TO: Chuanming Zong, Tianjin Center for Applied Mathematics (TCAM)
Tianjin, China
FM: Bruce E. Camber
RE: Your articles, Can You Pave the Plane with Identical Tiles? (PDF), AMS (2020); Mysteries in Packing Regular Tetrahedra with Jeffrey Lagarias, (PDF), AMS (2012); The kissing number, blocking number and covering number of a convex body, in Goodman, Pach, Pollack (eds.), Surveys on Discrete and Computational Geometry: Twenty Years Later (AMS-IMS-SIAM Joint Summer Research Conference, June 2006, Snowbird, Utah), Contemporary Mathematics, 453, Providence, RI: American Mathematical Society, pp. 529–548, doi:10.1090/conm/453/08812 (2008) as well as your articles in ArXiv (19) such as On Lattice Coverings by Simplices, 2015 (PDF); your awards such as 2015 AMS Levi L. Conant Prize; your books such as Sphere packings, Springer, 1999, and The Cube-A Window to Convex and Discrete Geometry, 2009. Also your homepage including those on Tianjin, Mathematics Genealogy Project, ResearchGate, X-twitter dblp, Google Scholar, inspireHEP and Wikipedia, i.e. Keller’s conjecture, H. F. Blichfeldt, Kissing Number
References within this website to your work: 6 September 2024: https://81018.com/too-simple/
January 14, 2021: https://81018.com/precis/#Zong______________ May 26, 2020: https://81018.com/biased/#R3-2
May 5, 2020: https://81018.com/biased/#Aristotle _____________._April 2020: https://81018.com/fqxi-aristotle/
____________ https://81018.com/biased/#1b ____________________. March 2020: https://81018.com/imperfection/
October 2018: https://81018.com/realization6/ __________.__. ___January 2016: https://81018.com/number/#En7
Most recent email: 13 May 2026
Attachment: Geometric Origin of the Fine-Structure Constant – A Two-Page Summary
Dear Professor Chuanming Zong,
Your 2012 paper with Jeffrey Lagarias — ‘Mysteries in Packing Regular Tetrahedra’ — is a foundational reference in the attached work, and I am writing to ask whether the mathematical observation at its center is of interest to you.
The observation is this: in a discrete base-2 framework anchored at the Planck scale, the proton charge radius falls at almost exactly Notation 65.496 — the geometric mean of two consecutive doubling scales, meaning ℓ_P × 2^65.5 ≈ 0.843 fm, within current measurement uncertainty. The classical electron radius falls at Notation 67.24. From there, approximately 70 doublings reach Notation 137, the integer nearest to α⁻¹ ≈ 137.036.
The proposed mechanism is the angular deficit that you and Professor Zong documented so clearly: δ = 2π − 5arccos(1/3) ≈ 7.356°. In our framework this gap is active at every notation from Notation 4 onward — it does not freeze in but propagates continuously, producing what we conjecture is a resonant minimum in accumulated geometric tension at Notation 137.
The central open problem is a rigorous derivation connecting δ to α⁻¹ through the base-2 doubling structure. We are not claiming that derivation exists; we are claiming the observation is precise enough to warrant either finding it or demonstrating conclusively why it cannot exist.
The two-page summary is attached. The full paper is at https://81018.com/geometric-origins-137/
I would be grateful for your mathematical assessment of whether the Notation 65.496 result — the proton at the geometric mean of two consecutive notations — is the kind of observation your work on tetrahedral packing would lead you to take seriously, or whether you see an immediate reason it must be coincidental.
With great respect for your work,
Bruce
Sixth email: 6 September 2024
Dear Prof. Dr. Chuanming Zong,
I’m writing about Aristotle’s Gap and, of course, your work with Jeffrey Lagarias is mentioned. Is my brief summary a fair summation; your judgment is what counts:
“Enter Lagarias and Zong. Neither Lagarias nor Zong expressed an interest in speculating about the nature of the gap and how it might manifest as a physical phenomenon. In December 2022 I wrote to Martin Bridson (Oxford mathematician and head of the Clay Institute of Mathematics) to ask his opinion.[6]
If you would like me to edit that in any way, I am happy to do so.
