Just Eight Key Ideas


Concepts For A Simple Model Of The Universe

1. 202 base-2 notations (doublings, containers, clusters, domains, layers, sets and/or steps)
2. This Quiet Expansion Is A Natural Inflation. Here’s A Sampling.
3. The speed of light has variable speeds and there is a wide variegation of light.
4. Facelift for Space-time
5. Rebirth of the Finite-Infinite Studies
6. Incommensurables, transcendentals and irrational numbers get real.
7. Continuity-symmetry-harmony creates order, relations and dynamics!
8. A Place For Everything, Everywhere, Throughout All Time

Fabric-Space-Time1. These 202 base-2 notations easily encapsulate the universe.

Multiply by 2. Start with the smallest units of measurement, the Planck Time and Planck Length, multiply each by 2, then the result by 2, and continue right up to the largest possible measurements, the Age of the Universe and the Observable Universe. Though both are still constantly expanding, your result will quickly be a totally, mathematically-integrated model of the universe that includes everything, everywhere, for all time in just 202 notations.

Just one more time: Most scientists and scholars now believe that Planck Length and Planck Time are the very smallest possible measurements. If you double them, then double them over and over again, in just 202 doublings, you will have reached the size of our Universe and you will have gone from the beginning of time to this present moment in time.  It sounds impossible, but the math is the math.

•  Rather special. Though simple logic, simple math, it’s still just a model. As a challenge, take this model as a given, to explore in what ways it could be an outline for the initial structures of the universe. Also, if simple logic and math do not work from the Planck Scale to the CERN scale to the Large Scale Universe, then logic and math should not work anywhere. Continuity is continuity. Order is order.

Eventually we added Planck Charge and Planck Mass to our doublings and continued to study how the numbers and the inherent logic worked together.  We have been exploring this model since December 2011 and have discovered that it seems quite sustainable! No scientists or scholars have seriously challenged it. Of course, it is idiosyncratic and it does raise rather unusual questions. Notwithstanding, the universe within just 202 doublings (clusters, containers, domains, layers, notations, sets and/or steps), so far, actually does seem to cohere 1 (Footnotes/endnotes at bottom).

•  Sweet. By the 31st notation, Planck Mass has increased to 103 pounds. That may be an ideal weight for some, yet in this notation, space and time are still so small that science has always considered these values meaningless.  We do not. Though an entirely novel concept, we take it as a given that there was a time when the universe was just 103 pounds. Though necessarily defined by point free vertices, it already has more active ratios than our imaginations can grasp (1.23794×1027). It exceeded one million vertices by the eighth notation, one billion at the 11th, and one trillion at the 15th, and a quintillion vertices at the 17th notation.  To follow that quiet expansion, look at line 9 in the chart below.  More 1-67a

2. This Quiet Expansion is a Natural Inflation. Here’s a sampling.

•  In this model each notation is more than just a simple doubling. The next notation doubles in size and doubles in content and each notation it would appear is currently defining our universe. As we live with the model, grasping the first notation as the start of the universe and the current notation as our current moment in time, all the notations come alive as a process that gradually builds upon itself, every notation changing and growing because of the others. To think about this very different sense of space-time, I did a sampling of six of the notations somewhat evenly spaced across the 202 notations. Of course, these Planck scale doublings are all predictive numbers. Within these six notations that were sampled, each demonstrates a natural inflation that is a quiet expansion.  A sampling  Prime numbers...2

•  Being natural is good. If one were to wonder about the basis for bifurcation theory, chemical bonding, cellular development, and isotropy-and-homogeneity, we may have found a new opening to explore. Then, if one were ask about the source of this thrust, again I think we may have found another good opening to explore.

The alternative is not pretty, elegant, or logical. Any one of the many theories that use the big bang logic of infinitely dense and infinitely hot are incomplete, unnatural, and riddled with questions. Worse yet, the big bang philosophy and psychology have trashed our culture with a de facto nihilism and dystopia.  It is bad physics. It is bad mathematics. And, it’s bad psychology.

