When might the first icosahedron of twenty tetrahedrons evolve?

Sphere to tetrahedron-actahedron couplet
by I. J. Onathunder

Start at the beginning. Applying a simple logic, two spheres define the second notation, four spheres define the third notation, eight spheres define the fourth, sixteen define the fifth, and 32 define the sixth. Observe these simple graphics generated by cubic-close packing of equal spheres.

The first notation, an expression of the finite-infinite relation, has a deep-seated order (continuity), relations (symmetry), and dynamics (harmony) and layers of tetrahedrons and octahedrons are readily being created. Within this computer-generated graphical image of sphere stacking, eighteen spheres are used. The question to explore is at what point in the profusion of spheres might 20 tetrahedrons bond within a common centerpoint? We will be asking the experts within neutron studies within solid state physics within condensed matter physics for help with an answer.

The first icosahedron of twenty tetrahedrons might evolve prior to the 64th notation, before the first particles have emerged, but probably not. By the 64th notation the simple math suggests over 1018 spheres and thus layers of tetrahedrons and octahedrons. Yet, densities may not yet allow much freedom of movement.

In the first second within Notation 143. There still may not be enough freedom of movement.

The emergent universe appears to be smooth.

The Primary Conjecture: The first notation is an expression of the finite-infinite relation defined by continuity-symmetry-harmony and is expressed as infinitesimal sphere that is defined by the Planck scale (or its symbolic equivalent) and that one sphere per Planck unit of time is generated…

Can you imagine the first icosahedron?

Icosahedron of twenty tetrahedrons sharing a common center point

Five octahedrons with a gap create the base. An icosahedron made of twenty (20) tetrahedrons sits on top.

An icosahedron made of twenty tetrahedrons sits on the five-octahedra with a gap. This image will be further developed. All the tape will be removed from both tetrahedrons and octahedrons. The convergence of three edges of the icosahedron over the five-octahedral gap will be more clearly displayed.

Then, the “when, where and how” will begin to be explored. -BEC

Let us introduce the twenty-tetrahedral icosahedron in place of the five-tetrahedral cluster. The complexity and potential functionality of this cluster increases exponentially.

In May 2022 we began making a study of the cluster of fifteen sharing a common centerpoint (with the hexagonals within each octahedron) as if it would make an interesting gate within circuitry of the infinitesimal.

Gaps of an Icosahedron of twenty tetrahedrons sharing a center point

These twenty tetrahedrons share a common center point and represent the greatest number of tetrahedrons sharing a centerpoint within quantum-or-imperfect geometries.

We also call this gap geometry, imperfect geometry, quantum geometry or squishy geometry.

Fifteen objects share a center point using a five-octahedral cluster, five-tetrahedral cluster with another five-tetrahedral cluster on top. Within the octahedron there are four smaller octahedrons and eight tetrahedrons that perfectly share a common centerpoint.

Icosahedron: 20 tetrahedrons with gaps, all around a common centerpoint
Major gap cover-up

Worksheet: An icosahedron with two groups of five tetrahedrons (in dots) facing each other and ten tetrahedrons circling the middle (solid orange line).

There is another way of looking at the twenty tetrahedrons with three groups of five tetrahedrons that leaves a small group of four and a single tetrahedron that are easily accounted.

Rotational symmetry. In any direction there is one face to face with a single edge exposed, then a convergence of five with a rotational symmetry as shown below.

More to come…


Quantum geometries: A polyhedron with 20 faces, one of the five Platonic solids, there are many historical studies. The focus is on the imperfect, the geometric gaps of the twenty  tetrahedrons, extended here (five images below) as three groups of five, a group of four and single. In another configuration there are two groups of five and a ring of ten separating them. Notwithstanding, there is rotational symmetry.

Taken all together students called it, “Squishy geometry.” And, it is!

In high school we speculated with the students. “This may be a part of the beginnings of quantum geometries within quantum physics.” Five tetrahedrons bound by red plastic tape, the magnetic balls remind us that no space is actually “empty”and that singularities exist on paper only.


In this second image or five tetrahedrons (same object), bound by blue tape, we can see part of the five tetrahedrons bound by red (above) through the clear plastic. There are in fact 20 tetrahedrons all bound together. They all share a common centerpoint. 


With this third image, 15 of the 20 tetrahedrons have been displayed. In this image you can see through the clear plastic tetrahedrons — there are reflections of the five bound by red at the top and reflections of the blue on the bottom right. That leaves five more tetrahedron to find.


We find a cluster of of four tetrahedrons, here bound by clear tape and you can see wisps of the red, blue and green groups on the edges. This four-and-one configuration (the one being just below) with the three groups of five tetrahedrons harbors a special type of asymmetry and discontinuity. We’ll eventually be studying it closely.


Yes, and finally, very clearly, all by itself, is a solitary tetrahedron. The red, blue and green tape of the abutting tetrahedral sets are readily discerned. Also, one can follow ten tetrahedrons around the icosahedron and two groups of five on either side.

The icosahedral cluster may be considered a transformational nexus; and in future articles, that concept will be explored further.

There are many references to the icosahedron within this site.  Use the “Find” function (CMD F) and enter “icosahedron” to go to each reference more quickly.

There is also a study of the tetrahedron, the octahedron, harmony and their combinations.

Adopted and inherited biases…


from Aristotle, Newton & Hawking


Introduction. Three thought leaders of our common history were also leaders in their own day. They held their ground when challenged. Throughout the years, their work became sacrosanct. Yet, among all the concepts they each introduced, I believe that they held onto a key conceptual mistake that still blocks us even today.

As a people we long for heroes and leaders; and, these three were ready to accommodate. Headstrong geniuses of their time, once they got into the limelight, they did not easily share it. They were not about encouraging others to discover their own gifts. They were more about imparting their genius to their adoring publics.

But, all three were fundamentally wrong about a foundational concept. They’ve thrown off generations of scholars. They’ve held us all back; and now it’s time to correct their mistakes, forgive them, and get on a path to breakthrough to new levels of insight.


Aristotle (384–322 BC, Athens)1 was wrong about a most-basic geometric fact. Obviously he could not have had perfect tetrahedrons within his toolbox. If he did, he would have known that one cannot perfectly tile and tessellate the universe with just tetrahedrons. He thought it was possible.

There are obvious gaps. Using the very tightest configuration of just five tetrahedrons sharing a simple edge, a most fundamentally important geometric gap is created. Simple logic tells us that it is a relatively early gap in physicality. Aristotle never saw this 7.35+ degree gap; and to his dying day, he promulgated an error as a truth.

There’s always so much more to learn.

Aristotle had such stature that this error was repeated by scholars for over 1800 years. Even today, not many people know about the gap. That should change. Our children should see it and begin to appreciate it profoundly.

What is it? I believe this simple gap is the beginning of the geometry of quantum fluctuations. That’s huge, but there is so much more. First, we know this — it is necessarily created by just five tetrahedrons which also outline a face of the dodecahedron, and define the primary faces of the icosahedron and the Pentakis dodecahedron. Aristotle’s mentor, Plato, defined the five basic solids — the tetrahedron, hexahedron (aka cube), octahedron, dodecahedron and icosahedron.

That gap has everything to do with basic structure. It just may also have everything to do with creativity, individuality, consciousness…

Enter Jeffrey C. Lagarias & Chuanming Zong. In 2012 they wrote a most-definitive article about the gap. These two mathematicians provide the background and an introduction to the people in the 1400s who observed and noted Aristotle’s mistake. Then, drawing from the 1926 research of D. J. Struik, they cite Johannes Müller von Königsberg (aka Regiomontanus, 1436–1476) as the first to recognize the error. The first to document it was by Paulus van Middelburg (1445–1534), a professor of astrology in Padua. Even though Aristotle’s error had finally been observed and analyzed, people focused on the fact that it was an 1800-year mistake. They also focused on the concepts within cubic-close packing of tetrahedrons and spheres. Over the years Kepler, Minkowski, Hilbert, and Hales — just to name a few — contributed insights to analyze technical aspects regarding packing densities.

