TO: Salvatore Torquato, Lewis Bernard Professor of Natural Sciences, Princeton Institute for the Science and Technology of Materials, Princeton, New Jersey
FM: Bruce E. Camber
RE: Articles especially ; ArXiv (149) especially The structure factor of primes (2018), Hyperuniform States of Matter (2018); your CV (PDF); your Google Scholar and Group Homepage; your homepage and publications; your Wikipedia; and your YouTube videos, especially Hyperuniformity in many-particle systems and its generalizations
This page: https://81018.com/2014/03/10/salvatore/
Fifth email 28 December 2025
Dear Prof. Dr. Salvatore Torquato:
You know that your sphere-packing research is directly foundational to our framework. You know that I’ve been at it for over 14 years. It is high time to have your expertise to evaluate whether our most recent geometric claims have any merit.
The core observation: Starting from a single Planck-scale sphere and applying systematic base-2 doubling, specific gauge symmetries appear to emerge at notations where the sphere count, packing geometry, and observed symmetry dimensions converge:
- Notation 2-3: 4-8 spheres → tetrahedral/octahedral → SU(2) (3 generators)
- Notation 8: 256 spheres → FCC packing establishes → SU(3) (8 generators, “eightfold way”)
- Notation 24: 16.7M spheres at 10⁻²⁸ m → 24-dimensional necessity → SU(5) (24 generators at GUT scale)
The question for you: Does FCC packing at 2⁸ = 256 spheres genuinely produce eight-fold symmetry patterns that could template the eight gluons of QCD? Or is this numerology?
Your work on optimal packing densities and the Kepler conjecture proof gives you unique authority to evaluate whether sphere-packing geometry at specific exponential scales can force dimensional symmetries.
Framework documentation:
- Geometric claims: https://81018.com/gauge-symmetries/
- Notation-by-notation analysis: https://81018.com/notations-0-24/
- Complete 202-step map: https://81018.com/base-2-map/
What I’m requesting:
- Your assessment of whether the sphere-packing geometry supports or contradicts these claims
- If this speculative geometry is even somewhat sound, would you consider co-authoring a paper focusing on the geometric foundations?
- Alternatively, would you endorse an arXiv submission so the physics community can evaluate it?
I’ve corresponded with 500+ physicists since 2011, but today you’re the only sphere-packing specialist I’m approaching. That’s oversight on my part, since this framework literally stands or falls on packing geometry.
I’m in Florida (not that far from Princeton) and would be happy to discuss in person if that’s more efficient than email exchanges.
Thank you for considering this. Even if the conclusion is “interesting numerology but not physics,” that feedback from T.H.E. sphere-packing authority would be invaluable.
Warmest regards,
Bruce
Bruce E. Camber
camber@81018.com
https://81018.com
Fourth email: 7 June 2025
Dear Prof. Dr. Salvatore Torquato:
I think you might appreciate how Grok is toning me down to bring it all out very slowly:
- https://81018.com/qualitative-expansion
- https://81018.com/big-ideas/
- https://81018.com/paradigm-shift/
Related to it, you’re the first person I’ll ask: Could the four hexagonal plates intrinsic to the octahedron be the four primary irrational numbers, one plate per number, and act kind of like a ship’s stabilizers? Possible? https://81018.com/irrationals/
Now, I hope you are well and doing fine.
I wish you a magnificent summer. It seems that Eric Weisstein of Wolfram MathWorld successfully named the five tetrahedral gap after Aristotle. I’ve asked before, I know, might you have a name for the five-octahedral gap. Everybody missed it. Also, nobody has yet volunteered a name for the five-tetrahedral, five octahedral, five tetrahedral stack. It’s turtles all the way down: https://81018.com/15-2/
With a smile,
Bruce
Third email: 1 December 2022
Prof. Dr. Salvatore Torquato:
Yes, it is too simple. The five octahedral gap is overlooked. I have asked dozens of people now and everyone has been puzzled. All our computer graphics programs appear to ignore it or compensate for it. Even the construction kits like Zometool do not account for it. Isn’t that fascinating?
So, what’s next? Can you write about it and get a larger group discussing it?
If it is in any way related to quantum fluctuations — and with my models, there is a nervousness with those models — it’s significant. We can actually make those constructions dance and bounce all around!
Would you like to have a set of models made of the clear plastic that we use?