•  Sweet.  (1) The quiet expansion continues its doublings until finally at the 67th the folks at the CERN labs (Geneva) are able to pick up something and our core sciences start to evolve.  (2) Truly sweet is that this natural inflation approximates all the cosmological epochs that were defined by big bang theorists; however, in this model there is no bang.     (3) The 67 notations provides more than enough “room” for every possible system of mathematics to be applied. Remember the 64 doublings within the Checkerboard story!  More  And more...

3. Light has variable speeds and many faces.

Planck’s Equation

•  Planck Length divided by Planck Time equals the speed of light. And, as you might anticipate, with every doubling, that approximation of the speed of light naturally continues. See line 10 within the horizontally-scrolled chart. More

•  That is very special. Light, Time and Length are being redefined by each other and by Charge and Mass.3

Please take note that between the 143rd and 144th notation, the universe is finally one second old and Planck Length has grown to the size defined by the distance that light travels within that second.

Seasonal look at an equation that resets the universe.

It is simple math and simple logic yet it re-engages us to expand and begin to understand a rather extended spectrum of light, of course, well beyond the visible spectrum and well beyond where it has been defined to date.

This work has given new meaning to an old song (1977), “You light up my life.” The very final refrain goes something like this: “It can’t be wrong when it feels so right…” More

•  Sweet.  Here is the first time light, defined by another very simple mathematical formula, can be found within a simple mathematical progression. The formula from Planck predates Einstein’s!  More

4. Space-time is long overdue for a facelift.

•  The Gnarls and Knots of Space-and-Time. Definitively cast in stone in 1686 when Newton released his blockbuster, Principia, time-and-space took on an absolute frame of reference and this view became our commonsense worldview.  It became a realism (and naturalism) whereby our external reality is assumed to exist as it is given and experienced. Both concepts — space-and-time — have been problematic and now some really smart scholars want to throw them both out; we believe it would be better to redefine them in light of our model with its 202 notations. Here, the universe begins within the Planck Scale. Our challenge is to figure out how the universe evolved within those initial notations, and then how it continued to the 202nd notation and to this very second within our expanding universe.

•  It is a very special and rather non-intuitive challenge. There are so many people today who still affirm Newton. They unwittingly do so when one of them looks up into the clear, night skies, observing the panoply of stars, and says, “It goes on forever.” Those wrestling with this new model would say, “We are looking out at our expanding universe. It is still expanding within the 202nd doublings and has been for about 13.82 billion years.”  What happens is that no notation goes away and recedes into a past. Instead, each notation participates in the current definition of the universe and space-and-time, as the fabric of the universe, absorbs every imprint for all time-and-space.

•  Sweet. This new sense of space-time gives it a currency and vibrancy.  The universe has its own clock and as of 2013, the universe was 436 quadrillion, 117 trillion, 076 billion, 640 million  seconds (436,117,076,640,000,000) old.

Let’s turn on the clock.  There are about 31,556,952 seconds per year so at some time we will be adding at least four years of seconds, somewhere over 126 million additional seconds, to bring the total to the current time. A hundred years is just 3.1556 billion seconds. Extend that to 1000 years (31.556 billion seconds), a million years (31.556 trillion seconds), and a billion years (31.556 quadrillon seconds).  Our earth and solar system are approximately 4.5 to 4.6 billion years old.  The estimates for the Milky Way galaxy are about 13.7 billion years (yes, among the original galaxies formed).  Yet, none of it is old; it is all current. All the cosmological epochs are now understood to be ongoing processes of the universe.   More…

5. Let us try to rebirth finite-infinite studies.

 Science and mathematics have made great use the concept of the infinite; philosophers and theologians have as well.  Yet, over the years we have seen how these disciplines tend not to work very well together, especially those elitists among them, particularly among those religionists and theologians who too frequently don’t work well with anybody except their own.

It is no wonder that a few very bold scholars would like to throw out the concept of infinity. It has an awkward personality. It is too big for most of those yellow academic school buses.