In 2015 Lagarias and Zong were recognized for their work. That is all very interesting, however, we are still looking for the scholars who have asked and answered the question, “What is the net-net effect of that natural gap on our understanding of ourselves and our universe?” 

Such questions should never be ignored, so let’s speculate a little.

Projections about the meaning of it all. We turn to our outline of the universe — the 202 base-2 notations from the Planck scale to this current time. Yet, we specially consider the uniqueness of the first 67 infinitesimal notations which are mostly below the thresholds of measurement. Notation-67 is the threshold of wave-particle duality. Notation-76 is the current limit of a measurement of a unit of time. If Notation-0 defines a finite-infinite cusp, these 67 notations are a new field for exploration.

Infinite-finite-and-Hilbert. Within this model there is a thrust created, a finite-infinite bridge best characterized by functions of continuity, symmetry, and harmony, three most-basic facets of the sphere. Quite contrary to the work and logic of David Hilbert, it would seem that the face of the infinite is within the finite. First, there are simple perfections where everything fits with no gaps. Although as simple as possible, granted, it becomes complex rather quickly.

What works survives. Every possible geometric combination that works provides form, function, structure, and then substance, relations, and networks of relations. What works best, survives. The universe, the penultimate opportunist, is creating something big that requires solid foundations. Perhaps somewhere around Notation-50, our universe begins to experiment with those five tetrahedrons with its built-in gap. Out of an abundance of shapes and configurations, the five tetrahedral structure is surrounded by perfectly manifesting forms and structures. Within a moment, that gap comes alive. Perhaps as early as Notation-50, the gap becomes a structural system, and then becomes a systemic fluctuation. Just a guess, the first expression of these systemic fluctuations just might be considered a primitive consciousness. By Notation-67, when it can be measured and “observed”, it will be defined as a quantum fluctuation.

Notation-50 and systemic fluctuations. Here we could postulate the beginning of identity, individuality, creativity, undecidability and unpredictability — a transmogrification from the perfect to the imperfect and indeterminant. Here may well be the birth of life as we experience it firsthand. The perfect is still there, yet it is now beginning to be masked with color, charge, flavors, sounds, and an assortment of other patinas.

A simple mistake by a legendary man has been hiding one of the most substantial mysteries of our time. It is time to absorb it and begin to absorb the new realities that it has been hiding.

We’ll always have a lot to learn.


Issac Newton (1642 – 1726, Cambridge, England)2 was wrong about a philosophical orientation adopted by the world as its commonsense perception of space and time yet that opinion does not integrate with tested formulas by Max Planck and Albert Einstein. Newton was bold to proclaim that space and time are absolute, the very fabric of our essential reality.

It certainly feels true. When you look up into the clear night sky, it goes on forever. Doesn’t it? And, the answer is, “No, it only does as far as the current expansion.”

In 1687 Isaac Newton finished his landmark, three-volume book, Philosophiae Naturalis Principia Mathematica. Better known as just the Principia, it helped to firm up the foundations for what we now know as the scientific method. Though glimpsed by science and mathematics (1) dating back to Babylonian astronomy (c. 1830 BCE) and the Egyptian medical schools (c. 1600 BCE), and then (2) seriously enhanced by Aristotle and the logic within his treatise, The Organon, and then (3) energized with the work of Copernicus, Johannes Kepler, and Galileo, one can say with some confidence that science as we know it today consistently grew out of Newton’s Principia.

Yet, within this landmark writing came his most important contribution to the disinformation of the world’s culture: absolute space and time. These absolutes will not begin to recede as a footnote in our intellectual history until a better orientation is adopted by most people. That is a problem because, to date, alternatives have been non-intuitive. Leibniz came close in 1716 within his indirect dialogues with Newton through Samuel Clarke — Leibniz said space and time are relational, derivative and finite. So we ask, “If not the container for all that is, what is?”

For many that question is about one’s belief in God.

We try not to engage in “God Talk” on this website. One’s personal belief systems are largely a factor of family systems. Our attention is focused on universal systems and their constants.

Planck Time

Enter Max Ernst Ludwig Planck (1858 – 1947, Kiel, Berlin)3 In 1899 Max Planck developed the equations to render base unit numbers of length, time, and mass that were defined by universal physical constants. Although largely ignored throughout his lifetime, this may well be his most important work. One of the earliest analyses of that work began in 2001 by Frank Wilczek. It was published in Physics Today in three parts. Titled, Climbing Mt. Planck I, II, and IIIa key calculation was overlooked.

Too simple for most, Planck had tied Planck Length and Planck Time together: Planck Time is equal to Planck Length divided by the speed of light. Of course, his little formula for Planck Time, can readily be re-written; the speed of light is equal to Planck Length divided by Planck Time.

That formula works! It worked in 1899. Using Planck’s numbers, the value is 299,792,422 meters per second. Without fanfare or celebrations, Max Planck had defined the speed of light using the mathematics of his equations a full 73 years before the National Institutes for Standards and Technology (NIST) accepted a slightly closer estimate, 299,792,456.2 meters per second defined in 1972 by K.M. Evenson and his group within the National Bureau of Standards in Boulder, Colorado.

Planck’s numbers are realthey work with real laboratory measurements. To date, the academy virtually ignores them. Applying base-2 creates a natural progression of those numbers and the first 67 notations have only been marginally explored. Here is an even more logical way to study the earliest universe where space and time are clearly derivative. The question is, “…derivative of what?  …light?”

In 1905, Max Planck advised a young Albert Einstein as he began to tie mass and energy together. Yet, at no time has the academy started with Einstein’s sacred formulation, e=mc2, the very first step of the Planck scale.

So, what comes first? If we look into the finite-infinite relation from the point of view of the sphere, well-removed from particles and waves, we begin to see what just might be facets of light that could well be more fundamental than space and time. Finally, Newton’s absolutes did not seem quite so absolute.

A simple door with simple logic opens a new path to explore. Another “hiding in plain sight” story, we have been looking at this door since 1899. We seem to have a difficult time opening that door and walking down that extraordinary path on the other side. It follows continuity, symmetry and harmony and puts our unique time within this dimensionality into a whole new light.

Newton was the second Lucasian Professor and Hawking was the 17th.


Stephen William Hawking (1942-2018, Oxford, Cambridge)4 captured the world’s imagination. He was a superstar. Everybody knew his name. In 1973 a young Stephen Hawking and George F. R. Ellis co-authored The Large Scale Structure of Space-Time at the University of Cambridge in England. Yes, although looking at the large-scale structure, Hawking and Ellis made a mistake at the get-go:

“The subject of this book is the structure of space-time on lengthscales from 10-13 cm, the radius of an elementary particle, up to 1028 cm, the radius of the universe.”

They missed the real foundations. They missed the core structures. They missed all the really cool stuff from 10-13 cm down to and including the Planck Length at 10-33 cm. Within our base-2 outline of the universe, that range is from Notations 73-to-75 down to Notation-0.

By 1980 the big bang theory was clearly on the ascendancy. By 1988 with the publication of his book, A Brief History Of Time, especially with its rapid rise to multi-millions of books sold, Hawking was also on the ascendancy as the primary spokesperson for big bang cosmology.

In 2016, he rhetorically asked his basic question:

Where did the universe come from?” He immediately continues:
The answer, as most people can tell you, is the big bang. Everything in existence, expanding exponentially in every direction, from an infinitely small, infinitely hot, infinitely dense point, creating a cosmos filled with energy and matter. But what does that really mean and where did it all begin?” -from the PBS-TV series, “Genius” aired in May 2016. (My emphases)

He was wrong. But, until he died on March 14, 2018, the big bang seemed to be the best answer even though it was fraught with problems and open questions.

When it comes to theories and mathematics, simple is better than complex.

