by Bruce Camber (under construction as of July 18, 2022)

If the Planck base units are meaningful, and that seems to have been settled in the past twenty or so years, does it describe a starting point for spacetime? I have naively concluded that it does. When efficacy of Planck’s constant was questioned, I dropped back to the 1874 work of George Johnstone Stoney and suggest the “actual numbers” are symbolic although in the range of a real reality.

So, from where does everything come?

I redefined infinity in light of three most basic facts of pi, and say, “continuity-symmetry-harmony.” The most controversial facets of infinity are considered personal. Of course, that may well be just too naive in light of the work of Hilbert, Gödel and Woodin. Yet, if we can show from where imperfections originate, maybe we have something different. Assuming perfection is the sphere and pi, the first imperfections may well be geometric; and, Aristotle may have unwittingly missed that key marker, five tetrahedrons creating a gap. It seems nobody recognized the five-octahedral gap. It is too simple, but it looks good coupled with the five-tetrahedral gap. I think there is something to it. That little cluster looks like it could function as a logic gate within computer programming. If we were to consider a twenty-tetrahedral icosahedron in place of a five-tetrahedral unit, complexity-and-possibility are multiplied. Then we consider a Pentakis dodecahedron and combinations of all of the above.

Abstract: Most people are unaware that Aristotle made a mistake that was not caught for about 1800 years.* That mistake has also been largely ignored by academia. It had to be re-discovered at least twice yet then only within limited scholarly circles. If this gap, created by five tetrahedrons (and any gap with pentagonal structures), is not better understood, we limit our knowledge and understanding of the universe and ourselves. These fundamental gaps in geometry deserve more attention. Imagine if the gaps can be ordered based upon their adoption within other configurations and formulas. Imagine if the gaps have a possible role within quantum fluctuations. Also, the role of a geometry without gaps shall be more closely examined. It might be considered a domain of perfection (out of our 202 base-2 notations) that have (1).higher densities, (2).shorter time sequences, and (3).no-gap, simple geometries.†

Introduction. Aristotle thought the universe could be tiled and tessellated with tetrahedrons. It cannot. Within his context, it requires both the tetrahedron and octahedron to fill a three-dimensional space perfectly (without gaps).

Aristotle’s mistake is quickly discerned with just five tetrahedrons, all sharing a common edge, two center points, and at least one face with another tetrahedron. There will be a 7.356103172453456+ degree gap between the first and fifth tetrahedron. For this study, it is called a five-tetrahedral gap. Within our work, that analysis began in 2016.^{1}

There is also a five-octahedral gap created by five octahedrons sharing a center point (three octahedrons share two faces and two octahedrons share just one face). It’s a real gap and our analysis of it began in May 2022.

Gap Geometry-Physics-and-Chemistry.^{2} Also, there are the icosahedral gaps when an icosahedron is constructed with twenty tetrahedrons. In our high school classes it was called imperfect geometry, squishy geometry, and sometimes, quantumgeometry. With four 7.35610+ degree gaps, one could easily squish or otherwise compress our classroom models. If evenly distributed over the twenty tetrahedrons, there is at least a 1.47+ degree gap between each tetrahedron.

These rarely-discussed gaps have been a key part of our analysis of numbers, geometries, chaos theory, fractals, and quantum fluctuations. It is also possible that these gaps are part of the dynamics of creativity, openness, indeterminism, uniqueness and human will.

These gaps are not arbitrary. Each is considered to be a very different kind of standard and each will be logically defined (“measured”) and categorized. Stretching, we include the mass gap of the Yang-Mills theory. The nature of mass and a very wide variety of hypothetical particles is open. Within the context of the first 64 notations out of the 202 that encapsulate everything-everywhere-for-all-time, more basic networks of more basic relations may be defined.

Our initial step will be to ask each IGS to create a five-tetrahedral object and a five-octahedral object. The icosahedron (twenty tetrahedrons pictured) has almost 30 degrees of gaps to include! Because these systems use two flavors of programming — continuity or determinism — we will ask for a sample from each.

We will do the same for geometric construction kits. There are over twenty popular sets. It is currently anticipated that none of these sets will display a gap.

We will report the results if any other system, hardware of software, can re-create a gap.

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Out of 202 base-2 notations: The most-infinitesimal scale — the 64 notations from the Planck scale to the scale for particle physics — can only be defined by logic and functions that are scale invariant, particularly key dimensionless constants.^{4} There are more than enough variables to speculate and postulate about the gap’s emergence within spacetime. The first notations are by definition the most dense with the shortest time sequences, and the most-simple geometries. With Planck’s base units, if we postulate one infinitesimal sphere per infinitesimal unit of time — that’s 539 tredecillion spheres per second — the dynamics of sphere stacking and cubic-close packing of equal spheres open and the most-basic tetrahedral-octahedral configurations emerge. It is a subject of extended studies by scholars like F. C. Frank and J. S. Kasper^{5} (circa 1957) and more recently by scholars like John Conway, Salvatore Torquato, and Jonathan Doye.^{6}

A place for those 202 base-2 notations to be studied.^{8} Since December.2011, we’ve known that the 202 notations mathematically and geometrically encapsulate the universe. Although starting naively and simply,.the discovery process was exciting albeit confusing. Our roots were from within a high school geometry class and we had, and continue to have, a step learning curve. This model is idiosyncratic. It flies in the face of current theory. Our letters, emails, and instant messages to scholars around the world have asked for help. We were prepared to hear someone say, “Been there, done that, and here’s why it is wrong.” Nobody did. So now, after ten years of study, we conclude there is something right about these charts and our goal has become to find out what that is.