I’d be glad to send a few models to you so you can see-and-experience that “nervousness” to which I am referring. Thanks!
Warmly,
Bruce
Second email: 15 July 2022
Dear Prof. Dr. Salvatore Torquato:
I have not found references online to a five-octahedral gap much like the five-tetrahedral gap that Aristotle missed and, of course, you and John Conway did a major study of it. Have you studied or are you aware of any studies of the five-octahedral gap?
Here is a picture of both gaps together: https://81018.com/15-2/. It is much too simple, but for that reason perhaps it has been overlooked.
Thank you.
Warmly,
Bruce
PS. We’re making a study of that cluster of fifteen sharing a common centerpoint (with the hexagonals within each octahedron). It would make an interesting gate within circuitry of the infinitesimal. If we introduce the twenty-tetrahedral icosahedron in place of the five-tetrahedral cluster, its complexity and potential functionality increases exponentially.
PPS. I have also updated my working page of references to your work: https://81018.com/torquato/ As usual, any change, additions or deletions will be accommodated and any private correspondence from you will not be included unless requested or permission is granted to do so. Thanks again for all that you do. -BEC
First email: Mar 10, 2014, 8:54 PM
REFERENCES:
1. Thank you: http://www.pnas.org/content/108/27/11009.abstract?sid=a37de813-198f-4f81-9641-ad2025190fd7
2. Beautiful: http://chemlabs.princeton.edu/torquato/research/maximally-dense-packings/
3. Hypostatic Jammed Packings (2006): http://pi.math.cornell.edu/~connelly/Hypostatic.pdf
Dear Prof. Dr. Salvatore Torquato:
Thank you, thank you, thank you for your work (referenced just above).
Back in 2001 I spent a very pleasant day with John Conway but he did accuse me of being hung up on the relation between the tetrahedron and octahedron. For more I’ll copy in part of the story below. Though I am late to discover your July 5, 2011 paper, I was so glad to discover it today. It adds fuel to the fire and opened the door to your work.
I am so glad to meet you through your writings. I have already inserted references to your work in two articles (referenced below).
After spending a bit more time with your writing, may I call you?
Thank you.
Warmly,
Bruce
PS. I’ve been working with clear plastic models — made the molds and made thousands of octahedrons and tetrahedrons — to delve into the issues of fragmentation and wholeness. David Bohm’s book by that title, has a prominent place in my library.
Our references about which you might be interested:
http://81018.com/home/
https://81018.com/2014/05/21/propaedeutics/
Here is what I said about John Conway:
“An earlier history began with the study of perfected states in space time.
Sometime in the Spring of 2001, at Princeton with geometer, John Conway, the discussion focused on the work of David Bohm, once a physicist from Birkbeck College, University of London. “What is a point? What is a line? What is a plane vis-a-vis the triangle? What is a tetrahedron?” Bohm’s book, Fragmentation & Wholeness, raised key questions about the nature of structure and thought. It occurred to me that I did not know what was perfectly and most simply enclosed by the tetrahedron. What were its most simple number of internal parts? Of course, John Conway, was amused by my simplicity. We talked about the four tetrahedrons and the octahedron in the center.
“I said, ‘We all should know these things as easily as we know 2 times 2. The kids should be playing with tetrahedrons and octahedrons, not just blocks.’
“What is most simply and perfectly enclosed within the octahedron?” There are six octahedrons in each corner and the eight tetrahedrons within each face. Known by many, it was not in our geometry textbook. Professor Conway asked, “Now, why are you so hung up on the octahedron?” Of course, I was at the beginning of this discovery process, talking to a person who had studied and developed conceptual richness throughout his lifetime. I was taking baby steps, and was still surprised and delighted to find so much within both objects. Also, at that time I had asked thousands of professionals — teachers, including geometry teachers, architects, biologists, and chemists — and no one knew the answer that John Conway so easily articulated. It was not long thereafter that we began discovering communities of people in virtually every academic discipline who easily knew that answer and were shaping new discussions about facets of geometry we never imagined existed.
“Of course, I blamed myself for getting hung up on the two most simple structures… “You’re just too simple and easily get hung up on simple things.”
_____
http://utable.wordpress.com/2014/03/04/sbs/
Everything Starts Most Simply. Therefore, Might It Follow That The Planck Length Becomes The Next Big Thing?