For most of Hawking’s life, Max Planck’s numbers were considered by the leading scholars of this world to be a curiosity. Dirac had his very-large numbers. Planck had his very-small numbers. Dirac’s were too big to matter and Planck’s were too small to be significant. Again, it wasn’t until 2001 that Wilczek introduced the world to the meaning and value of Planck’s numbers. Slowly, the academy began to test those waters; yet, it was much too late for Hawking to enter. His 1973 co-author, G.F.R. Ellis, on the other hand, was open to explore the failures and deeper problems within the concept of an unfathomably hot beginning.

An introduction to Turok: https://81018.com/lefschetz

It doesn’t work, and it’s a conundrum. The big bang theory has been backed up with the humor of a twelve-year television series (2007-to-2019) that is now in endless re-runs. Yet, ever so much more daunting is the mythopoetics of Hawking’s life.

Diffusing the big bang will not be easy, but diffuse it we must.

The logic and simplicity of the 202 notations. Going from the Planck units to the current expansion appears to have the most simple mathematics of any construct of the universe offered to date. It has a natural inflation. It starts superconductingly cold and naturally heats up and becomes superconductingly hot just in time to absorb the epochs of big bang cosmology.

The problem with our so-called Quiet Expansion is that its concept of space-and-time is non-intuitive. There is some light on this path. Others have been talking about the Now as well.

The Now. In this model, there is no past and no future, only the Now for the entire universe. All of the 202 time periods are still active and everything, everywhere for all time is related to everything, everywhere for all time. It is all constantly encoding and re-encoding the universe. 

Every thought-word-and-deed affects the look and feel of the universe.

And, because there are multiple paths throughout the 202 active notations (categories, clusters, containers, domains, doublings, groups, jumps, layers, periods, sets, steps…), in this model, it is not only a small world after all, it is also a small and intimate universe.


Currently there is no way around the naïveté within this three-point charge against three of the foremost scholars of our entire history. I expect each point will be hammered, yet it is only by such hammering can it all be shaped into real possibilities. Thanks. – BEC


Three sections follow: (1) Footnotes & Endnotes, (2) References, Reflections & Resources, and (3)_Miscellaneous Notes including emails and tweets.

Footnotes & Endnotes

Navigation: Please click only on the section number to return go back. This page is a working document and editing continues on the Footnotes & Endnotes, as well as the References & Resources and the Miscellaneous Notes and it will all actively continue to be edited and updated for the next several months. Thank you. – BEC

[1] Aristotle (384–322 BC, Athens).

“Be a free thinker and don’t accept everything you hear as truth.
Be critical and evaluate what you believe in.”

1a. Jeffrey C. Lagarias & Chuanming Zong, Mysteries in Packing Regular Tetrahedra (PDF), American Mathematical Society (AMS), December 2012. In 2015 Lagarias and Zong were awarded the 2015 AMS Levi L. Conant Prize at the Joint Mathematics Meetings.  And, there is more…

1bDirk J. Struik. If you do not have time to read the “Mysteries in Packing…”, you should know that Lagarias and Zong credit Struik, a Dutch-American and MIT mathematics professor, for reopening those discussions in the 1400s that broke the 1800+ year impasse. The primary reference: D. J. Struik, Het Probleem ‘De impletione loci’ (Dutch) (English: Translation by M Senechal), Nieuw Archief voor Wiskunde, Series 2, 15 (1926), no. 3, 121–137

1cThe geometric gap of 7.3561031+ degrees was first encountered within our work in July 2013 in the process of prioritizing numbers to answer the question, “What are the key numbers to create this universe?” This geometric gap was judged to be the fourth most important after (1) pi, (2) Kepler’s Conjecture, and (3) 0-and-1.

1dContinuity, symmetry and harmony. Pi has to come into being in some manner. The spheres of the Kepler conjecture have to originate somehow.

To answer the question, “Why is there something rather than nothing?” we assume that something is more fundamental than space and time, matter and energy. Here is our attempt to define the concepts that create a finite-infinite relation that gives rise to homogeneity-isotropy. Within this emerging model, the infinite is the qualitative; the finite is the quantitative. Instead of retiring the concept of infinity (Tegmark, 2012), in this model, it is the centerfold but with very specific definitions.

1eFrom systemic to quantum fluctuations. Because so many concepts are being introduced, these comments will become future postings and homepages within this site. In March 2020, I wrote up an overview of some of these concepts (PDF) to get some feedback from the FQXi peopleHere are the key claims.

Recognizing how idiosyncratic it is to associate the geometric gap with fluctuations, it is certainly a greater stretch to differentiate types of fluctuations. Yet, that study has begun and eventually we’ll be showing a video of what we call “squishy geometries” and the rather unusual motions created by tetrahedrons.

Systemic fluctuations. Those two words in May 2020 only had 569 references within a Google search. These fluctuations, admittedly a guess, originate with the five tetrahedral structure fully engulfed by perfected systems. With the emergence of particle physics between Notation-64 to Notation-67, they become part of the look-and-feel that define all physical systems. It is a stretch, for sure, however, we will continue to pursue it further. 

So, yes, there will always be more. Go to our References & Resources section.

[2] Issac Newton (1642 – 1726, Cambridge, England).

No great discovery was ever made without a bold guess.”

2a. Tested formulas by Max Planck and Albert Einstein defrock Newton’s absolute time and space pageantry. Follow all four values from Notation 1 to Notation 202, our universe is naturally exponential. Space and time are derivative and finite.
2bVery fabric of our essential reality. A new aether (ether) emerges. Described often in these pages, the subject has initially addressed (2017) as the fabric of the universe.
2c. Of course, Aristotle’s influence on the way we think runs deep. Newton credits Aristotle’s work, The Organon, within his Principia. Yet, we should ask which comes first, basic logic, or the continuity-symmetry-harmony, the heart of the structure of the universe.
2dLeibniz challenged Newton in 1715 and 1716. In his lifetime, Leibniz advocated for a relational view of the universe and it perhaps is the best foundation for an alternative approach.

[3] Max Ernst Ludwig Planck (1858 – 1947, Kiel, Berlin)

All matter originates and exists only by virtue of a force which brings the particle of an atom to vibration and holds this most minute solar system of the atom together.
We must assume behind this force the existence of a conscious and intelligent mind.
This mind is the matrix of all matter.

3a. A key calculation has been overlooked. The math is simple. The results dramatic.
3bKen M. Evenson et al (1972), “Speed of Light from Direct Frequency and Wavelength Measurements of the Methane-Stabilized Laser“, Physical Review Letters29 (19): 1346–49. Bibcode: 1972 PhRvL..29.1346E, doi:10.1103/PhysRevLett.29.1346). Quantum Electronics Division, National Bureau of Standards, Boulder, Colorado 80302)
3cMore fundamental than space and time. Simple logic redefines the finite-infinite relation and the nature of light, and the nature of space and time.
3dContinuitysymmetry and harmony are three facets of both the finite and infinite. It is the baseline of this model of the universe. Here it seems all are universals that are, in the same instant, dimensionless, dimensionful, and dimensional.

[4] Stephen Hawking (1942-2018, Oxford, Cambridge)

The greatest enemy of knowledge is not ignorance, it is the illusion of knowledge.”

4aThe Large Scale Structure of Space-Time is a classic; however, it is not easy reading because it is laden with formulas (PDF).
4bA Brief History Of Time: Hawking reviews the history, but does not rule out absolute time. He is after all, the 17th Lucasian professor, following in the footsteps of Sir Isaac Newton, the second Lucasian professor. Although scholars from around the world were calling for a re-evaluation of its growing status, there was increasingly less room for discussion. Its devotees accepted it as fact, not theory, and Hawking championed that big bang right to his dying day. More…

4cGeorge F. R. Ellis, Hawking’s 1973 co-author, had begun to recognize the deeper problems with conclusions from those early years of explorations at University of Cambridge. In 2012 in his collaboration with Roy Maartens and Malcolm MacCallum, (Relativistic Cosmology [PDF], Cambridge University Press), the big bang model is clearly under close scrutiny and every assumption is on the table: inflation, singularities, the most-recent measurements of the Hubble spacecraft of the cosmic background radiation, fine-tuning…. he is open to explore virtually every issue; yet with close to 50 years of analysis, he can not be absurd to himself. In February 2020, with colleague, A.A. Coley, the topic is, Theoretical Cosmology (PDF), virtually all the same issues are reviewed.

