Aristotle’s mistake, not recognized for 1800+ years, threw us all off.


From Perfection to Imperfection

By Bruce Camber  The sequel

A simple mathematical model of the universe,a unceremoniously developed in 2011 within a New Orleans high school,b has 202 base-2 notations which start at the first moment of time, Planck Time, and goes to the current age of the universe, today or the Now. Discussed in our prior homepage,c the first notations out of the 202, instantiate the perfections of continuity (numbers and order), symmetry (geometries and relations), and harmony (dynamics). These are the qualitative perfections of infinity and the quantitative perfections of a primordial, infinitesimal sphere. This sphere appears to be the first thing to manifest in space-and-time.d As sphere dynamics extend those perfections, a finite-infinite relation becomes emergent and is always active. Yet, a simple geometry of five tetrahedrons sharing a common edge, sometimes called a pentastar,e has a gap and therein is a potential that gives rise to imperfection. At some early notation, that potential becomes actual, a dynamic geometry manifests as quantum indeterminacy. By Notation-67 it becomes a fundamental face of all subsequent notations. It masks moments of perfection or perfected states within space-and-time. The deepest underlayment for these primordial, infinitesimal spheres is base-2 exponentiation through which all notations or domains organize. The ever-so-limited domains of perfection with no quantum fluctuations are well-hidden within the much larger domain of imperfections where quantum fluctuations have become dominant.

Introduction: On backing into a base-2 exponential model of the universe. In our high school geometry classes we were dividing the edges of a tetrahedron by 2 and connecting those new vertices; there are four smaller tetrahedrons, one in each corner, and the octahedron in the middle. Dividing the edges of the octahedron by 2, there is a smaller octahedron in each of the six corners and eight tetrahedrons, one in each face. All 14 objects share a common centerpoint. Also, all create plates of triangles, squares, and hexagonals.

Rather quickly, we thought, “We can tile and tessellate the universe.”

From our classroom models, going further and further within, in just 45 steps, we were down in the domain of particle physics. Within another 67 steps we were within the Planck scale. We then used the Planck length for our edge and multiplied by 2 to return to our classroom model in just 112 steps or doublings. We continued multiplying the edges by 2 until we were out to the age and size of the universe in just another 90 steps. A total of 202 notations mapped our universe. Notation-202 is 10.9816 billion years in duration. The cumulative aggregate from Notation-1 to Notation-201 is also 10.9816 billion years so just 2.82+ billion years of the 202nd notation have evolved (if we take as a given our universe is approximately 13.81 billion years from its start).

Planck Base Units. We learned about Max Planck’s 1899 definition of his four base units of time, length, mass and charge. We took their face values as a given, and asked the question, “What would the universe look like if the very first moment begins with the instantiation of those four values?” Of course, these values are the result of equations with additional values used by Max Planck to render his basic numbers.

Consider the four equations and their numbers for space (length), time, mass and charge:

Planck Base Units
Illustration 1. The four Planck base units, their formulas, and their discrete values (numbers)

We asked, “If the Planck Length and Planck Time are the smallest possible units of length and time, does it follow that these are also the first units of length and time? [1] Does it follow that these equations, with all their dimensionless constants, come together to become the very first moment of physicality?” Unwittingly we had opened the “CDM of the universe” and soon wondered if Steven Weinberg would consider our model a grand reductionism.” [2]

Our postulate is that the Planck’s units are really-real physical entities, not zero-dimensional point particles, but an actual entity defined by the Planck base units. So our next question was, “What would that entity look like?”

Every equation is in part defined by the speed of light, pi (π), and the Planck Constant.

Because our students were studying basic geometric structures, they had a few answers. Yet, after some discussion, the students of pi, circles, and spheres prevailed. We then assumed not one sphere, but an impossibly-fast, steady stream of spheres emerge. We then wondered what the next dynamic could be.