The first base-2 chart of the 202 notations was published in December 2011. We had a simple container within which to put everything. On most the lines we had something to study. Yet, most of the first 64 notations were blank.

By 2013 we were increasingly focused on the range from the Planck base units to particle physics.

If Langlands, Witten, S.J. Gates or any of our leading scholars make a claim on a notation or group of notations, the paradigm will shift. It is a challenge that begs people to hold back judgments of the unknown and to take thoughtful time to entertain the potentials of this domain defined by no less than 64 base-2 notations.

New speculations about these 64 notations have to go substantially beyond current work. To do that, three articles for ArXiv are being developed. Each will have multiple authors with long histories of pre-publishing through ArXiv. Outlines of these articles are given within many articles already online here, particularly STEM, this page, and New Ideas-New Concepts. So, yes, these new articles will extend and perhaps change the conclusions within previously-posted articles on this website. Thank you.

When an article is initially constructed, links often go outside this website. Most-often, within a couple of weeks of its first draft, those links become a footnote or an endnote.

[1] 7.356103172453456+ degree gap between the fifth and first tetrahedron: It has no formally-recognized name within the scientific community. We once called it a pentastar gap, but now more accurately and generally, the five-tetrahedral gap. We have also begun calling the other “simple” gap, the five-octahedral gap. Our casual introduction to this gap was in 2011. We began writing about it in 2016. That five-octahedral gap was first discerned in May 2022. These structural gaps are really real. And, they are part of the physics of the infinitesimal. Our challenge is to begin to figure out what difference these gaps make.

[2] Gap Geometries, Gap Physics and Gap Chemistry. The icosahedron, one of the basic five Platonic structures, can be created with twenty tetrahedrons. Of course, each set of five tetrahedrons has a total gap of 7.35610+ degrees. We are asking about the necessary mechanisms for attractors and repellers and for dimensionless constants to manifest. What might that circuitry be? Is it vertex-to-vertex, along the edges or from face-to-face (or plate-to-plate), or holistic? Our hope within our studies of Langlands programs, string-and-M theory, SUSY, and other related disciplines, is that we will find new insights to these questions. Our simple start is to look for connections to tetrahedrons-and-octahedrons and to base-2. We are also asking questions about infinity defined here as continuity-symmetry-harmony.

Our goal is to further define the first ten notations. Our hope is to affirm the most likely paths to the five-tetrahedral structure, then to the twenty-tetrahedrons as an icosahedron, and then to twelve sets of five tetrahedrons known as the Pentakis dodecahedron. By the way, that calculation of squishiness, a distribution of the four gaps, each 7.35610+ degrees, render 29.4244 degrees now spread over 20 tetrahedrons or approximately 1.47+ degrees per tetrahedron. In reality, it looks considerably looser.

From the geometric gaps to actual physical gaps such as the mass gap, to chemical gaps such as synaptic functions, there is an anharmonicity that is being explored. The synaptic function is re-engaged; systems are systems and analogies open new insights. There is a kind of dissonance with these gaps; however, here it can be constructive and productive. As a work in progress, the Wikipedia’s analysis of anharmonicity is instructive. If continuity, symmetry, harmony are the perfections of the sphere, the imperfections of the gap (discontinuity, asymmetry, and dissonance or anharmonicity) are not necessarily negative qualities.

Certainly human creativity requires room to breathe.

An unsolved mystery is the Yang–Mills existence and mass gap. One might readily conclude that is unsolved because it has not been considered within the grid of 202 notations, particularly with the grid of the first 64 notations, and most especially within the grid of the first ten notations where the paths, shapes, and textures of mass are initially developed.

[3] Visualizations. In our high school we tried using Mathematica, but it was too advanced for most, including me. We did use the Zometool; however, it was disconcerting not to be able to see the gaps. Our clear-plastic tetrahedrons and octahedrons were a better representation of the real realities. Math manipulatives, both as physical objects and computer-generated graphical models, are important teaching tools. So, with this article we have made an earnest commitment to explore both interactive geometry software (IGS) and a diversity of geometric construction kits. Wikipedia has a working summaries of these dynamic geometry environments (DGEs) whereby the features and results for continuity and determinism can be tested.

Also, see my personal correspondence, Jonathan Doye.