I took the most comfort from an article in 2017, Physics on the Edge, where he names all the key players and essentially shows how confoundingly muddled it all is. Our 2017 academy of scholars do not have clear answers.

References, reflections & resources

Please note: The primary links into this section are from the Endnotes & Footnotes from where there is the word, More… Links back to that More… are  from the [Numbers].

[1] Aristotle. Our work began in a high school geometry class. We knew it required a tetrahedron and octahedron to tile and tessellate the universe.  In 2011 we walked with Zeno deeper and deeper inside each object and learned a lot. You should know that our shapes were all perfectly made according to Plato’s specifications. 

Yes, in 1998 we manufactured our own tetrahedrons and octahedrons!

It was hard to believe that neither Aristotle nor 1800 years of scholars (at least 90 generations) did not have their own perfect tetrahedrons in their toolbox. We wondered if geometry had slid from importance or was Aristotle beyond criticism?

The icosahedrons and Pentakis dodecahedrons use the five-tetrahedral configuration; they have gaps, and we dubbed it “squishy” or quantum geometry. By 2011, now with many years of visceral experience, and within our new chart of the infinitesimal scales, we thought that such a pervasive gap had to be significant.

We began thinking of quantum fluctuations and then systemic fluctuations. 

[1a]  Jeffrey C. Lagarias Chuanming Zong. In 2011, just about the time we were beginning to explore the infinitesimal universe, Lagarias and Zong had begun writing the best little introduction that I’ve found to this geometric gap. It is a relatively short article (PDF) for the American Mathematical Society (AMS), December 2012. We appreciate that the AMS has made it readily available.

Also see: Lagarias, Clay Fellow Senior Talk, “Packing Space with Regular Tetrahedra“ and Chuanming ZongCan You Pave the Plane Nicely with Identical Tiles, 2018

The people of China and the USA — not the governments, but the people — must find common ground. One would think that mathematics and the sciences would give us an abundance of places with which to build ties that are greater than politics. Articles like this encourage us. As important as their personal relation is, these two are also building relations between the University of Michigan and Tianjin Center for Applied Mathematics (TCAM). Zong was initially at Peking National University. I believe that the work of Lagarias and Zong actually changes the quality of life for everyone and for everything within this universe.

So, it is incumbent on all of us to begin to understand this gap (See #19), the first in the universe. It just might teach us all to become more patient with each other, especially with our superficial historic differences.

[1b] Personal. For me, Aristotle was always secondary to Plato. I am still in my earliest stages of plowing beyond a perfunctory understanding of Aristotle. Just from this encounter, I am fascinated with him. It appears for some of the Aristotelian crowd, his understanding of the tetrahedron is a bit of an embarrassment. Substantial studies do not touch it. My interest was so piqued, I started simple — with the Wikipedia overview — and then went on to other authors who came up in specific searches. I empathize with the less well-known authors, people like Ric Machuga, a professor at a junior college (Butte College, Oroville, CA).  His book, Life, the Universe, and Everything: An Aristotelian Philosophy for a Scientific Age, was published in 2011.


A summary of the problems associated with sphere packing is the December 2015 article, Mathematical Optimization for Packing Problems, by Fernando Màrio de Oliveira Filho and Frank Vallentin

URL: http://wiki.siam.org/siag-op/images/siag-op/c/c4/ViewsAndNews-23-2.pdf


[1c] Zeno, Aristotle, Planck and Infinite Divisibility. I remember well the puzzled look of our students, when in 2011 I said, “Zeno has bumped into a limit called the Planck Length. We cannot divide-by-2 forever.” Planck gave the universe boundaries and logical conditions for those boundaries. Not entirely satisfied with that perception, a Russian by the name of Sergey Fedosin has taken another step: Infinite Hierarchical Nesting of MatterI always immediately look to see what their starting points are. Within that document, they do not discuss the Planck base units and so they miss the possibility of defining the domain from the Planck units to the particle physics in a highly textured manner.

More References, reflections & resources:
[2] Issac Newton
 did not have the advantage of Leonhard Euler‘s exponentiation. He created the concept after Newton had died. Of all possible manners of notations, base-2, is the most simple, yet it still lacks proper respect. The chessboard stories are told but under-appreciated.

The seemingly simple progression,  264 yields a large number,  18,446,744,073,709,551,616. If you were turn turn it into pennies, you could easily retire the world’s debt, all nations and all people… I tried to explain it to my sister-in-law

2202 is another story. Notationally, 6.42775218×1060 is the raw number. Once there is an amount associated with it, like infinitesimal spheres, it begins to open the imagination. 

Newton did not have Planck’s base units. He was arrogantly unsure of himself. This Lucasian Professor (#2) was confident, however, that space and time were absolute. It is profoundly part of our commonsense worldview. Unfortunately, however, I believe it is wrong. Indeed, the approach of Gottfried Leibniz will render a much richer view of our universe.

There are two living Lucasian professors, Michael Green (#18), and Michael Cates (#19). I’ll keep trying to develop a working relation with them, yet prior history tells me that I am not sophisticated enough for these people.  https://81018.com/uni/ https://81018.com/lucasian/

There are related postings within the website that need follow-up. Among them is: https://81018.com/ math/

More References, reflections & resources: 

[3] Max Ernst Ludwig Planck Within the complex of Max Planck institutes around the world, there have been several attempts to open discussions. In this section, we will look at some of those exchanges more closely.

Max Planck Innovation:  https://www.max-planck-innovation.com/max-planck-innovation/max-planck-society.html

More References, reflections & resources:
[4] Stephen Hawking
: There are many articles about the problems within big bang cosmology. A few of these papers will be selected and analyzed in light of the 202 notations. Our first emails to Stephen Hawking referenced our very early attempts to interpret our chart of just Planck Length and Planck Time doublings.

Mauricio Mondragon ;& Luis Lopeza, Space and time as containers, Space divisibility, and unrepeatability of events, 2007, 2012

Miscellaneous Notes

DIRK J. SRUIK: Aristoteles weiß, daß der Raum durch kongruente Würfel voll ausgefüllt werden kann, behauptet aber weiter, daß das auch mit Tetraedern gelinge. Verf. verfolgt diese falsche Behauptung, die auch für die Lehre vom Vakuum eine gewisse Bedeutung hat, durch die Geschichte der Mathematik. Der erste, der die Unrichtigkeit des Satzes nachweist, ist Regiomontanus. Aber Ramus und Snellius folgen wieder dem Aristoteles. Erst mit dem 16. Jahrhundert tritt völlige Klarheit ein (Benedetti, Blancani, Broscius). (V 3.)


 Ellis et al, Page 310, Chapter 12 – Structure formation and gravitational lensing

“The basic idea is that quantum fluctuations of the inflaton field behave like one-dimensional quantum harmonic oscillators (with time-varying mass). Zero-point fluctuations of a quantum harmonic oscillator induce a non-zero variance of the oscillator amplitude, ⟨xˆ2⟩ = /2ω. Similarly, the inflaton zero-point fluctuations generate a non- zero variance ⟨δφ2⟩. The fluctuation modes (with co-moving wave number k) are stretched from their original small scale (assumed to be above the Planck scale) by the rapid accelerating expansion of the universe, until their wavelength ak−1 exceeds the Hubble scale (when they are assumed to become classical fluctuations).”

• W. Patrick Hooper et al., , 2018, Platonic solids and high genus covers of lattice surfaces
We will be proposing one that has been reached by a few well-grounded scientist/scholars. Unfortunately, it still feels a bit more like science fiction, so we’ll come back to it within our final overview and conclusions.