Illustration 2: Kepler 1611

We decided to follow Kepler (1611) and his sphere-stacking exercise of that year. Analogically, a little like Kepler, we now have this infinitesimal, raw stacking of primordial spheres. It seemed that there could be many ways to count each notation. When does it become Notation-2? …Notation-3? If the first notation is defined by a sphere, what defines the next? Does it progress 2, 4, 8, 16, 32, 64, 128, 256, 512 and 1024?

We thought, “If there is no absolute space and time, the spheres can’t roll. There is nowhere to go.” So if the spheres fill space perfectly, what does that look like?

There already exists a rather deep science and mathematics of sphere stacking. There is another science of cubic-close packing of equal spheres. With our naive, rather cursory overview of both, we left our questions open. Within Notation-10 we guessed there could be as few as 256 spheres and as many as 67,108,864. We were analyzing the base numbers within our Chart, column 10, lines 8 & 9.  Within that range is the advice of Freeman Dyson, “Since space has three dimensions, the number of points goes up by a factor eight, not two, when you double the scale.” We had to learn about scaling vertices  and dimensional analysis and that gave us the high-end of our range.

Sphere to tetrahedron-octahedron couplet
Illustration 3: Cubic close packing

It is confusing; so we simply concluded, “Nothing is easy,” and went on to the next question, “What happens next?”

Our students had a quick answer, “The spheres come alive.”

First, there are those dynamics within cubic-close packing of equal spheres. The radii “discover” radii and triangulation begins (aka, triangulated coordination shells [3]. The discovery process continues and a tetrahedral layering begins enclosing octahedral cavities. There are structures within structures:

Indeed, the entire structure of spheres and tetrahedrons-and-octahedrons comes alive.

We concluded that any-and-every known, scientifically-defined dynamic that does not have a time or length dimension just might be applied within these primordial on-going events.

There is so much to learn here and we’ve only just begun.
The key point is that everything fits perfectly.
Pure geometry meets physics 101.

Spherical perfections. Creating perfect continuity, perfect symmetry, and perfect harmony, this infinitesimal universe begins taking shape and the ubiquitous Planck Sphere dominates [7]. It is all a quiet emergence within a simple perfection. Infinitesimal and way-too-small-to-measure, here are domains reached only by logic and mathematics. Actual physical measurements of a length do not begin until around the 67th base-2 notation; and, a unit of time, the attosecond, is not measured until the 84th base-2 notation. That is an extraordinarily large amount of intellectual space to tie logic, numbers and geometries together. Of course, it will be a challenge, but it just may be relatively straightforward for some of the scholars who have focused on the Langlands program or string theory throughout their professional careers.

Of those three facets of pi coming out of those never-ending, never-repeating numbers, the first is the face of continuity. It is a perfect ordering system that creates new sets of numbers and flow. Also, within those circles and spheres is a deep and abiding symmetry that gives rise to tetrahedrons and octahedrons — see Illustration 2 (above) — which is also a perfection. Those symmetries begin to discover symmetries and there is a simple harmony. Kepler’s music comes alive well before there is any range for human hearing!

It is a tangible perfection. And, it creates our homogeneous and isotropic universe. It is a study of perfected states in space-and-time. You will notice on every top-level page of this website are the words, CONTINUITY•SYMMETRY•HARMONY. Here is our definition of infinity and the foundations of perfection.


The expansion is now geometric, arithmetic, and exponential. And, we do project that Langlands scholars and string theorists have done a major amount of work to define this space and its automorphic forms [8]. Here is a discovery process whereby every equation within Langlands programs can be tested within a highly-structured environment. We need help with simple questions: “Might the radius of our infinitesimal sphere be a string?” We need help with more difficult questions, i.e. “Does this model reopen Witten’s equations of state within the Seiberg–Witten invariants?”[9]

Complexity and shortcuts: This simple base-2 ordering system quickly becomes complex. Each of the nineteen subsequent prime-number notations — 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, and 61 — could be used to test and instantiate more complex mathematics. The remaining prime numbers — 67,  71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 197 and 199 — also open new potentials.