[7] Quantum fluctuations. Our study of quantum fluctuations is still young and naive. For the many scholars within Langlands programs, string-and-M theory, SUSY, CDT, CST, LQG, SSM, and SFT, and those working with hundreds of hypothetical particles, a base-2 container for their work could be insultingly simple. Also, for all those who have held onto historic definitions of infinity, it may be difficult to redefine infinity quite so simply as continuity-symmetry-harmony (and to have time become the Now). For all those who have held onto Hawking’s concept of the big bang, it may be quite difficult to let it go. It’s ok. These kinds of transitions are difficult for everyone.

[8] Place where the 202 base-2 notations are studied. There is nothing easy about exponential notation. It’s non-intuitive within our current framework for thinking about spacetime. Those 202 notations that encapsulate the universe seem altogether too simplistic. With over four centuries of a Newtonian worldview, it is will not be easy. Yet, base-2 notation from the Planck scale to current time is upon us. It is time to make it a serious study.

Many geometers, chemists, and physicists know that five tetrahedrons sharing a common edge create a gap: https://81018.com/gap/ Most do not know that five octahedrons create the same gap; and that stacked, this gap is a beautiful thing to see: https://81018.com/15-2/

We have unsuccessfully searched for studies that explore the very nature of that gap. We’ve asked many scholars, “Might that gap be associated with quantum fluctuations? Could there be a geometry for quantum fluctuations?”

Might you have any insights that could help us grasp these realities more profoundly? Thank you.

Caltech (IQIM): Jason Alicea (June 29, 2022) and a few others University of Maryland (Joint Quantum Institute): Maissam Barkeshli and a few others

9:25 PM · Jun 4, 2022@DM_Rubenstein Our worldviews are too small; we need the universe and the only way to begin to get it is with mathematics that start with the Planck base units. There are 202 base-2 notations from Planck Time to this day. http://81018.com opens a start to a highly-integrated view of the universe.

7:05 PM · Jun 4, 2022. @theo__oneill How about a paradigm shift so we don’t get caught up in our own little worldviews (solipsism at best)? How about entertaining a base-2 parsing of the universe? There are just 202 notations: https://81018.com is a start. Your comments would be wonderful.

The world is increasingly nihilistic. Many places have become dystopian. We all need to do something.

We are preparing pages so we all become teachers of an integrated view of the universe whereby we all begin to profoundly understand that what we do each and every moment effects the quality of life within this universe.

This is a key document. Click on the “Back Arrow” (or “Left Arrow”) at the top of this page to go back to another key document. If you agree and you will begin to teach others about this integrated view of the universe, we will list you within our soon to-be-added “Teachers” page. Thank you.

Introduction to Quantum Fluctuations In Light Of Exponentiation: Our base-2 exponentiation chart^{[1]} from the Planck base units to the Observable Universe and Age of the Universe has 202 notations. It is logical and simple.^{[2]} Yet, to qualify as a potential alternative to big bang cosmology, it will have to do everything the big bang seems to do well, and go beyond it in addressing other open questions. One such question is regarding quantum fluctuations. Considered to be a random change of energy in a point in space described by the Heisenberg uncertainty principle, within this model that “point in space” is much larger than currently accepted. In our mathematical and geometrical model based on the Planck units, fluctuations are projected to be in the range of the Notations 55-to-67. The first “points” are transformation points between the finite and infinite smaller than the Planck base units. The so-called points within the core physics curriculum have values defined by the Planck base unit multiples from Notation-1 through to at least Notation-64, perhaps as high as Notation-67.

Background. In December 2011, this project began by following embedded geometries, smaller and smaller, by dividing the edges of the tetrahedron by 2 and then also the octahedron, connecting the new vertices. In 45 steps doing a Zeno-like progression further and further within, we were in the domain of particle physics. In another 67 steps within we were in the domain of the Planck scale. In our naïveté we then multiplied our little tetrahedron by 2, and in 90 steps were out to the age and size of the universe.

Certainly Max Planck and Albert Einstein could have done the simple math. Probably it was just too simple for them. It took the extreme naïveté of high school people to follow embedded geometries to the Planck scale then to apply base-2 notation to create a chart of 202 notations from the first moment of time to this day. It seems logical. It looks logical. Since 2011, we have asked hundreds of our best scholars for advice. Very little advice has been given. We’ve learned this much — geometry and mathematics are real realities and Aristotle made a mistake when he thought one could tile the universe with just tetrahedrons.

There are other gaps. Icosahedral gaps are dramatic. Given the lack of understanding in this realm of physics, we postulate and hypothesize that these gaps, once they’ve became systemic, become the earliest fluctuations in the universe.

The first-possible physical measurements in our universe happen in labs like CERN (Geneva, Switzerland). Estimates of the ionic radius involved in their heavy ion collisions range from 30-to-300 picometers or approximately 3.9077×10^{-11 }meters (Notation-81) to 3.12618×10^{-10 }meters (Notation-84). It appears that the collider experiments result in measurements within the scale of the 67th notation (2.38509×10^{-15} meters), the 66th notation (1.19254×10^{-15} meters) yet some say down as small as 1.16459×10^{-18} meters at the 56th notation. These and the other numbers in the 202 columns are real numbers, all multiples of the Planck base units.