• Alvaro G. LópezOn an electrodynamic origin of quantum fluctuations, ArXiv, 2020
Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Física, Universidad Rey Juan Carlos, Tulipán s/n, 28933 Móstoles, Madrid, Spain (Dated: January 31, 2020)

Here, space and time appear quite derivative. It appears that he was not ready to challenge absolute space and time. He defines a relation that begins with the Planck units. When we apply base-2, we are looking at the natural unfolding. The two formulas mass-energy equivalence and length-time equivalence are bound by light and appear to be bound to each other. 

Wikipediagauge theory is a type of field theory in which the Lagrangian does not change (is invariant) under local transformations from certain Lie groups. … If the symmetry group is non-commutative, then the gauge theory is referred to as non-abelian gauge theory, the usual example being the Yang–Mills theory.

• Chuanming Zong, Can You Pave the Plane Nicely with Identical Tiles, 2018

An Email to a therapist

You might want to ground your people within a very simple model of the universe. All our current models are too big for most of us and those models tend to cause great anxieties. We call our work, The Big Board – little universe. It mathematically connects everything, everywhere for all time within a functional schema that actually seems to be a much better model for cosmology, physics, and mathematics than the models we currently entertain. The universe can be parsed from the Planck base units (it’s our the start)to this current day within 202 base-2 notations (all simple doublings). Once people understand that we live in an exponential universe, this place we live and have our being becomes quite intimate, comforting and secure, plus we realize that we are an important part of the equation and what we do counts. We make a difference. For more, you might start with today’s homepage: http://81018.com

That homepage most-always has the most-recent work.

Please have a glance at the 202 notations, the chart: https://81018.com/chart/  Here is a short-cut: Review these claims: https://81018.com/checklist/; It is simple, simple, simple, so don’t let it appear otherwise.

And, yes, I am always open for questions!


June 9, 2020: A few sample tweets

Keeanga-Yamahtta Taylor, Princeton: @KeeangaYamahtta
You have intuited what science has failed to understand –
There is a profound integration of all things everywhere for all time.
https://81018.com/biased/ is my first analysis of Aristotle’s mistake (geometry), Newton’s mistake (space and time), and Hawking’s mistake (infinitely hot start): https://81018.com/biased/#Now

There will be much more to come. 

Please Note: I also sent a direct email. -BEC

Nature Magazine If you want to make a difference, teach us all something about the scientific foundations that we do not know, i.e. Aristotle’s geometry mistake, Newton’s space-time mistake, and Hawking’s lack of infinity: https://81018.com/biased/ It is all so tightly inter-related and we don’t see it.
Please note: Shall we re-submit this article to Nature? It was ignored.


Key Dates for Biased

This article was initiated on Wednesday, May 20, 2020.
Biased became a homepage or top-level post: Wednesday, June 3, 2020.
Last update: Friday, September 25, 2020
The Prior Homepage: https://81018.com/alternative/
The URL for this page: https://81018.com/biased/
A section on Aristotle and geometry: https://81018.com/biased/#Aristotle
A section on Newton and absolutes: https://81018.com/biased/#Newton
A section on Hawking and “infinitesimally hot” start: https://81018.com/biased/#Hawking
The tagline: We reach for the stars, but we’re conceptually blocked…

Back on December 19, 2011…

Left Yellow Arrow
Right Yellow Arrow

Pages: Agree | Gravity| Hope | Hypostasis Mistakes | PI (π) | Questions | Sphere | STEM | Up

Our Big Board-little universe chart began and
evolved as a highly-ordered Quiet Expansion.

by Bruce E. Camber, a first draft

Big Board - little universe was a Quiet Expansion of the universe, not from a big bang.

New Orleans: Five high school classes of students — mostly for geometry — were greeted with the chart on the left. It was huge, over seven feet high. Click on the image to see it! It is a mathematical map of the universe.

It didn’t take long for the students to get the knack of it. They were quickly going further and further inside new and unknown spaces. They quickly got smaller than a molecule, then a cell, and atom, and a particle.

Those students had been cajoled to go inside that tetrahedron and octahedron. In your mind’s eye you can do extraordinary things. Shrinking smaller and smaller is one of them. Dividing edges by 2 and connecting the new vertices is another.

Down through the DNA of biology and into the chemistries and its periodic table. Then very quickly we were into particles and waves and fluctuations. Our goal was to reach those natural units that Max Planck defined in 1899. We understood these were the the smallest possible units of space and time. We had come down 45 steps to the atom and then uncovered 67 more steps to those Planck units.

The first layer of the inside of a tetrahedron. There is so much to learn.
Four hexagonal plates are in every octahedron. Five octahedrons together sharing a common centerpoint creates a pivotal gap.

Encapsulated Tetrahedrons and octahedrons. All the way down smaller and smaller, each notation is active. It was difficult to discern the archetypal forms within a notation; yet, at Notation-0 it appears to be an infinitesimally-small, sphere.

We hypothesize that an infinitesimal sphere instantiates at Notation-0, the first space-time moment. The EOU within Notation-202 is the Edge of the Universe which is the current expansion of tredecillions of spheres per second.
From the Big Board - little universe chart from December 19, 2011
Eight Notations: Data from the larger cells, the egg within Notation-103 to the smaller at Notation-97.

Vibrant and dynamic, it appears to be the instantiation of the Planck base units and so much more.

From our magical observation deck, we could see those spheres stacking, with new functions with each doubling. From here we could readily look up the 112 steps back into the classroom. In just 90 additional magical doublings or steps or notations, we could see that we would be out to the edge of the universe watching the current expansion.

It happens so fast: Tredecillions of spheres per second. We’ve had to double check our numbers over and over again. In just over a second from Planck Time, we were out to Notation-143. In just over a year, Notation-169. And in just over 1000 years, Notation-179. A million years is just over Notation-189 and a billion years, just over 199. If we assume there is one infinitesimal sphere per unit of Planck Time and Planck Length, there are 539 tredecillion spheres per second. If we use Stoney’s numbers, there are 4605 spheres per second.

Here is our universe from the smallest to the largest in 202 steps or doublings or notations. It has its own special logic such that all notations are ongoing, interdependent, and forever.

What’s this all about? Yes, dynamic and logical, this model has numbers and geometries. It has the Planck units. It has a simple algebra; and it is constantly filling with infinitesimal spheres. It is the first time we could see the universe on one highly-ordered, fully-integrated chart. We searched online for experts to help us interpret our emergent model, but we couldn’t find such our chart in any textbook or anywhere on the web. So, we turned to the living scholars who would know. Many complimented our work and said something like, “Your chart uses base-2 exponential notation to parse the universe from the smallest to the largest possible measurements.”

Top to bottom: Five tetrahedrons over five octahedrons over five tetrahedrons

We learned that the work was unique. We also learned that the results did not jive with current cosmological theory. This model posits a very smooth, yet highly-integrated beginning of the universe. Here the geometries all fit together perfectly. These geometries tile-and-tessellate without gaps. Yet, we knew that a five-tetrahedral cluster made an object with a gap. Octahedral clusters do the same. Together they make a geometry that has not been discussed in any of the literature. We propose that it is a geometry of quantum fluctuations. It is squishy geometry.

Over our heads and inundated with new information. I consulted with old acquaintances who were scholars — John Conway, Phil Davis, Freeman Dyson, and Lisa Randall, Then we began learning through the work of new people like Frank Wilczek and Stephon Alexander. Work by Jeffery Lagarias and Chaunming Zong, Mysteries in Packing Regular Tetrahedra (PDF), was an indictment on the academic community. Aristotle’s mistake had been ignored and it continues to be ignored within academia. That’s a profound mistake and causes one to pause and ask, “What are some of our other profound mistakes?

Here’s my quick introduction to three:
1. Sphere-to-tetrahedron-octahedron dynamics: Cubic close packing of equal spheres: https://81018.com/ccp/ Scholars have focused on packing densities; very few have focused on the process by which tetrahedrons and octahedrons are created from sphere stacking.

