for Planck Time and Planck Length were so precise that simple mathematics gave him the most accurate definition of the speed of light with the smallest possible statistical error at the earliest possible date. Planck Length divided by Planck Time is equal to 299,792,422.79 meters/second if Planck Length is 1.616255×10-35 meters and Planck Time is 5.391247×10-44 seconds. The laboratory measurement in a vacuum would not be settled until 1975 and 1983: 299,792,458. Planck’s equations are a mathematical accounting for the first moment in time. To come up to the current time for the entire universe, we apply base-2 exponential notation. Within just 202 doublings we are at the age of the universe today, the Now. Between the 143rd and 144th notation, at just one second, the length is the distance light travels in a second. This chart is based on light. That 202nd notation is 10.9816 billion years. The 201st doubling is 5.4908 billion years, and the aggregate total of notations 1- to-200 is also 5.4908 billion years. If the universe is generally-accepted to be 13.81 billion years, about 2.8284 billion years of the 202nd notation has unfolded. Mathematically validated, the raw data of this chart needs deep analysis. Scholars from around the world are being invited to help us to evaluate the sanctity of its logic.

proton-smallWe celebrate 2020 with thanks to CERN artist and scientist, Daniel Dominguez’s imaginative rendering of quarks and gluons within the proton. Perhaps we’ll have better vision or maybe a new vision about the planckscale. Just maybe we can know how the planckscale relates to the rest of the universe. We have a few ideas which we gladly share here.

Introduction. Our base-2 chart from the Planck scale to the current age and size of the universe defines the longest, most-meaningful, continuity equations in the shortest number of steps or doublings. There is an order throughout our universe and here we find a basis for homogeneity. A planckscale, primordial version of Poincaré’s sphere is assumed to be the most-simple expression of Planck Length and Planck Time. Sphere stacking is introduced and the geometries of tetrahedrons and octahedrons emerge. There are now several layers of relations that are best described by symmetry functions. Here we find a basis for isotropy.

The most basic functions of the universe begin to emerge: the Lorentz transformations, Poincaré spheres and period-doubling bifurcations. There is a convergence of formulas in what has been improperly described as a singularity. The dynamics of these interacting symmetries are best described by harmonic analyses, harmonic series and Fourier transform. There is a time and place for every dimensionless constant.

The first ten notations build on each other and are the foundations for all that follow.

Using base-2 exponentiation. There are just 202 notations that encapsulate the universe from the first moment of time, Planck Time, to this moment in time, the Now, approximately 13.81 billion years later.  Of the 10.9816 billion years of the 202nd notation, about 2.82 billion years has become active. The 201st notation is 5.490 billion years and all the parameters which define it are active. Every notation is always active.

Continuities. The most-simple, most-used dimensionless constant, π (pi), is never-ending, never-repeating, always the same, a simple ratio and a number that has been carried out trillions of spaces.  It is a dimensionless constant like no other.  Circles and spheres take on a shape, size, dimensionality, and even a mass and energy.  Part of π (pi) does not. It remains dimensionless.  That quality of pi, continuity-order, is assumed to be what defines the infinite and infinity.

Symmetries. At the same time, a hypostatic topology emerges. With real numbers, all extensions of the Planck base units by doublings, the symmetries of circles, spheres, and sphere stacking emerge. From sphere stacking, tetrahedrons and octahedrons emerge. With just these two Platonic solids, the other geometries unfold.  It is all hypostatic simply because it cannot be measured directly. It is all happening below our thresholds for measurement.

Harmonies. With continuity and symmetry as the core of this cosmology and ontology, the dream of Pythagoras, Copernicus, Kepler, Newton, Einstein, Wilczek and so many others — the harmony of the universe — is realized within a hypostatic domain between the planckscale and the CERN -scale of measurements.

Finite-Infinite. That quality of π (pi), symmetry-relations, is assumed to be what defines the infinite and infinity. This finite-infinite relation is a primary real. Continuity-symmetry-harmony are manifest within the finite, within the infinite, and within the bridge that connects the two.

Everything, everywhere, for all time. Because everything builds on each other, the first notation may well be “behaving” the same as it did within the very initial instant. The progression through to the 196th notation, has a cumulative total of 300 million years. These notations provide data that is not available in any other manner within science today.

Work is underway to interpret this base-2 or “quiet expansion” data in light of current work within cosmology including big bang cosmology and the beginnings of large structure formation.  This model gives us everything we have now, but cleans up basic problems and opens new avenues for exploration…


• Big bang  singularity dumps the qualitative for narcissism, solipsism, nihilism, and dystopianism.
Think of things in terms of the entire universe! You can!
    Everything, everywhere for all time is encapsulated in 202 base-2 notations.
Just 436,117,076,900,000,000 seconds (13.8 billion years) There’s more
This pretty little universe, described by two words, homogeneity and isotropy

Planck Length to the atom is like the atom to the Solar System.

What if…  Yes, let’s explore the question, “Is it a real model? Why now?

Ways to open up the entire universe

First, there is a finite-infinite passageway.
Second, get to know light! Grasp a new space-time.
Third, see the two other faces of light: matter-energy (and you).
Fourth, there is no dark matter and dark energy; it’s a continuum.

Basic concepts and simple facts

Speed of light confirmed mathematically. So now, what is the nature of light?
• Our base-2 exponential model (goes to a little history). Mathematically-
integrated universe in just 202 notations (goes to the actual numbers).
Absolutes go back in the box.
Sphere stacking, tetrahedrons, and octahedrons come out of the box.

Go out on the edge. Warning, still rough…

Planckspheres, spheres, a natural inflation, and a doubling,
Cubic close packing and period doubling bifurcation
Fourier transform at the Planck scale
Poincare, Gauss, Hilbert, Euler

We all have a lot to learn about the infinitesimal!

References & Resources:

Initiated in private on December 6, 2019
As an unreferenced posting: Late Sunday, December 9, 2019
A first-draft homepage: Tuesday, December 11, 2019
Most active editing: December 10-22, 2019