•  We suggest a shakedown. First, let us shake out all the concepts that have anything to do with faith statements — these are too often argumentative, emotional, and sometimes even hostile. Within the quiet expansion model, there are several conceptual assumptions that begin to bind the finite and infinite in special-and-heretofore unexplored ways. These concepts include: continuity-order, symmetry-relations, and harmony-dynamics.  Though an unusual combinations of words, some element of each can be readily understood.

 We can do this. The word combinations may be a little gray, but we’ve all used each word at one time or another. Taken together and in this sequence, these words become very rare concepts because each reflects a face of the finite (our beingness, whether it is science, mathematics, psychology, philosophy or religion) and also a face of the infinite.  More   Foundations…   Ethics-and-values…

6. Incommensurables, transcendentals and irrational numbers

 Numbers. Perfect, unshakeable continuity occurs within numbers that are called incommensurables, transcendentals and the irrational. These numbers are never-ending and never-repeating. Our old friend, yes, a very dear friend of every circle and sphere, is Pi.  Its very close cousin is e, for Euler’s number.  Euler’s equation also defines the universe.  It is the heart of our exponential universe. Also, meet Pi once again.

•  Through our Looking Glass. Most looking glasses are mirrors. This looking glass can also be a magnifying glass. Isn’t it unusual that a number can seemingly go on forever without repeating itself and with no special pattern or repetition?  Doesn’t that define a certain kind of uniqueness?  …diversity?  Within this class of numbers there is a group that is also known as dimensionless constants.  There are quite a lot of them. Yet, in all the discussions about the many scholarly blogs, papers, articles and books, there does not appear to be any discussion about how these numbers must define some aspect of infinity.  Isn’t that an oversight? If we take it as a given within our new definition of infinity, this very special kind of continuity and order can also be understood as uniqueness and diversity. That’s rather special.

Now of all the scholars I know, perhaps Clifford Pickover or John Barrow have started a process of looking at infinity in this simpler way; we’ll be digging further into their scholarship. More…

But, for now, let’s see if we can take it one step further. Let us take as a given that these rather unique numbers (1) originate within the infinite, (2) tell us about the nature of the infinite, and (3) define the transformation between the finite and infinite.

•  That’s different and very sweet.  So, we now have an ordered-chaos and a discontinuous-continuity defining our finite universe.  The next step is to look at the symmetry/relations and harmony/dynamics.

7. Continuity-symmetry-harmony create order, relations and dynamics!

Sphere to tetrahedron-octahedron couplet
Creative Commons Share-Alike 2.5 Generic license

•   Pi leads the way. Leading the way out of the infinite is pi. Perfect equal circles, then spheres, then multiple spheres, then stacks of equal and doubled spheres. We are probably within the first ten notations or doublings.  And then, a radius discovers a radius and the first line is created. Then the radii begin discovering radii and lines and dimensionless, point free, vertices are created.  Vertices discover vertices and the first triangles are created. Triangles discover triangles and the first tetrahedron is created. Lines, vertices and triangles discover each other and then the first octahedron is created within the tetrahedron. And, quickly, hexagonal plates are discovered. Triangular plates are discovered.  And diversity begins to move into high gear in a perfectly simple, yet increasingly complex manner. Symmetries abound. Perfection is in the air. But the imperfect and asymmetrical are not far behind.  More…

•   Dynamics.  Euler’s number is somehow holding pi’s hand! The two are working those continuity and symmetry relations and are making a universe one step at a time. More…

•   That is a very special challenge. All shapes and forms and sizes and every manner of symmetry relations are being created. The pulse of energy known as Planck Charge creates a thrust, an energy, a direction, a motion, and soon our little universe is pulsing with activity.  More…

First, called the Planck Scale, it transmogrifies into what some call a Grand Unification.  Here is the first doubling. But for all intents and purposes, it is within a self-discovery mode. Yet, rather quickly (understatement), that self-discovery process organizes and begins what we call the Inflationary Processes right up to and including the 67th notation.  Now declared to be the Electroweak Processes, Planck Charge is huge. Planck Mass is huge. Planck Length has become manageable. Planck Time is still too small to be considered anything other than a placeholder.  More…

