A potential geometry for quantum fluctuations starts with Aristotle.

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Gaps, Fluctuations and Perfected States
by Bruce E. Camber (a working first draft)

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, quantum geometry. 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.

The Pentakis dodecahedronal gaps are also considered.

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.


Visualizations: Geometric software systems and geometric construction kits.3
We are just now starting to learn about the most popular interactive geometry software (IGS) and their dynamic geometry environments (DGEs) that are created. Wikipedia has a working summary of over twenty competitive programs. We are learning that this object on the right with its gap is quite possibly a first!

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.


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. Kasper5 (circa 1957) and more recently by scholars like John Conway, Salvatore Torquato, and Jonathan Doye.6

The first column (pictured) from our 2014 desktop version of the original chart opened our first discussion about Langlands programs. Our question, “Can Langlands programs work within 202 base-2 notations that are defined by continuity-symmetry-harmony? Can it accommodate these gaps?”

These questions are asked of string-and-M theory, SUSY, CDT, CST, LQG, SSM, SFT and hypothetical particles. I believe all of this work can be located on this grid of 64 notations. And, in their midst will also be quantum fluctuations and the gaps.7

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.


Endnotes & Footnotes __

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.

* Mysteries in Packing Regular Tetrahedra, Jeffrey Lagarias, Chaunming Zong, (PDF), AMS, 2012
A domain of perfection or https://81018.com/perfection/

[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.

[4] Key Dimensionless Constants. In 2011 when we started, John Baez had identified 26 constants necessary for our study of the Standard Model of Particle Physics. As we researched further we found the 2006 work of Wilczek, Tegmark, Rees, and Aguirre where they had identified 31 constants in search of a theory of everything (TOE). Wikipedia identified 104 dimensionless constants. The National Institute for Standards & Technology (NIST) had identified over 300 (PDF – 2008, SI Report). Most interestingly, a Canadian (Quebec) geometer, Simon Plouffe, has identified, through algorithmic programming, 215 million mathematical constants (as of August 2017) which includes pi, Euler’s number (I am still chasing claims about more11.3 billion, computer-generated, mathematical constants). The Standard Model for Particle Physics (Wikipedia) is well known. Its weaknesses are, too. The first 64 notations open a huge infrastructure whereby the model can become a continuum from the Planck scale to the standard cosmological model.

[5] Frederick Charles Frank and John Simon Kasper. Frank, F. C.; Kasper, J. S. (1958-03-10). “Complex alloy structures regarded as sphere packings. I. Definitions and basic principles”Acta CrystallographicaInternational Union of Crystallography (IUCr). 11 (3): 184–190. doi:10.1107/s0365110x58000487ISSN 0365-110X. Also see: Frank–Kasper phases retrieved from Wikipedia on June 24, 2022.

[6] Jonathan P.K. Doye. Jonathan P. K. Doye, A model metal potential exhibiting polytetrahedral clusters, https://doi.org/10.48550/arXiv.cond-mat/0301374, 2003

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 programsstring-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.


References & Resources

Okuma, R., Kofu, M., Asai, S. et al. Dimensional reduction by geometrical frustration in a cubic antiferromagnet composed of tetrahedral clustersNature Communications, 12, 4382 (2021). https://doi.org/10.1038/s41467-021-24636-1, , (e)

• Quantum fluctuating geometries and the information paradox (PDF), Rodolfo Gambini, Rodrigo Eyheralde, Miguel Campiglia, Jorge Pullin, 2017

• What every physicist should know about string theory, Ed Witten, Physics Today, 68, 11, 38 (2015)https://doi.org/10.1063/PT.3.2980

•  John Willard Milnor (1985). “On the concept of attractor”. Communications in Mathematical Physics99 (2): 177–195. doi:10.1007/BF01212280S2CID 120688149.

•  John Willard Milnor is one of the five mathematicians to have won the Fields Medal, the Wolf Prize, and the Abel Prize (the others being SerreThompsonDeligne, and Margulis.)

•  Bell’s Theorem, Non-Computability and Conformal Cyclic Cosmology: A Top-Down Approach to Quantum Gravity, T.N. Palmer, arXiv:2108.10902, August 2021



• Dharam Vir Ahluwalia, Center for the Studies of the Glass Bead Game, June 7, 2022
Johannes Buchner, June 5, 2022
• David J. Gross, Kavli Institute, University of California at Santa Barbara, June 5, 2022
• Espen Gaarder HaugNorwegian University of Life Sciences (NMBU), May 24, 202
• National Science Foundation, Washington, DC, May 31, 2022
• Monika Schleier-Smith, Stanford University, May 30, 2022
Matthew J. Strassler, June 6, 2022
• Paola Zizzi, University of Padua, June 5, 2022 @ 4:24 PM

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/

My study of that gap is here: https://81018.com/geometries/

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



8:08 AM · Jun 9, 2022 @7homaslin @nattyover Congrats. You all have been doing sensational work for ten years. Natalie has been excellent, but many others are as well. I’ve written a letter to you here — https://81018.com/quanta/#Lin The top of that page is my tribute to the magazine and to Simons people.

3:00 PM · Jun 6, 2022, @MattStrassler Can you help us unfold this base-2 chart of the universe: https://81018.com/chart/ The current homepage is always my latest struggle with it all: https://81018.com/. PS. Note that at the Center for the Fundamental Laws of Nature, we have begun following the work of Andrew Strominger, Cumrun Vafa and you. Perhaps you all can help Lisa Randall and Howard Georgi.

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.



• This page became the homepage on June 14, 2022.
• This page was initiated on Friday, May 6, 2022 at 7:34 PM, National Space Day.
• The last update was Saturday, July 8, 2022.
• The URL for this file is https://81018.com/geometries/
• Related URLs include: https://81018.com/idea/ https://81018.com/editors/
• The headline for this article: From Perfected States to Gaps & Fluctuations
• First byline is: The Geometries of Quantum Fluctuations: Re-visiting Aristotle and Others.
• The current byline is: A possible geometry of quantum fluctuations starting with Aristotle.

Future work:
3.3 inches (8.39 cm): 1.2980×1033 units (Notation 112) Read: “1.298 decillion touchpoints”
6.6 inches (16.7835 cm): 2.60×1033 units (Notation 113)
13.22 inches (33.56 cm): 5.1922×1033 units (Notation 114)
26.43 inches (67.1343 cm): 1.03845×1034 units (Notation 115) Read: “10.38 decillion touchpoints”
52.75591 inches(1.34 meters) 2.08×1034 units (Notation 116)
105.72 inches (2.68 meters or 8.8 feet): 4.15×1034 units (Notation 117)