Our little universe gets smaller. Base-2 is just one dimension of this expansion. This universe appears to be opportunistic so may well figure out how to use base-3, base-5, base-7, base-11 and base-13. What might be the linear components? How would linearity relate to the dynamics of base-2? Might there be simple coupling that introduces yet even more complex functions for this universe? [10] How would John Wheeler’s infamous wormholes work? How do we constructively open our imaginations and its science fiction?

Within this model all notations are active all the time. They build off of each other.

Finite-infinite. This system is slowly evolving with its own rules and axioms that are grounded in what will be a problematic statement for many people — the origins of these perfections are not finite. Opening the finite-infinite relation is an age-old enterprise fraught with tensions so please allow me to close that door rather quickly by emphasizing our simple definition of the infinite: “The infinite is the qualitative expression of continuity, symmetry and harmony whereby continuity begets order, symmetry begets relations, and harmony begets dynamics.” The finite is the quantitative expression. That’s it. Nothing more. Anything else anybody wants to impute is their business; it is probably not relevant here.

Within a little over one second, the base-2 expansion is out to the 144th notation. Planck Length is 360,424.632 kilometers. Planck mass is a hefty 4.8537×1034 kilograms. Just to put that in context, the mean average distance between the earth and the moon is 384,402 km (238,856 mi). It is 356,500 km (221,500 mi) at the perigee and 406,700 km (252,700 mi) at the apogee. The sun’s mass is around 1.989 × 1030 kg so at this notation, the density of the universe has analogies to a neutron star. The universe as we know it begins to take shape between Notation-196 and 197.  At 10,829,559,004,640,000 seconds, Notation-196, the emergence is at 343.15 million year mark.

Just within these 202 notations, here are a few highlights of this base-2 model:

  • One second, between Notation-143 and 144, Planck Length is 299,792± km.
  • A year is between Notation-168 and 169.
  • The first 1000 years is between Notation-178 and 179.
  • The first million years is between Notations-188 and 189.
  • And, the first billion years is between Notations-198 and 199.

This model is primarily about the very early universe.

An all-natural imperfection. Within this process, always being filled with Planck Spheres and constantly testing every flavor and texture of geometry, a key construction is a five-tetrahedral cluster.

Although Aristotle thought it was a perfect configuration [11] , among many others, chemists in the 1950s recognized his mistake and calculated that gap. It is an important gap. Pentagonal, icosahedral and Pentakis-dodecahedral structures have such a gap (or a stretched imperfect surface and angle). In the 1960s the first concepts around aperiodic tilings were introduced. In 1976 Roger Penrose introduced his unique tilings and Alan Mackay followed up experimentally to show how a two-dimensional Fourier transform (with rather sharp Dirac delta peaks) manifests a fivefold symmetry.  In 1982 Daniel Shechtman began his public-struggle to open the exploration of quasicrystals. Each one is related to the other.

The first quantum fluctuations. With this discussion, we push that gap down into our infinitesimal and primordial work with Planck spheres. We postulate that there are many potential dynamics to begin the first quantum fluctuations as it moves up those notations. At some point, the pentastar becomes part of a system and begins to move. It is dynamic. It can start and stop. It can move around and up and down notations.

We do know that by Notation-67 it has become part of the fabric of the notations and so begins the measurements that are characterized as undecidability (subjects), uncomputability (relations), and unpredictability (objects).

We are in search of answers to the question, “When and where do these fluctuations manifest?” We are begging for help. These are all new studies for us. Our simple history begins in 2011. Our more critical history didn’t really begin until 2016. Notwithstanding, we are speculative people and believe the fluctuations actually fluctuate, first between notations, then between sets of notations, and then within groups of notations.

Yes, since 2011 we have been asking scholars for advice about it all. It is a very different, entirely idiosyncratic model based on eighteen divergent points of view. Notwithstanding,  there is much more to explore, and so we will continue as best we can.