Today these numbers are all just simple mathematics and imputed geometries and this system of numbers is not yet recognized by the scholarly community to be meaningful or significant.

Potential Significance: By themselves these numbers create a rich panoply of possibility. We believe it would be unimaginative not to try to do something with them, so we enter the domain of the mysterious quantum fluctuations. Unlike current theorists, their “out of nothing” is for us as many as 67 notations where distinctions between notation 2-to-66 are expected to be as great as those within the next 67 notations (67-134), and then again within the next 67 (134-201) that follow.

Pi And Never-ending Numbers Without a Pattern. The animated image (on the right) challenges us in unusual ways. Looking for other possible parts of the puzzle, we accept the logic and necessity of a singularity, but conclude that it is a dynamic transformational nexus between the finite and infinite whereby there is a complete unification of all the basic forces of nature, the Planck base units (aka Planck scale), the dimensionless constants, and a very special class of circles and spheres.

That is our working hypothesis. We further postulate that this working premise creates an environment to build a panoply of bridges from the Planck scale to all current physical theories whereby each notation is a domain for unique predictive values. The challenge will be to recognize the mechanisms that would confirm or deny these hypotheses and postulations.

The universe appears to be logically divided into three scales. The small scale from notations 2-67, the human-scale from 67-to-134, and the large scale from 134-to-201.

The next big observation is that the progression of epochs defined by big bang cosmology is well-defined by a natural inflation within base-2 exponentiation. Simple logic seems to suggest that each currently-defined big bang epoch can readily become a process within the base-2 model. And, significantly, there is no need for an actual big bang; natural inflation is quite enough.

Within our Top Ten Reasons to Put Big Bang Cosmology In Time Out, we now turn to a few of the many enigmatic questions^{[3]} within cosmology. Perhaps within the base-2 model we can find answers or something new.

SPECULATIONS:

More Postulations. These first 67 notations could be the deep systems structure of the universe that give rise to richly-textured quantum fluctuations, cosmic microwave background radiation (CMBR), homogeneity and isotropy, dark energy and dark matter, magnetism, and gravity (defined by General Relativity). Big bang cosmology has no answers. Given that the base-2 model is simple, it may hold rather simple answers. This article is our first to enter this configuration space starting at notations 0,1 with a point (vertex), then a line, a circle, a sphere, a tetrahedron, and then tetrahedrons and octahedrons. That animated image (top left) demonstrates a process that apparently has not been studied much beyond the problem of stacking cannonballs. (Thomas Hales followed up Thomas Harriot’s work)^{[4]}

The Art: Using dimensional analysis within the first 21 notations, there are over a quintillion base-8 vertices with which to experiment and build initial systems! Sophus Lie could make millions of his E8 structures. Within just the first ten notations, there are 134,217,728 scaling vertices. Possible analogues include the protein machines or nanomachines within the body. Other analogues could include genetic code machines.

The first ten notations. We’ll call them axiomatic machines (systems) that extend the continuities inherently defined by Planck base units (length, time, mass, and charge). `These axiomatic machines are also defined by the speed of light (or special relativity), the gravitational constant (or general relativity), the reduced Planck constant (or ħ or quantum mechanics), the Coulomb constant (or ε_{0 }or electric charge or electromagnetism), and the Boltzmann constant (or k_{B} or temperature). All “things” within these notations are uniquely defined by pi. These are not physical things; these are ontological things, or perhaps axiomatic things. We will be working on the nomenclature as we go along this path.

So, let’s build some simple mechanisms, then work with the art (and magic) to bring all that richly-textured phenomena of the universe to life! Let’s see what we can do with the most artistic and unpredictable of that group, quantum fluctuations.

Transformation Nexus. Once the singularity is multiplied by 2 (the form) or by 8 (the function), it is no longer the Planck singularity or Planck Epoch; it becomes the first possible transformation nexus between the finite and infinite.

Could it be that from this beginning, everything, everywhere, shares these eight vertices. “Ridiculous? Poppycock?“

We ask that you hold those judgments in abeyance. To get into this model will require stretching, suspending judgment and imagining in rather new ways. With eight vertices the universe might organize those vertices as five tetrahedrons with a small space between the fifth and first tetrahedron. There is one extra vertex. It is elegant. It is imperfect. It creates degrees of freedom (7.356103 degree gap or 0.12838822 radians).

Couldn’t this be the birthplace, the earliest beginning of the quantum fluctuation?

Could this possibly be a key process that is extended throughout all 202 notations? Of course, those who know molecular geometry readily recognize the figure.