•   Sweet.  The rest of the cosmological epochs seem to follow rather naturally in line with the continued expansions (doublings and notations) of these Planck base units. Even Planck Temperature, which was placed at the top of the scale because everything begins simply, is now substantial enough to warm things up and to begin looking like our little universe.  More    And more…

8. A Place For Everything, Everywhere, Throughout All Time

•   The first 60 or so notations will never be observed by our machines. These domains are much too small to be picked up by our measuring devices. If there can be no validation by our machines, then we must rely on logic and mathematics. This special domain has a very special word to describe it; it’s call hypostatic. We actually create a real reality when shared by another. It is a really real reality when shared by many.  As more people adopt that structure, we press meaning and purpose into it.  Then we ask, “Is this helpful? Does it seem to describe a real reality for people within our time?”

•   Yes, this is very special challenge for all of us. Because everything, everywhere, for all time must be logically included on this grid — nothing is exempt —  we will hypostatize a place for philosophy, psychology, mathematics, quantum fluctuations, the cosmological constant, religions, the mind, consciousness, sleep, and qualities. More

•   So we have a bit of work to do.  And, we recognize that this work just started in December 2011 and it is still quite naive and young.  The initial goal was to give our high school students a very special chart to integrate Science, Technology, Engineering and Mathematics (also known as a STEM tool).  Today this project is known as the Big Board -little universe.

That was cool. But now, the real work begins. We know we are onto something and have opened a door that is worth exploring. We’ve begun to study some of the older problems within science with such names as inertial frame of reference, point processes, scalar fields and scalar field theory, and path integral formulation. Many of those subjects are listed on line 11 of our horizontally-scrolled chart.

We struggle with it all most everyday so your advice, insight, and help are most welcomed! Thank you.

Footnotes & Endnotes

1 Time: It is good to be reminded that the first second of the universe is between notation 143 and 144. The estimated 500 million years necessary for galaxy formation takes us up in between notations 197 and 198.

2 Number groups:  We will be studying every prime number as a possibility for a new mathematical system. The obvious, primes from 2 to 199 (3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 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, 193, 197), prioritize themselves! The challenge will be to logically build the sequence of systems. It is anticipated that the work of Grothendieck and Langlands (IAS) will contain a few keys. More

3  Quantum fluctuations and imperfections.  We believe the geometry of imperfection begins with the pentastar and the pentastar gap.  Though possible within notations 1-to-40, simple logic suggests that these quantum geometries do not manifest until Systems (first column, eight row) begin organizing.  More

PS.  There may be more footnotes inserted in the forthcoming days. If you are observing changes in the post, it is because this post is edited every day. It will change, and undoubtedly grow a little longer.

Special Thanks

Friends from New Orleans came for a visit. Their two images have been in my mind’s eye as I worked on this page. I have been constantly asking myself, “Is there a way to write this up so they understand it?”   -BEC


We welcome visitors from Belgium, Brazil, Canada, Cambodia, China,  France, Germany, Honduras, the Netherlands, New Zealand, Portugal, Serbia, Singapore, South Africa and Turkey. Yes, all visitors to this site, we welcome you all.

Thumbs Up! Or, “Please Comment.”

We can only grow by helping each other with constructive criticism or reinforcement. A quick little “Thumbs Up” is helpful. If it is “Thumbs down” please send a comment. If you want to just use numbers, -1 for Do not understand, -2 for Confusing and -3 for Disagree.

Our most simple assumptions and insights for an alternative theory:

•  The finite-infinite relation can always be understood more profoundly.
•  The key numbers, functions and geometries recognized by our most-respected scholars all begin their work to define the finite-infinite relations within the very first moment that defines space, time and light.
•  Pi and Euler’s number are the first steps within this understanding and define the very beingness of the universe.
•  The Planck numbers are also part of these first steps and all subsequent steps. More…
•  Multiplying the Planck units by 2 over and over and over again creates a grid or a matrix within which to begin to understand numbers, functions and geometries. The principles of continuity and symmetry become penultimate. Though space-and-time-light are infinitesimal and thought to be meaningless, this is the beginning of a natural inflation and thrust of the universe.