2. Structures created by basic geometries. First, the octahedron within every tetrahedron is a key. The four interlocking hexagonal plates within every octahedron is another key. Those gaps created by clusters of five tetrahedral clusters and five octahedral clusters are also a key. The gaps created by clusters of twenty tetrahedrons (an icosahedron) are keys as well. It is all unfinished business, whereby we all, especially our scholars, should focus on the place and purpose of each gap. Are these gaps related to quantum fluctuations? That’s a major discussion.

3. The very nature of pi (π): I am no scholar but the mathematics of infinity seems to be a penultimate challenge. There are so few discussions of the place and importance of pi (π) and infinitesimal spheres. I believe that the open questions about the very nature of homogeneity and isotropy of the universe are keys and that Planck units or Stoney units (or the ISO’s equivalent units) will define an approximate rate of expansion so tredecillions of infinitesimal spheres per second fill the universe from the start through to the Now.

These three points are so idiosyncratic, it’ll take time to engage them, absorb them, then use them. It is, however, all very approachable with high school students. We even had our AP class of sixth grade savants get immersed in it; but once our graduates started circling back, we realized that it was too disruptive within the current curriculums. The only hope is within the special integrity of people like you. Might you have any advice for us? How do we proceed? Thank you.


PS. There were many new ideas and presuppositions that emerged along the way. Then, we organized them as a checklist to start and grow the universe. -BEC


Endnotes & Footnotes
Most of key points already have pages within this website; however, new footnotes may yet emerge.


References & Resources
Key references and resources will be added over time.

• All the webpages, week by week, going back to 2016.


A few of the emails to our many scholars.

• Martin Bridson, Oxford and Clay Institute, December 17, 2022
• Levent Alpoge, Harvard, Cambridge, Massachusetts, December 16, 2022
• Richard J. Fitzgerald, AIP and Univ. Texas-Austin, December 16, 2022
• Alan Guth, MIT, Cambridge, Massachusetts, December 13, 2022
• Gil Lonzarich, Cavendish Lab, Cambridge University, December 12, 2022
• Sankar Das Sarma, University of Maryland, December 11, 2022
• Ana Caraiani, Hausdorff Chair BonnImperial College London, Thurs, 8 Dec, 2022 8:43 AM
Orli Dahan, Tel-Hai College, Israel, 7 December 2022 @ 2:42 PM
Elizabeth Gibney, Nature magazine, Tues, Dec 6, 20227:57 PM


Often using Twitter. Criticized, now also using Parler. New IM will be added.


Participate…     You are always invited.


Keys to this page, December-19-2022

• This page became the homepage on Monday, December 19, 2022.
• The last update was 21 December 2022.
• This page was initiated on 18 December 2022.
• The URL for this file is https://81018.com/December-19-2022
• The first headline for this article: An Alternate Point of View Evolved
Also: Our Big Board-little universe chart began and evolved as a Quiet Expansion, not a Big Bang.
• First byline is: Eleven years ago, from December 19, 2011 to December 19, 2022…

On following the work of Jason Yust of Boston University

Jason Douglas Yust, Boston University

Books: Organized Time, OUP, 2018; Mathematics and Computation in Music, 2013
Google Scholar
Homepage(s): Researchgate
Publications: Degree Difference, 2020
Society for Mathematics and Computation in Music (SMCM)

Where Jason Yust is quoted within this website:

  1. Harmony of the Universe
  2. Justin Timberlake and continuity-symmetry-harmony
3 August 2022 at 3:21 AM

Dear Prof. Dr. Jason Yust:

Your work on the Discrete Fourier Transform within music theory came to my attention and pulled me in deeper and deeper. Nicely done. Inspiring. Also, “Congrats” on your Brown start. I wonder if you ever took a class with Phil Davis in applied mathematics. Phil was my go-to person on spheres!

Also, roots of mine go back to BU. First, with Robert Cohen and Marx Wartofsky in the physics department, and then Harry Oliver and J. Robert Nelson in the School of Theology.

I am developing a resource page of links to your work and it will be here when its done: https://81018.com/yust/ (this page)

My orientation has been called idiosyncratic, notwithstanding, here are very different orientations that you may appreciate:

  1. There are 202 base-2 notations from the first moment of time (assumed to be defined by Planck’s units of time and length).
  2. All notations necessarily appear to be active refining space and time, and the finite-infinite relation.
  3. If so, the first 64 notations are below all possibilities of measurement.
  4. I am in the process of introducing a “geometry of the gap.” It’s the place where quantum fluctuations are measurable. Here are a few more references: https://81018.com/icosahedron/ https://81018.com/icosahedron-2/#Close-up https://81018.com/15-2/ https://81018.com/gap/

Thanks again for all that you do.

Warm regards,



Quoted in following:

1. The Musical Arrow of Time – arXiv, Q Xu, 2022

Worldviews are too intransigent. A fully-integrated UniverseView is needed.

Concepts key to an Integrative Universe
by Bruce E. Camber A working draft, in process on this late August day

Intransigence becomes hostility. People from every nation and walk of life are not willing to see a broader, more-inclusive context to grasp the meaning-and-value of life. Limited worldviews clash. Temperatures flare and people fight. A better way is an integrative model of the entire universe within which to know ourselves, our world, and a bit more about our universe.

This outline of an integrative universe uses eight concepts:

#1 Infinite qualities — continuity, symmetry and harmony — shape every-and-all finite quantities, each a dynamic equation and the beginning of spacetime.

The finite-infinite relation is best understood by expanding our understanding of pi. We first learn the first simple equation, the circumference-to-diameter ratio. That’s a start, then it builds from there. As noted in the summary of pi, “…It is our oldest, most-used, mathematical constant, and the ultimate basis for all equations, especially those describing a fundamental principle of our universe.”

There are three facets of pi, however, that are not finite or quantitative so we assume (hypothesize and/or hypostatize) these facets define the infinite and the qualitative.

Continuity is our first facet of infinity. It is the very nature of order. Within the finite it looks like a string of numbers and feels like time. Pi qualifies; it’s an equation that has never-ending results that are always the same and always changing.

Symmetry is the second facet of infinity. It looks like geometries and is the very nature of a relation. Within the finite it feels like space. Pi qualifies; it’s a symmetry that generates symmetries. It’s an equation that generates equations.

Harmony is the third facet of infinity. It is the very nature of dynamics; and within the finite, it is always cyclical (periodicity) and experienced as space-time moments. Pi’s numbers, geometries, and equations (Fourier transform and others) are here within an eternal dance and there’s a domain of perfection which may be experienced as a moment of perfection.

Ultimately, pi is the face of both sides of every equation, one is qualitative (infinite) and the other is quantitative (finite). Let those natural, dynamic relations be natural. Let all your relations breathe and come alive. Be open and engage the harmony of the universe.

#2 There are foundations within mathematics to integrate our Universe.

Are Planck Length and Planck Time real? Among the scholars in this area, they seem to say, “Real enough.”
Are their numbers real? Infinitesimally small, we are prone to say that these are symbolically real and “close enough.”
What manifests first? Pi drives the finite-infinite equations; we consider a size/time invariant sphere that is defined by those Planck base units (numbers). We’ve also used Stoney’s numbers. Both sets of numbers are symbolic placeholders until there is a new consensus among scholars, NIST and ISO.

#3 There are 202 base-2 notations that encapsulate the universe.

Our Story. In December 2011, our high school geometry classes unwittingly made a first pass at defining the universe using base-2 notation starting at the Planck base units. We believed the scholars that the Planck units for length and time were the smallest possible units of space and time. Our conjecture was that they would also be the very first units of spacetime.

We decided to explore. To get down that small, we followed a 4D path inspired by Zeno. We divided the edges of a tetrahedron by 2, and then its internal octahedron and four smaller tetrahedrons, and continued dividing by 2. There were just 112 steps within to the Planck scale. We thought it might feel a bit like Alice’s fall into that rabbit hole as in Lewis Carroll’s Wonderland. But, our walk was highly-ordered, systematic, yet most magical. As we went down deeper and deeper within, it was not at all confusing. Even our shrinking in size each step seemed quite natural.