In the next doubling there are 64 vertices. Seven-plus-1 is established. The next 13 vertices extending from “The Seven” create a mathematically whole figure, an absolute figure of six tetrahedrons whereby there is potential action because the base is a perfect structure, a hexagonal plate.

Each cannonball is a vertex. Or, is it an infinitesimal circle or sphere? …both? Given that we are working with the five basic Planck units with real numbers that define each unit, the vertices are 134,217,728 scaling vertices. a real manifestation within space-and-time.

The animated image (gif for “Graphics Interchange Format”) of the cannonball stack tells a powerful story.

The all-important quantum fluctuation seems like an art form on one hand and a fundamental energy source on the other. Open this conceptual roadblock and we find a truly golden rainbow. Although each roadblock is a face of this small-scale universe, perhaps quantum fluctuations would best be known as the art of the universe!

The world’s leading cosmological theory is challenged. Though our little model is little-known today, that is about to change. Big bang cosmology is going to be placed on ice and assigned to timeout.

The Art, Mechanics & Systems

Let’s start at the very beginning: Notations 1-10

PLEASE OPEN THE CHART IN A NEW WINDOW.

Each of the Planck base units, considered to be part of the so-called singularity, has a real number associated with it. Though the formulas for deriving the numbers of each Planck base are readily understood by many, watching how the numbers progress along their scale is new.

In the first ten notations, we first assumed the numbers represented vertices that will double and slowly become structure.

Endnotes & Footnotes There may not be many because all these points already have pages within this website.

References & Resources As these references are studied, key references and resources will be added.within this website.

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Emails There will be emails to many of our scholars about the key points within this posting.

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IM There will also be many instant messages to thought leaders about the key points.

2:25 PM · Nov 7, 2022 Geraint F. Lewis, @Cosmic_Horizons@Sydney_Uni Go inside the tetrahedron (divide the edges by 2, connect new vertices) and in 67 steps you’ll be at the Planck base units. Now multiply by 2, 202 times and you have the universe: https://81018.com A little math and geometry go a long way!

Pisa, Italy. You can almost hear him think, “If we don’t have a predictive theory of quantum gravity, we’ll make it up.” Damiano Anselmi, associate professor of physics at University of Pisa, created a website, Renormalization, with nine or so other scholar/authors to take it on. The thrust of their work is around a hypothetical particle they’ve named the fakeon.

If the challenge is to develop a predictive theory of quantum gravity, what parameters and boundary conditions would best engage such a theory?

• Would this model of the universe best start at the Planck base units? • Might the first manifestation within space-time be an infinitesimal sphere? • Might one sphere per unit of Planck Time-and-Length be our cosmological constant?

And what if those spheres were infused at get go — https://81018.com/ccp/— with cubic-close packing, fully activating the Fourier, and perhaps even with a bit of Smale and Milnor‘s attractors and repellers, all generating basic geometries? • Could the universe be fundamentally exponential? • Does base-2 create an initial outline that maps our universe?

Also, it appears that this model has one foot in the infinite — https://81018.com/csh/ — so it is a rather different model that requires a redefinition of time, space, and infinity to start.

Abstract. Base-2 exponentiation symbolically begins with the Planck scale, the first possible moment in time, and in 202 doublings defines this current moment in time.^{[*]} Base-2 groups diverse studies such as exponentiation, bifurcation theory, fractal geometries, and cellular growth. It’s becoming a working model based on simple logic, geometries, and algebra, initially based on spheres, tetrahedrons, and octahedrons. It begins with natural units and is driven by pi (π). Its 202 predictive, base-2 notations totally absorb big bang cosmology.[†] This mathematically-integrated view of the Universe addresses key issues that big bang cosmology cannot. Five examples follow.

I. Smoothness. The James Webb Space Telescope images have raised anxieties that the early universe is too smooth for our working understanding of the dynamics of big bang cosmology.^{[1]} The transition from smooth to lumpy is not a problem within our 202 base-2 notations. In this model geometric gaps do not become systemic until later in the unfolding of the universe.^{[2]} The earliest notations begin perfectly smooth because it starts with the most simple geometries that tile and tessellate this universe perfectly.^{[3]}

II. Time: The first minute, first second, and first zeptosecond. The first minute of the universe is within Notations 149-150.^{[4]} The first second is between Notation 143-and-144. Zeptoseconds (1×10^{-21}) are between Notation 65-and-67. And, Notations 1-64 are opened for definition!

Eminent scholars, Sean Carroll ^{[5]} of John Hopkins and James Peebles ^{[6]} of Princeton (Nobel laureate, 2019, and a pioneer of cosmic microwave background radiation research) articulate the importance of defining the first minute of the universe. In June 2020 twenty-seven scholars wrote “The First Three Seconds: A Review of Possible Expansion Histories of the Early Universe” (arXiv:2006.16182 [astro-ph.co]). It focused on emergence, yet they were unable to define the first minutes of our universe.