The Finite and the Infinite

Most recent update: Sunday, 8 October 2017   Prior homepage  Next homepage   On Time

Stephen Hawking and Max Tegmark
Stephen Hawking – Max Tegmark

The relation between the Finite and the Infinite is rather hotly debated, mostly between people who equate the infinite with God and those who have no place for any kind of god. Yet, the infinite also creates conundrums within science. In 2014, MIT physicist, Max Tegmark advocated, “I’m betting that we also need to let go of it.” He wants to retire what he and others call an incorrect assumption – infinity. His friend, Stephen Hawking, opened that door with his 1983 No Boundary Proposal that holds that the universe has no beginning or end.1

“Not so fast y’all” as it is said along the banks of the Mississippi River in New Orleans. With these two thought leaders, Hawking and Tegmark, it should be acknowledged that both are being speculative and that their concepts are brilliantly incomplete.

In the Big Board-little universe model using base-2 notation from the Planck Time to the Age of the universe, the entire physical universe is contained within just over 200 notations that are highly-integrated and totally-predictive. Notations are also known as clusters, doublings, groups, sets or steps. Within the first second of the universe, there is more than enough “natural inflation” from the Planck Charge to get “things” going. As a result of studying and working with this model since December 2011, there are many-many facets to explore, however one of the most important is that this model logically suggests that time is derivative and that the finite and the infinite are perhaps best understood in terms of continuity, symmetry and harmony.

Continuity. Though an unusual way to define infinity, even with quantum indeterminacy, continuity throughout the universe is the bedrock of science, logic, and rational thought. Numbers clarify this continuity. If we were to carry our measurements out a billion places, the universe and its systems around us would still replicate day after day with utmost precision.

Within this model, continuity is more fundamental than time; it begets time. It is the initial condition of order.  More…


The complexity of a single molecule


Symmetry. The second face of the infinite is symmetry. Though so much of life is asymmetrical, the deepest examination of any physical thing begins to reveal deeper symmetries. Numbered relations define those symmetries, relations are created, and the universe appears to be tiled and tessellated deeply within every notation throughout the model. Here symmetry is more fundamental than space; it begets space.  More…

Harmony. Speculating, it is hypostatized that two symmetries begin interacting within a notation and then across notations, dynamics are created, and though not quite perfect, the interaction of the symmetries perfects the moment for the observer or for the notations involved. Therefore, we have moments of perfection within our experiences of the universe. More…

Our studies. At this point in our studies, there is not much more we can say about how the infinite defines the model and what the model says about the very nature of the infinite. These three insights, although reflective of the model, in part come out of a study of a moment of perfection in 1972,3 then from studies of the book, Finite and Infinite: A Philosophical Essay (Austin Farrer, Oxford, Dacre Press, Westminster, 1943), and from an application to a business model.4    More…

These three qualities became the bedrock for our model of the universe and for discussions about the shared nature of the finite and infinite.

What difference does it make? First, it is a clear contrast to the nihilism of big bang cosmology. Building in strength and popularity over the past 30 years, that nihilism has had a lot to do the fraying of our little world. So much is out of control and spinning apart. Money is not the issue. What we believe and how we believe is. Hope is. Charity is. Integrity is.

What is 5000 to 13.8 billion years?

The finite and infinite relation has been the focus of humanity for as long as we have been recording our ever-so-short history. In light of 13.8+ billion years, five thousand years of records is, of course, quite short. We’ve just begun to make sense of it all.

Today in history. The finite is usually associated with physical, limited things. The infinite is often capitalized and associated with godly things, the eternal and everlasting. To our knowledge, Max Tegmark is the first theoretical physicist who has suggested that the concept of the infinite be abandoned. His rationale is that it gets in the way. He cannot make it work for the science he wants to create. Within these many articles, we hope to convince him, Hawking, Guth and so many others to re-engage our simple definition of the infinite. It does not require a religion or religious beliefs. Notwithstanding, it also does not necessarily fly in the face of those who believe in a much more robustly-defined Infinite.