We rebounded back in the classroom by multiplying the Planck Length by 2. Later we would add Planck Time. And then even later, Planck Mass and Planck Charge. It was extraordinary going from that smallest unit out to the edge of the universe to watch the current expansion, all in just 202 base-2 notations. Yes, from the smallest to the largest sizes and from the first moment of time to 13.81+ billion years later, we had encapsulated everything-everywhere-for-all-time.

#4 It’s an answer to big bang cosmology… but it’s just too simple.

Exponentiation. The entire universe, from the smallest possible measurement to the largest in 202 notations, stretches credulity yet it’s 100% mathematical and predictive. All notations are profoundly related and always dynamic. We were so new and naive about it all, we asked questions of the thought leaders of big bang cosmology, Hawking, Ellis, Guth, Steinhardt… and so many others. “What are we doing wrong?” Nobody was willing to guide us, so we placed the time line for the big bang and our big board, side-by -side. We found only a microsecond’s difference with Hawking cosmology. We were beginning to learn about the problems, so when a scholar labeled our model, idiosyncratic, we knew that his judgement was quick and somewhat flippant. There was — and still is — just too much here to consider.

It has taken ten years to begin to understand why change is difficult to engage. First, there is so, so much vested in the Hawking model. It has stood strong for many years. It began building in 1973 when Hawking and Ellis wrote a thesis, The Large Scale Structure of Space-Time. Every new citation, each new book and movie and video, making reference to Hawking’s big bang cosmology, created a muscular defense around Hawking and his model. Yet, that core belief system could not answer a growing number of questions. It had to choose to ignore others. Second, no new model came along. Still, eventually some of our best scholars broke rank and called more stridently for a new paradigm.

We were late to that party and we had no scholarly credentials, yet our emergent model had clear, simple, and compelling mathematics and logic. Nobody argued that.

#5 The first 64 notations are the foundations.

Those first 64 notations. Although impossibly small, here is a huge infinitesimal domain that is well below the current thresholds of measurement. Unwittingly, all 64 notations, albeit a most-speculative domain, had never been considered. The more we read and studied about the mathematics of Langlands programs, those earliest notations seemed like a natural home. It also seemed like string and M-theory could benefit. Then we began learning about SUSY, and studies like causal set theory (CST), loop quantum gravity (LQG), spectral standard model (SSM), and others. All could benefit. Then came all the hypothetical particles and what we called the Moonshine outliers. They all needed a place to begin working with the two Standard Models. Most naively we raised our hand, “Over here!” Yet, I would guess that we were too simple, too basic, and our grasp so superficial, nobody dared to get too close. We could readily taint their work! I understood… and understand even today.

Consider the obvious. These 64 notations have dimensionality. The conjectured infinitesimal spheres are not “one-dimensional space entities or membranes of higher-dimensional extensions existing in higher-dimensional spaces.” We might say as above, so below, considering that we started with numbers and basic geometries and carry it forward throughout the entire universe.

A new geometry. In May 2022, our simple clear-plastic models opened a new door. We had plenty of images of a five-tetrahedral gap and that work was well-known within small circles of scholars. With various five-tetrahedral models on my desk for several years, one day I asked myself, “Could there be a five-octahedral gap?” In minutes the first models were made; and within the month, we had our first pass at an explanation. A most-challenging composite is a five-tetrahedral gap on the top and bottom with the five octahedral gap in the middle. In June 2022 we began inviting scholars within pure geometry to help interpret where these gaps fit within the larger scholarly models of the universe. My simple thought was that these basic geometries, especially the three with basic gaps — tetrahedrons, octahedrons, and icosahedrons — could be part of the transition from Standard Model of Particle Physics to a new, different, and very-special science of the infinitesimal. The earliest infinitesimal architecture, we conjectured from Notation-0 to Notation-64, would give us that smooth-most-perfect start of the universe and then open a domain, Notations 65-to-67, for quantum fluctuations (our 2017 speculations).

#6 Proposed: Geometries of Quantum Fluctuations

Very few talk about a geometry of quantum fluctuations. When those words were placed in quotes within an online search, the only references that came up in September 2021 were to this website. Yet, when students made tetrahedrons, octahedrons and icosahedrons with those clear plastic models, they naturally dubbed it squishy geometry (also: https://81018.com/squishy/) and quantum geometry.

When pressed on the possible application of these gaps, our scholars seem to avoid those discussions. We can avoid it no longer. It is time to engage the gaps and all their implications for mathematics, physics, chemistry and biology (i.e. synapses).

#7 Pi, One Sphere Per Unit of Planck Time, Cubic Close Packing…

The expanding definition of pi as the actual bridge between the dimensionless constants of the infinite and all finite quantities is a key. Pi and spheres go hand and glove. The first sphere emerges, then one infinitesimal sphere per unit of Planck time. In the first second, we are out to Notation-143 and no less than 539 tredecillion (1042) infinitesimal spheres. In the first year within Notation 169, we would multiply 539 tredecillion by 31,556,952 (seconds per year). Exponential notation has taken over, yet there is a linearity deep within each notation.

For better or worse, the universe has begun.

#8 Next steps: Building consensus

Where do we go from here? Mathematicians and physicists are opening pathways to this domain. It is the domain of finite-infinite transformations where there just happens to be a fair amount of activity. From the esoteric to the basics, magazines like Quanta explore the edges of knowledge. New people are introduced everyday. People like Philipp Dumitrescu call into question the very nature of time. Mary Gaillard, a particle theorist at University of California – Berkeley, asks about the very nature of mass. Peter Scholze pushes forward with his perfectoids in Langlands programs. Although those who define infinity in other ways may disagree, it all seems to boil down to the finite-infinite relation. Many mathematicians are attempting to get beyond David Hilbert and Kurt Gödel and the limitations created by their logic that never entertained the first 64 notations and the perfected states within continuity, symmetry and harmony.


Let’s get beyond our worldviews.

Let us look beyond our little worldviews and consider the universe. In the process of exploring our universe, it was gratifying to find that the International Astronomical Union (IAU) and the United Nations promulgating Universe Awareness, a group that got started in 2004 through the initial work of George Kildare Miley, an Irish-Dutch astronomer and professor at Leiden University’s Observatory. Miley was the Director from 1996 to 2003. Once this page has been gone through several edits, I will introduced these folks to this work and references. Already oriented to a view of the universe, the question is, “Will they be open to the 202 base-2 notations that encapsulate this ever-expanding universe?”

There are many living scholars who have had vision and courage who have helped us begin to break out of our own limited worldviews, people like Frank Wilczek, Robert Langlands, George Ellis, Edward Witten, Helen Quinn, Paul Steinhardt, Sylvester Gates, Alain Connes, and Salvatore Torquato. Of course, there have been many others.

So, as you may well imagine, there will be many more scholars to come who will lead us. -BEC


Endnotes & Footnotes
These Endnotes are placeholders. I anticipate feedback that will open the discussion. Footnotes may be added. -BEC

  1. Infinite qualities — continuity, symmetry and harmony — shape all
    finite quantities. All dynamic equations, here is the beginning of spacetime