III. Quantum fluctuations and quantum physics. The study of a geometry of quantum fluctuations over the years has been limited.^{[11]} A most-simple place to start is to recognize the tetrahedral gap that Aristotle did not see.^{[12]} That gap is real. And, there are others that are ignored. Add to it our octahedral gap and then the icosahedral gaps.^{[13]} Here are possibilities for fluctuations. Big bang cosmology has no simple geometry and certainly no geometry for chaos and indeterminacy. Our mathematically-integrated view of the Universe begins with the inherent geometries of doublings that perfectly tile and tessellate; plus, it has a geometry for fluctuations that is inherently indeterminant. One of our primary studies is to grasp the varieties of gaps and the probabilities for each manifestation within a given notation.

IV. A natural inflation and dark energy and dark matter. Dark matter and dark energy are below all possible thresholds of direct measurement. The indirect effects have been well-regarded since the work of Lord Kelvin (1884), Jacobus Kapteyn (1922), Jan Oort (1932) and Fritz Zwicky (1933). That mystery prevails.^{[14]} Big bang cosmology blocks the view of both the geometries of perfection and of imperfection. Initially there are no gaps as spacetime is dynamically created. Within this mathematically-integrated view of the Universe, a natural inflation starts with the most simple infinitesimal sphere, sphere stacking, and cubic-close packing of equal spheres. Tetrahedrons and octahedrons naturally and smoothly tile and tessellate the earliest universe.

If taken as a given that one infinitesimal sphere is generated per unit of length and time, the rate would range from 539-to-4609 tredecillion spheres per second.^{[15]} Of course, that range is established by Planck Time and Stoney Time.

V. Finite-infinite, quantitative-qualitative. Big bang cosmology has no finite-infinite relation and makes no distinction between the quantitative and the qualitative. It does not posit the smallest physical unit of spacetime. It makes no statements about pi (π), nor recognizes the essential faces of pi (π): continuity (ordering), symmetry (relating) and harmony (making dynamic).^{[16]} Naturally, the ubiquitous dimensionless constant that generates the natural units for Planck and Stoney is pi (π). We hypothesize that these three faces of pi (π) define the qualitative and the infinite. Within big bang cosmology the Planck Epoch and a singularity are the baseline whereas within our mathematically-integrated view of the Universe, it is the simple perfections of the infinitesimal sphere, then spheres, and then tetrahedrons and octahedrons.

[1] Smooth to lumpy. An early article, The Lumpy Universe, by a group within NASA’s Goddard Space Flight Center is written for students from 14 years old and up. The Imagine Team is an effort headed by Dr. Barbara Mattson and the segment, Imagine the Universe, addresses the current concerns within cosmology and astrophysics. It raises those issues that are pushing the scientific-academic community to re-imagine big bang cosmology.

[2] Systemic gaps and fluctuations. Too dense and too fast to accommodate fluctuations, it is hypothesized that quantum fluctuations do not initially manifest until much later than the first minute which is within Notation-144. It is also hypothesized that these gaps, once systemic, will work back into prior notations and are currently at least within Notation-67. Further, it is hypothesized that there is also a geometry of gaps for consciousness and the mind.

[3] Tile and tessellate this universe with no gaps. Throughout mathematical and geometric history, tilings and tessellations have challenged the human mind. Kepler was the first to make it a systematic study (Harmonices Mundi). The simplest 3D tessellationis with the tetrahedron and octahedron. Within this model it shapes the first instant of the universe resulting first in a sphere, then a systematic expansion of spheres (see #15 below), sphere stacking, and then what is known as close-cubic packing of equal spheres. Here is the starting point within Notation-0.

[4] Notations 1-202: This chart of 202 notations began in 2011. It became a horizontally-scrolled chart in 2016. It may take several generations to fill out even the first level of notations. It may well take many more generations to tie all the notations together in a dynamic unit. Notwithstanding, elementary school children can (and have) effectively engaged the chart to see the order within our universe.

[5] Sean Carroll. He is listed among the scholars to whom we turn for insight and help. Based on the videos we have watched and from his writings, we readily speculate that Sean is a bit like Steven Weinberg. When Weinberg got impatient with a person with whom he disagreed, he would say, “Utter nonsense.” I suspect the response from Sean Carroll would be similar. Yet, we’ll continue to write to him!

[6] James Peebles. Of course, most Nobel laureates are swamped with correspondence. There are just too many people hoping to grab their attention. James Peebles is older now and still very active looking at astronomical phenomena that are “off the beaten track.” In that light, our strategy has been to catch him by surprise in hopes that he might rethink the concepts of space-time-and-infinity with us.

[7] Max Planck. There are current challenges to Max Planck’s base units yet the results of those challenges are still being evaluated. We are following the work of John Ralston, Espen Gaarder Haug, and Tim Palmer as each impacts our thinking about the Planck constant, the Planck base units, and the dimensionless constants.

[9] Redefining Time. The most disconcerting notion to emerge from this work is to discover that in some manner of speaking all time is now. Although physicists like Carlo Rovelli, Max Tegmark, and Julian Barbour, claim time is an illusion, within the 202 notations linear time is associated with just that period of time experienced prior to sleep (when it is recompile into the whole). We say that time is exponential; time is an equation and all notations are always active.