We can all begin to tolerate each other.

This is our simple introduction to a very large topic and we will return to this page often to expand its range and its depth.

1 A general introduction: See the prior homepage. Also, this Wikipedia article about the 1983 Hartle-Hawking State (wave function) provides even more information.
2 This construct was introduced within these pages at the end of the year, December 2015.
3 This construct goes back to work in 1972 at both Synectics Education Systems in Cambridge, Massachusetts and the Harvard Philomorphs with Arthur Loeb and Buckminster Fuller.
4 The original construct was used as the foundation of a business model for a weekly television series, Small Business School, that Bruce Camber and Hattie Bryant started in 1994. It aired on PBS-TV stations throughout the USA and on the Voice of America around the world for over 50 seasons (2012).


Related homepages:

Key references: Communications

Reserved for an analysis of the work of Salvatore Torquato…

Salvatore Torquato, Lewis Bernard Professor of Natural Sciences
Princeton Institute for the Science and Technology of Materials, Princeton, New Jersey

ArXiv (149): The structure factor of primes (2018), Hyperuniform States of Matter (2018)
Google Scholar
Group Homepage
YouTube: Hyperuniformity in many-particle systems and its generalizations

First email: Mar 10, 2014, 8:54 PM
1. Thank you: http://www.pnas.org/content/108/27/11009.abstract?sid=a37de813-198f-4f81-9641-ad2025190fd7
2. Beautiful: http://chemlabs.princeton.edu/torquato/research/maximally-dense-packings/
3. Hypostatic Jammed Packings (2006): http://pi.math.cornell.edu/~connelly/Hypostatic.pdf

Dear Prof. Dr. Salvatore Torquato:

Thank you, thank you, thank you for your work (referenced just above).

Back in 2001 I spent a very pleasant day with John Conway but he did accuse me of being hung up on the relation between the tetrahedron and octahedron. For more I’ll copy in part of the story below. Though I am late to discover your July 5, 2011 paper, I was so glad to discover it today. It adds fuel to the fire and opened the door to your work.

I am so glad to meet you through your writings. I have already inserted references to your work in two articles (referenced below).

After spending a bit more time with your writing, may I call you?

Thank you.



PS. I’ve been working with clear plastic models — made the molds and made thousands of octahedrons and tetrahedrons — to delve into the issues of fragmentation and wholeness. David Bohm’s book by that title, has a prominent place in my library.

Our references about which you might be interested:

Here is what I said about John Conway:

“An earlier history began with the study of perfected states in space time.
Sometime in the Spring of 2001, at Princeton with geometer, John Conway, the discussion focused on the work of David Bohm, once a physicist from Birkbeck College, University of London. “What is a point? What is a line? What is a plane vis-a-vis the triangle? What is a tetrahedron?” Bohm’s book, Fragmentation & Wholeness, raised key questions about the nature of structure and thought. It occurred to me that I did not know what was perfectly and most simply enclosed by the tetrahedron. What were its most simple number of internal parts? Of course, John Conway, was amused by my simplicity. We talked about the four tetrahedrons and the octahedron in the center.

“I said, ‘We all should know these things as easily as we know 2 times 2. The kids should be playing with tetrahedrons and octahedrons, not just blocks.’

“What is most simply and perfectly enclosed within the octahedron?” There are six octahedrons in each corner and the eight tetrahedrons within each face. Known by many, it was not in our geometry textbook. Professor Conway asked, “Now, why are you so hung up on the octahedron?” Of course, I was at the beginning of this discovery process, talking to a person who had studied and developed conceptual richness throughout his lifetime. I was taking baby steps, and was still surprised and delighted to find so much within both objects. Also, at that time I had asked thousands of professionals — teachers, including geometry teachers, architects, biologists, and chemists — and no one knew the answer that John Conway so easily articulated. It was not long thereafter that we began discovering communities of people in virtually every academic discipline who easily knew that answer and were shaping new discussions about facets of geometry we never imagined existed.

“Of course, I blamed myself for getting hung up on the two most simple structures… “You’re just too simple and easily get hung up on simple things.”