Please note that this is still a rough first draft. -BEC


Endnotes, Footnotes, References, and Resources

Abstract: All Embedded Links Go To Pages Within The Website
a A simple mathematical model of the universe
b December 19, 2011 within a New Orleans high school
c Prior homepage: Starts With A Mathematical Perfection
d Space-and-time:  Emergence
e Pentastar: Five Tetrahedrons and a Gap

[1] Although there are links throughout this website to the December 19, 2011 story of our high school geometry students and their teachers chasing tetrahedrons and octahedrons down into the Planck scale and then out to the age and size of the universe, here are other key links to tell a bit more of that story:

  • Because of its systematic ordering, this project was initially considered a STEM tool.
  • When we could find no place within our grid for Plato’s Eidos and forms, Aristotle’s Ousia, binary operations, pointfree geometries, Langlands programs, string theory or loop quantum gravity (please see line 11 of our horizontal chart), we decided, “That’ll all be within the first 67 notations.” We knew then there would be an endless amount of work to do within this model.
  • One of the earliest stories about this enterprise

[2] We started as everything does, simple. We take little steps and ask simple questions. We try to respect all prior scholarship. When we become confused, we step back to something more simple. So, it was with deep respect that we engaged the CDM approach to the universe and read that the most-distinguished  Steven Weinberg (book, Facing Up)  might call this model, “a grand reductionism.” We continue wrestling with his work and with these other scholars:

[3] Our initial studies of the work of F. C. Frank (H. H. Wills Physics Laboratory, University of Bristol, England) and J. S. Kasper (General Electric Company’s Research Laboratory, Schenectady, N.Y.) opened the concepts within cubic-close packing of equal spheres, the triangulated coordination shells, and the emergence of the tetrahedron from just four spheres. That all opened the way to engage The Physics of Quasicrystals edited by P J Steinhardt and S Ostlund. We struggle to grasp the work of scholars within this area:

[4] Our first introduction to the Fourier transform was through Steven Strogatz on Pi day in 2015. His article for The New Yorker Magazine resonated at that time and it still does today. Now we are attempting to really dig into the Fourier work. Of course, we have a long way to go. Here are some of the scholars to whom we are currently turning for help:

[5] Linear transformations are part of the dynamics within a notation. Yet, there is a homogeneity with all the contiguous planckspheres so geometries may readily extend within notations and across notations. It seems that the dynamics of all geometric models of the universe may hold insightful keys. With that mindset, we are open to all studies of space and time symmetry:

[6]  In 1980 I worked with Jean-Pierre Vigier and Olivier Costa de Beauregard at the Institut Henri Poincaré. Our focus was solely on the EPR paradox, Bell’s theorem, and the experimental work of Alain Aspect at the SupOptique or “IOGS” in d’Orsay (just outside of Paris). Never did we look back at the work of Henri Poincaré. Today, a focus is on the Poincaré sphere and its underlying Lorentzian symmetry as a geometrical representation of Lorentz transformations. We continue working with these scholar’s ideas:

[7] The word, Planck sphere, is a key, core concept and we will continue to research it until we find the best possible resources that go back as early as possible. To date, we start with John Wheeler’s work with quantum foam; it could hold a key. Here are others:

[8] Who are the scholars to whom we can turn to learn about automorphic forms? Of course, there are the scholars within the Langlands programs and string theory. They have done sustained work since the 1970s and they have done a major amount of work to define its automorphic forms:

[9] One of the world’s leading scholars within string theory is Ed Witten. He is also a gentleman. Because the majority of his career has been in the shadow of  big bang cosmology, his work has had an impossible starting point with which to contend. There is no easy migration to a theory that pushes time-space-and-light together at the Planck scale, and then again with mass-and-charge at the next level (c2). It will be fascinating to see if they will do better within a cold start that redefines the historic æther, and gives their discipline the radius of the plancksphere within Notation-1 and every plancksphere through Notation-67, and at least the first 67 notations of each subsequent notation.

