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Possible infinitesimal building block of the Universe

by Bruce E. Camber (between particle physics and the Planck scale)

**Introduction**. Because I could not find references on the web to this five-octahedral gap (with the five-tetrahedral gap above and below it), on June 16, 2022, I sent a quick note to Jonathan Doye, Professor of Theoretical Chemistry at Oxford.^{1} I asked, “Are there studies of this five-octahedral gap? I’m not finding any articles about it.”

In May 2022, I had started my initial analysis of five-octahedral gap.^{2}

**Five-octahedra around a centerpoint create a gap**. Among all the living scholars of the world, Jonathan Doye would know if it had been documented. My guess — it had escaped scrutiny. Geometric computer programs do not account for it, nor do the geometric construction kits like Zometool.

Scholars have made a big deal about Aristotle’s failure to see the five-tetrahedral gap. Perhaps the five-octahedral gap is an even a bigger deal! It seems we all have failed to see the octahedral gap and its possible working relation with all other gaps including the twenty-tetrahedrons to make an icosahedron and sixty-tetrahedrons to make the Pentakis Dodecahedron. It does not appear that there has been a scholarly discussion of the possible application of those gaps within quantum physics, particularly for quantum fluctuations.

Scholars are busy and it is vacation time in England, so I have continued to ask others questions about this object. On July 5, I asked a few of those whose work I was reading. The short list included June Huh of Princeton University, Brian Cox of BBC-Science and Royal Society (a tweet), and Maryna Viazovska of the Institute of Mathematics (EPFL) of Lausanne, Switzerland. I’m always reading the works of younger scholars so also sent notes to Oliver Janzer of Zurich’s ETH and Julia Collins, School of Science, Edith Cowan University in Australia.^{3}

On July 6, 2022 I turned my attention to scholars at the** **Institute for Advanced Studies. There would be many who would quickly know if these gaps are significant. There are people like Robert Langlands, Ed Witten, Karen Uhlenbeck, and Nathan Seiberg. Then, I began thinking of the work of Jacob Lurie and Natalie Paquette, younger scholars associated with IAS and among the thought leaders of our time.^{4}

Similar emails went to mathematicians (geometers) at Harvard, MIT, Brown, Chicago, Stanford, Texas, and Berkeley. I will continue this list — there will be many more — on a separate page. Although based on a person’s work, there will be certain themes. For example, this note to Stanley Deser of Brandeis (Massachusetts) is a result of reading Deser’s micro article, *Before Before*, in *Inference* (2022) so the focus shifts to the very start of the universe, the first instants.

What do you think? Thanks. -BEC

PS. When associated with a potential infinitesimal geometry, particularly a geometry quantum fluctuations, the simplicity of these gaps becomes increasingly important. This image increasingly looks like it is analogous to a computer gate, but perhaps it is really the other way around. -BEC

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**[1] Email**. An email to Jonathan Doye, June 16, 2022

**[2] May 2022**. Our initial step in more formally considering gap geometries and the nature of quantum fluctuations.

**[3] Scholars**. To date, Huh, Viazovska, Janzer and Collins have a reference page on this site. A page for Cox will be forthcoming.

**[4] IAS Scholars**. These are the best of the best, but it doesn’t make them infallible. If their starting points are off, particularly if they hold onto a Hawking-like big bang cosmology, they’ll not have any conceptual space to start at the very beginning, the first instance of spacetime. To acknowledge the 64 notations prior to fluctuations, particles, and waves will require a bit of introspection and perhaps some levity and even joy (relief).

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

• Inner and Outer Sphere Mechanisms and Charge Transfer Reactions, UC Davis, 2022

• Thomas Hales, Univ. Pittsburgh: A formal proof of the Kepler conjecture (Jan. 2015) (PDF)

• John Harrison: Complete publications list

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

• Edward N. Zalta, Principia Logico-Metaphysica, Stanford University, with Daniel Kirchner, Freie Universität Berlin and Uri Nodelman, Stanford University. May 5, 2022

http://mally.stanford.edu/principia.pdf

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

in light of this work. Emails this week include to the following:

• Prof. Dr. Simon Plouffe, Quebec and France, July 17, 2022

• John J. Harrison, Cambridge and Amazon Web Services, July 11, 2022

• Prof. Dr. Vladimir Drinfeld, University of Chicago, July 7, 2022

• Prof. Dr. William Goldman, University of Maryland, July 7, 2022

• Dr. Toyin Alli, University of Georgia, July 7, 2022

• Prof. Dr. Thomas Banchoff, Brown University, July 7, 2022

• Prof. Dr. W. Hugh Woodin, Harvard University, July 7, 2022

• Dr. Boris Lishak, University of Sydney, July 7, 2022

• Prof. Dr. Jeanetta Jackson, Tennessee State University, July 7, 2022

• Dr. Heidi Schweingruber, National Academy of Science, July 7, 2022

• Prof. Dr. Noam Elkies, Harvard University, July 7, 2022

• Prof. Dr. Dan Knopf, University of Texas, Austin, July 7, 2022

• Prof. Dr. Deidra Coleman, Wofford College, July 7, 2022

• Prof. Dr. Salvatore Torquato, Princeton University, July 6, 2022

• Dr. Julia Collins, ECU, Joondalup, Australia, July 5, 2022

• Dr. Oliver Janzer, ETH Zurich, July 5, 2022

There will also be many instant messages to thought leaders about basic geometry.

7:17 PM · Jul 10, 2022 @maanow Virtual components should always be an option and encouraged in every way so informal meetings happen all the time. I’d love to have a study group of our STEM tool, 202 base-2 notations from the Planck base units at the start to today: https://81018.com/stem/

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Participate

You are always invited.

- Might this image be telling us something about quantum fluctuations?
- From notation 1 to notation 202, where might if manifest in the physical world?
- Might attractors and repellers, as well as dimensionless constants, be part of the dynamics?

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• This page will eventually become a homepage. It is still “under construction” even today.

• The last update was July 20, 2022.

• This page was initiated on June 29, 2022.

• The URL for this file is https://81018.com/15-2/

• The headline for this article: Maybe a basic building block of the Universe.

• First byline is: Please consider this gap into the centerpoint created by five tetrahedra on the top, five octahedra in the middle, and five tetrahedra on the bottom.

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