[11] Quantum fluctuations. A search on the words, “geometry of quantum fluctuations,” renders just ten results on October 5, 2022. There has been no geometry of fluctuations based on simple geometries of gaps. Our page, geometries, is our announcement of our intention to pursue this domain of study beginning with the geometries of gaps within the most simple structures.

[12] Aristotle’s blindspot. This history serves to remind us to be somewhat reserved about our conclusions. Life is short, but our words may well live on. Aristotle’s arrogance is all of us. Aristotle’s mistake is profoundly all of us.

[13] Gaps. The initial study of the five-octahedral gap was part of the /geometries/ efforts in May 2022 when we announced our intention to pursue this domain of study. We were even able to create a model of what we believe is the first five-octahedral gap — 81018.com/2022/05/19/five/ — that then coupled with the tetrahedral gaps — 81018.com/gap/ — becomes a new object — 81018.com/15-1/ — which has been tentatively named, triantahedron, for a thirty-sided object. We’ll see if it works.

[15] 539-to-4609 tredecillion spheres per second. When I surveyed the initial formula in September 2021, it seemed simple and logical enough. Yet, to begin to imagine the emergence of 539 tredecillion spheres per second is simply beyond my grasp. This is the opening of a new domain of study. The range is a placeholder for more refined figures to come.

[16] Continuity-symmetry-harmony. The heart of this study began with simple geometries but backed into the infinitesimal sphere by backing into the very nature of pi (π) and finding the three faces of a perfected state in spacetime, continuity, symmetry, and harmony, all deeply inherent within pi. This is our essential story and it will be told and retold many more times to come.

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References & Resources As these references are studied, key references and resources will be added.

• Reason to believe by Rod Stewart, Lyrics by Tim Hardin

If I listened long enough to you I’d find a way to believe that it’s all true Knowing that you lied straight-faced while I cried Still, I look to find a reason to believe

Someone like you makes it hard to live without Somebody else Someone like you makes it easy to give Never think about myself

If I gave you time to change my mind I’d try to leave all the past behind Knowing that you lied straight-faced while I cried Still, I look to find a reason to believe

Please note: These are all the lyrics; two paragraphs repeat. Life is fair; it breaks everyone’s heart be it with love or beliefs (ideas and concepts).

• Diego Rapoport, The Geometry of Quantum Fluctuations II (Quantum Gravity and Ergodicity), in “Proceedings of the International Workshops on the Frontiers of Mathematics, Physics and Biology”, Monteroduni, Italy, August 1995 vol. 2, G. Tsagas (ed.), Hadronic Press and Ukraine Academy of Sciences, 1996

• Two Questions: (1) Is the smoothness of the JWST’s earliest universe a cause for concern? It seems some are concerned that it cannot be supported by big bang cosmology. Sean Carroll says, “…dark energy made the universe smooth out and accelerate, but it didn’t stick around for long.” (2) Is the Guth-defined inflation cause for concern? Most of the old guard like ‘t Hooft, seem fine with the one-time exception that inflation was an expansion of space itself so extralogical assumptions apply.

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Emails This is a representative selection of our notes to scholars asking about the substance of this article.

Editor’s note: Continuity is the first principle of order. The second is symmetry whereby a relation is created. The third is harmony, where there are multiple symmetries within a dynamic moment. We need all three, but continuity will at least get us on a level playing field.

Just as an aside, it appears that the adoption cycles of a major idea require no less than three generations and quantum theory is no exception. New theories have new truths but are not absolute truth and not “all truth for all time.”

S. Carlip, Department of Physics, University of California, Davis, CA 95616, USA R. A. Mosna and J. P. M. Pitelli, Departamento de Matematica Aplicada, Universidade Estadual de Campinas, 13083-859, Campinas, Sao Paulo, Brazil

First email: Sunday, August 1, 2022

TO: Joao Paulo Manoel Pitelli cc: Ricardo A. Mosna, Steve Carlip

Gentlemen:

Your work — Carlip–Mosna–Pitelli — regarding geometric fluctuations has come to my attention. There are not too many articles that have geometry and quantum fluctuations in the same sentence. So, very quickly, I saved it out so I could read it at my leisure and study all your references.

When Patricio got his PhD, I went back into a business that I had started six years earlier (so my background within academia is incomplete). I returned to my earlier work quite by accident when helping a nephew by taking his geometry classes for a few days. That was back in 2011. We were having fun with embedded geometries when we rather unwittingly uncovered the fact that there are just 202 base-2 notations from the Planck scale to the current time (and size of the universe). We thought it was a good STEM tool. For years, the first 64 notations up to particle physics eluded us. We could not imagine what was there. Then, we learned a little about Langlands programs and I returned to memories of late night discussions about string theory with Patricio. More recently I uncovered an octahedral gap commensurate with the five tetrahedral gap. Together they struck me as a possible gate in quantum computing. I also began thinking about transitions to non-Gaussianity within those first 64 notations.