[10] Of course, base-2 is the first exponential expansion of this model such that no point within the universe, right from the first instant, is more than 202 notational steps away. Yet, I believe our opportunistic universe will also test base-3. If necessarily held in check by base-2, there may step by three to Notation-6, then Notation-9 and so on, such that there is an aggregate of 67-steps through to Notation-201. Base-5, if necessarily held in check by base-2, might be 40 steps through to Notation-200.

How do these other bases manifest? What is their relation to the prime number and base-2?

Take base-7. If necessarily held in check by base-2, would it still provide 28-steps to Notation-196? Take base-11; if necessarily held in check by base-2, might there be just 18 steps to Notation-198? Of course, we have more questions than answers.  and base-13 just fifteen steps through to Notation-195. These clusters of notations possibly may well introduce even more complex functions.

  • The largest square of a prime, 1313 is 169, obviously under 202; and, 1717 (the next prime) is over (289). So, although included within the base-2 progression,  are Notation-13 and Notation-169 fully-relational with base-2? What does it look like?

[11] Imperfection. One of history’s greatest thinkers made a most fundamental mistake that was repeated for about 1800 years. That is a mistake of epic proportions. We are all taught to have such great respect for scholars, yet sometimes it holds us back. Aristotle (384–322 BCE), one of the greatest Greek philosophers and a polymath, obviously had imperfect models of the tetrahedron, otherwise he would have seen and felt a geometric  gap. Five tetrahedrons all sharing one common edge opens a gap.

My first discussion about it was in 2016.

1800 years later. The greats that followed Aristotle repeated his mistake and failed to grasp a most-essential quality of simple geometry. One of our primary source articles is “Mysteries in Packing Regular Tetrahedra (PDF)” by Jeffrey C. Lagarias and Chuanming Zong. They relied heavily on the Dutch article by D. J. Struik, Het Probleem ‘De impletione loci’, Nieuw Archief voor Wiskunde, Series 2, 15 (1925–1928), no. 3, 121–137.

We’ve all got to reflect on this mistake.

Two chemists, F.C. Frank and J.S. Kasper with their article, Complex Alloy Structures Regarded as Sphere Packings, took it further and calculated that gap.

We are undoubtedly among a very few who claim that this gap is the basis for quantum indeterminacy, imperfections, free will, unpredictability, undecidability and uncomputability, so yes, they is much more to come!


Notes. Throughout the  process, many new resources are uncovered. It takes time to do a critical review. Much is discounted and deleted. Some things are not. Though new to us, the statements seem to capture a truth, so for a few weeks they are left down here below the footnotes. Some of these comments will be used with the editing process for this article. Some of them may become part of the next article.  -BEC


“Knowledge-building in cosmology, more than in any other field, should begin with visions of the reality, and passing to have a technical form whenever concepts and relations in between are translated into a mathematical structure.”  –F. J. Amaral Vieira (Acarau State U., Sobral


Mores references and resources being reviewed:

Clusters, groups, and sets can readily be parsed giving us a way of checking the equations of state, the physical numbers, and potential relations.

Sets are dynamic while groups are not.

Time symmetry in modern physics, Andrew Holster, New Zealand


Discrete Calculations of Charge and Gravity with Planck Spinning Spheres and Kaluza Spinning Spheres, Michael John Sarnowski


What? Be careful here. “The fundamental ratio of surface to volume quantizations is what yields mass. The proton has 1040 on surface, 1060 in volume, it’s ratio is 10-20 * planck mass = proton mass.” Nassim Haramein and physicists at the Resonance Science Foundation and the Hawaii Institute for Unified Physics.  (


Bagdoo, Russell, Quebec City, The World in an equation

As for As for Gödel’s undecidability and incompleteness theorems, within the first 64 notations, extending the mythopoetics of the Chessboard and the Wheat stories, there are nineteen prime numbers to initiate new mathematics within that base-2 foundation.


This page is Imperfection

This page was initiated on 28 February 2020
First time as a homepage date: 3 March 2020
Last edit: Sunday, 23 May 2022
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