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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Five-tetrahedrons, five-octahedrons, five tetrahedrons and their gap

It didn’t have a name, so we named it!* These fifteen objects are fundamentally defined by the 7.35610+ degree gap.

Is it an 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 because 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 even more important. 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 of quantum fluctuations, the simplicity of these gaps becomes increasingly important. This image increasingly looks like it is analogous to a computer’s logic gate, but perhaps it is really the other way around. -BEC


Endnotes & Footnotes

[*] Naming this object. In June 2022, we began asking the question, “Does it have a name?” Nobody seemed to know. On November 22, stopping for lunch in the little town of Unadilla, Georgia at a little Subway sandwich shoppe, I was telling Hattie how no scholars had a name for the object. She asked, “How many sides does it have?” As you can see, it is not a straight-forward answer. Thinking about our squishy geometry where the gap could be eliminated, I said, “20.” I should have said, 30, trianta. On February 1, 2023, it became informally known as the triantahedron. It is defined by fifteen objects (five octahedrons and ten tetrahedrons, and by the 7.35610+ degree gap (creating two more faces, but we closed it up for this naming). More…

[1] Email. An email to Oxford scholar, 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. There is also a page for Brian Cox.

[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).


References & Resources

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. PittsburghA 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).

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

•  Nikolay Bogolyubov and John George Valatin transformation: quantum harmonic oscillator



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:


You are always invited.

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


Keys to this page, 15-2

• 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
• 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.