    The keys: Qualitative-Quantitative. Continuity, symmetry and harmony are the qualitative; it follows that real numbers that are generated by dimensionless constants constitute the quantitative. So, students, without fully grasping the most-sophisticated work of Hawking, Hilbert, Gödel and so many others over the decades, can understand the outside parameters defined by 202 base-2 notations, then become increasingly sophisticated as they add more and more textures to it. So, we will tarry on unless, of course, we hit a wall where the feedback is deafening and complete.
  2. Foundations for the mathematics to integrate our Universe
    We begin with real numbers. These are the best numbers we have today. They could readily be refined, yet the conceptual boundaries defined by base-2 will be little changed. Notwithstanding, we hold that no page within this website is ever finished. Each can be improved.
  3. The 202 base-2 notations that encapsulate the universe
    Our chart emerged over a five-year period. It took us that long to believe it hadn’t been done and to engage the fullness of it. Beginning in July 2016, the chart stimulated the development of this website for research and another for our secondary schools. That is was all highly-ordered and systematic was surprising and reinforced our basic geometries.
  4. An Answer to Big Bang Cosmology… It’s just too simple and expansive.
    It can’t go on forever. Aristotle had an 1800 year old mistake. We’re less than 100 years into this mistake (Lemaître, 1932; Hawking, 1990). We can begin to clean it up in our lifetime.
     The Large Scale Structure of Space-Time (PDF), S. Hawking and G. EllisCambridge,1973
  5. The First 64 Notations as the Foundations for Everything
    So many possibilities open up, an empowering creativity could become contagious. Ethics could begin to break out all over.
  6. Emergent Geometries of Quantum Fluctuations
    When we know there is far more room to expand, we will. Today we have an index of a bit more than a million total words but collectively use less than 170,000 words, and personally limit ourselves to somewhere around 25,000 words. According to Simon Plouffe, there are 215,000,000 dimensionless constants; to grasp that level of subtlety will require new words and new studies. Entirely new fields of study will emerge.
  7. Pi, One Sphere Per Unit of Planck Time, then Cubic Close Packing…
    A re-engagement with pi and an exponential universe such that every expression understood today will become part of standard curriculum. We know from our teaching and testing with 6th grade students that they can readily grasp these concepts and begin using them immediately.
  8. Next steps: Building consensus
    New leadership worldwide could well be empowered. We already have witnessed how younger scholars have been empowered. Getting the attention of today’s leading thinkers is more difficult. However, people making breakthroughs like Peter Scholze or Philipp Dumitrescu may be more open to simplicity. Then, some within their emeritus status, like Mary Gaillard (Berkeley), may become incrementally bolder and not be so quick to judge the new and the simple.
    Dynamical topological phase realized in a trapped-ion quantum simulator, Philipp T. Dumitrescu, Justin G. Bohnet, John P. Gaebler, Aaron Hankin, David Hayes, Ajesh Kumar, Brian Neyenhuis, Romain Vasseur & Andrew C. Potter, Nature, V.607, pp.463–467, July 20,2022
    International Astronomical Union (IAU) and the United Nations promulgate Universe Awareness, a group that got started in 2004 through the initial work of George Kildare Miley, an Irish-Dutch astronomer and professor at Leiden University‘s Observatory (Director, 1996 to 2003).


References & Resources
A few of the key related works that are studied but not within an endnote or footnote, are added here.

ArXiv: Dark Photon Stars: Formation and Role as Dark Matter Substructure, March 2022 with references to Witten, Wilczek, Kolb, Dimopoulos, Preskill, Fairbairn, Hogan (Carl J.), Garcia-Garcia, and others
Alain Connes, Noncommutativity and Physics: A non-technical review, July 25, 2022 (PDF)
George Ellis, Emergence of time, 2019 with Barbara Drossel and The physics of infinity, Nature Physics, V. 14, Issue 8, p.770-772, 2018
Sylvester Gates, Supersymmetry and Representation Theory in Low Dimensions, Dec. 2020
Robert Langlands, Langlands Program, Trace Formulas, and their GeometrizationEdward Frenkel, 2014
Helen Quinn, BOSE NAS, 20218
Scientific American: The Universe’s Unseen Dimensions, The visible universe could lie on a membrane floating within a higher-dimensional space, (PDF), Georgi Dvali, Nima Arkani-Hamed and Savas Dimopoulos, 20028
Paul Steinhardt
Salvatore Torquato
Frank Wilczek
Edward Witten


There will always be emails to our scholars with questions about their work.

Craig J. Hogan, University of Chicago, August 4, 2022 at 2:00 PM
Jürgen Jost, Max Planck Institute for Mathematics, Leipzig, August 4, 2022 at 11:18 AM
Sir Peter Knight, Imperial College London, August 2, 2022, at 3:320 PM
Peter Scholze, Max Planck Institute for Mathematic, Bonn, August 1, 2022 at 4:51 PM
Steve J. Carlip, UC-Davis, July 31, 2022
George Ellis, Cape Town, South Africa on July 27, 2022 at 5:01 PM
Helen Quinn, Stanford, on July 27, 2022 at 11:118 AM
Possible: Frank Wilczek, MIT / Paul Steinhardt, Princeton / Sylvester Gates, Brown
Alain Connes, IHES, Paris / Salvatore Torquato, Princeton / Edward Witten, IAS
Robert Langlands, IAS


Here will also be many instant messages to thought leaders about key points.

UNESCO, International Banks, Vladimir Putin, Ukraine, Pope, China,


You are always invited to lead a program, Each One – Teach Two.


Keys to this page, as-above-so-below

• This page became a homepage on July 28, 2022.
• The prior homepage is https://81018.com/starting-point/
• The last update was Friday August 26, 2022.
• This page was initiated on July 12, 2022.
• The URL for this file is https://81018.com/as-above-so-below/
• Current headline: Eight Concepts toward an Integrative Universe
• Earlier headlines for this article: The Mathematically-Integrated View of the Universe
• Current byline: Worldviews are too intransigent. A fully-integrated UniverseView is needed.
Other bylines: A good revolution in our time: Old Worldviews to New UniverseView!
The geometry of quantum fluctuationsThe First 64 Notations Out of 202 Are Key
Essential Key: Three Basic Geometries of Quantum Fluctuations
All of us are getting too hostile and intransigent. Let’s embrace the universe.
Let the next revolution begin! Independence from absolute time!









On following the work of Salvatore Torquato

Initial focus: Packing, tiling, and covering with tetrahedra, J. H. ConwayS. Torquato, PNAS V.103  No. 28, 2006

Salvatore TorquatoLewis Bernard Professor of Natural Sciences,
Princeton Institute for the Science and Technology of MaterialsPrincetonNew Jersey

ArXiv (156):  The structure factor of primes (2018)Hyperuniform States of Matter (2018)
Homepage(s): Complex Materials Theory Group, Google Scholar, inSPIREHEP, Prabook, Wikipedia
Publications: Resume, Books
Packing, tiling, and covering with tetrahedra, J. H. ConwayS. Torquato, PNAS V.103 No.28, 2006
New family of tilings of three-dimensional Euclidean space by tetrahedra and octahedra, PNAS, July 5, 2011 108 (27) 11009-11012; https://doi.org/10.1073/pnas.1105594108
Twitter: ICERM, Physical Review, Quanta Magazine and many more
YouTubeHyperuniformity in many-particle systems and its generalizations

Third email: 1 December 2022 at 1:11 PM

Dear Prof. Dr. Salvatore Torquato:

Yes, it is too simple. The five octahedral gap is overlooked. I have asked dozens of people now and everyone has been puzzled. All our computer graphics programs appear to ignore it or compensate for it. Even the construction kits like Zometool do not account for it. Isn’t that fascinating?

So, what’s next? Can you write about it and get a larger group discussing it?

If it is in any way related to quantum fluctuations — and with my models, there is a nervousness with those models — it’s significant.  We can actually make those constructions dance and bounce all around!

Would you like to have a set of models made of the clear plastic that we use? I’d be glad to send a few models to you so you can see-and-experience that “nervousness” to which I am referring.  Thanks!



Second email: July 6, 2022, 5:56 PM (Updated: July 15)

Dear Prof. Dr. Salvatore Torquato:

I have not found references online to a five-octahedral gap much like the five-tetrahedral gap that Aristotle missed and, of course, you and John Conway did a major study of it. Have you studied or are you aware of any studies of the five-octahedral gap?

Here is a picture of both gaps together: https://81018.com/15-2/. It is much too simple, but for that reason perhaps it has been overlooked.

Thank you.



PS. We’re making a study of that cluster of fifteen sharing a common centerpoint (with the hexagonals within each octahedron). It would make an interesting gate within circuitry of the infinitesimal. If we introduce the twenty-tetrahedral icosahedron in place of the five-tetrahedral cluster, its complexity and potential functionality increases exponentially. -BEC

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 August 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.

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 who was a physicist at 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… scolding myself, “You’re just too simple and easily get hung up on simple things.”