I fully agree that our work is entirely odd, a wiffle ball coming out of left field. But I thought you’d be interested to see this page about that it: https://81018.com/geometries/ Of course, I would be most fascinated with your initial comments, no matter how harsh or direct you’d like to be!

Theoretical physics and theoretical geometries: Squishy or quantum geometries necessarily involve the five-tetrahedral gap or the five-octahedral gap or both together. It is a dynamic interface between perfected geometries with no gaps and the Standard Model for Particle Physics. Although just a conjecture or a speculative hunch, this domain is possibly best defined by Notation-50 though to Notation-80 where the Periodic Table begins. Further conjectured is that the physical phenomena at distance scales based on the Planck length define perfected or “no-gap” geometries in a range between Notation-1 and Notation-50. It is believed that quantum fluctuations may readily surge down into Notations 50-to-30 and that a perfected state can surge all the way up into Notation-202. For more, see: • Numbers, geometries, equations • From Perfected States to Gaps & Fluctuations • Icosahedrons • Started: June 13, 2022. Much more to come…

Geometric or Pentagonal Gap: The first little-known gap that we studied was created by five tetrahedrons. Beginning in 2016, it has been a key part of our analysis of numbers, geometries, chaos, quantum fluctuations, and human will. Is it possible that this little gap could be the basis for diversity, creativity, openness, indeterminism, and uniqueness? [1] In 2022 we also began to consider the pentagonal gap uncovered by five octahedrons sharing a centerpoint and each octahedron sharing at least one face. That gap was hidden by the geometry construction sets and computer programs. There are also the icosahedronal gaps, and the Pentakis dodecahedral gaps that are extensions of the five-tetrahedrons.

History: Aristotle had it wrong [2]; he claimed that he could perfectly tessellate the universe with tetrahedrons. [3] The mistake was not identified for over 1800 years. However, with five perfectly-shaped tetrahedrons, the imperfection is easily observed. That gap is a seminal. Tetrahedrons have four vertices. Octahedrons have six. Five regular tetrahedrons have seven or eight vertices. Cubes have eight. This progression is important.

The gap appears to be transcendental, non-repeating, and never-ending.

Of course, together with a ratio of two tetrahedrons to one octahedron, you can create a whole, ordered, rational, and perfect object that will perfectly tile and tessellate the entire universe.

The indeterminate and chaotic reside somewhere deep within the structure of the universe. We believe that place just may begin right here with the “Pentastar Gap.”

Here may well be the basis for broken symmetries. Of course, for many readers, this will be quite a stretch. That’s okay. For more, we’ll study chaotic maps, the classification of discontinuities, imperfections, and quantum fluctuations. More…

The Pentastar logo by Chrysler has a fan club! Now that Fiat’s involved, the Pentastar logo is being flushed out of their branding. There is a group of over 13,000+ advocating that they continue using it. I suggested in a note to them, “Chrysler made it an artform! The gap is the beginning of quantum fluctuations. It is the beginning of the geometry of living.”

[1]Numbers: The first analysis within this website of the pentastar gap was done in on January 8, 2016 within an articles, “Numbers: Creating Our Universe From Scratch.” The subject was re-engaged on July 4, 2018 within an article on scientific revolutions.

Right in the top level expression of the institute, there is a special challenge. On their website, it proudly announces to the first time visitor, that here they are “Solving the universe’s greatest mysteries, from quantum to cosmos.”

In 1899 Max Planck defined the meaning of quantum; he was mathematical. He used dimensionless constants. Although the Planck constant, according to some, may be mistaken, the four Planck base units — length, mass, charge, and time — are part of the ISO standards today. Where the Perimeter Institute fails is that their work has a very limited understanding of the domain between the beginning of the Planck scale at Notation-0 and Notation-1, and the measurement of quantum fluctuations at Notation-67 (perhaps down to Notation-64).

In all fairness, most of the scientific and scholarly community ignore those first 64 notations, yet they still may qualify as quantum physics. My intuition is telling me that the earliest of those 64 notations may be qualified as “hypostatic physics” that actually defines a type of perfection, i.e. there is perfect continuity, symmetry and harmony, and that “quantum physics” will be reserved for the imperfect. Possible.

By the way, my first encounter with the work of the Perimeter Institute was their 2016 conference on the nature of time, Time in Cosmology, June 27-29. Then, I got to learn about Neil Turok.

The Finite-Infinite relations… if we only could agree that the infinite is that same perfection that is found within the sphere and not bring emotional, colorful, and historic discussions to bear on it, more progress might be made.

There is much more to come… of course, the radical, idiosyncratic assertion here is that there is a domain of perfection within the earliest notations simply because compression, compactification and densities are so heighten, it is easier to have that perfection than to have fluctuations and differences. So, indeed, there is much more to come.