Warm regards,
Bruce
Fifth email: 26 April 2024 (updated)
Dear Prof. Dr. Chuanming Zong,
Although I recently sent you a note, I hope you do not object to another note so soon after that.
My first email to you was over ten years ago. And, I so appreciate the work you’ve done. My work is so idiosyncratic by comparison! Yet, as a mathematician, at least you know my direction. If you have any ideas or thoughts about fine-tuning this work, please do not hesitate to be critical of it. I am trying hard to.find critical feedback. Thank you.
Warmly,
Bruce
Fourth email: 7 March 2024
RE: Two quick questions
Dear Prof. Dr. Chuanming Zong,
- Have you or Prof. Lagarias ever considered publishing your AMS Mysteries paper within ArXiv?
- Would you help to write an article about the nature of the gap and how it manifests in spacetime? If five scholars were involved and each took a different look at the gaps – tetrahedral, octahedral, icosahedral and pentakis-dodecahedral. It may become a well-received, highly-quoted article.
I have been talking with the ArXiv folks about the gap and how it is under-analyzed by their authors.
I hope you have been well.
As you can imagine, I think the ramifications of all your work have implications that transcend nations and history.
Thank you.
Warmly,
Bruce
PS. Last summer we were in northern Italy and southern Switzerland climbing around their smaller mountains — https://81018.com/bec/ — with a picture of my wife and me at that time. -BEC
Third email: Wednesday, May 28, 2020
Dear Prof. Dr. Chuanming Zong:
First, let me congratulate you on your new location. Wonderful. It appears that you are still within 100 miles of Beijing. That’s excellent.
I am still quoting you after all these years (see above). Because the citations were getting so numerous, I created references page for you and Prof. J. Lagarias. My page for you: https://81018.com/2020/05/27/zong/
In these days and times, my most important conclusion is here about all our work, collectively and individually: https://81018.com/biased/#R3-2 Of course, if you would like anything changed, deleted, or added, I will be glad to accommodate your request. Thank you.
Warm regards,
Bruce
Second email: Wednesday, January 8, 2014
Your paper is sensational. It is exactly what I needed to be assured that Frank-Kaspers (1959) and others were not leading us astray (https://scripts.iucr.org/cgi-bin/paper?S0365110X58000487).
Your mathematics and analysis are spot on.
Let me share my reasons for my enthusiasm below this note to you. Thanks.
-Bruce
PS. Your work helps us with #2 and #4 below:
1. The universe is mathematically very small.
Using base-2 exponential notation from the Planck Length
to the Observable Universe, there are 202 base-2 notations,
steps or doublings. NASA’s Joe Kolecki and J.P Luminet
(Paris Observatory) helped us with the calculations. Our work began
in our high school geometry classes with a tetrahedron. We divided
the edges by 2, connected the new vertices and found the octahedron
in the middle and four tetrahedrons in each corner. Dividing the octahedron
we found the eight tetrahedron in each face and the six octahedron, one
in each corner. We kept going inside until we found the Planck Length.
Then it was easy to standardize the measurements by just multiplying
the Planck Length by 2. In 202 notations we go from the smallest
to the largest possible measurements of a length.
2. The very small scale universe is an amazingly complex place.
Assuming the Planck Length is a “singularity” of one vertex, we also
noted the expansion of vertices. By the 60th notation, of course, there are
over a quintillion vertices and at 61st notation well over 3 quintillion more
vertices. Yet, it must start most simply and here we believe the work
within cellular automaton and the principles of computational equivalence
could have a great impact. The mathematics of the most simple is being
done. We also believe A.N. Whitehead’s point-free geometries should
have applicability.
3. This little universe is readily tiled by the simplest structures.
The universe can be simply and readily tiled with the four hexagonal plates
within the octahedron and by the tetrahedral-octahedral-tetrahedral chains.
4. And, the universe is delightfully imperfect.
In 1959, Frank/Kaspers discerned the 7.38 degree gap with a simple
construction of five tetrahedrons (seven vertices) looking a lot like
the Chrysler logo. We have several icosahedron models made with
20 tetrahedrons. We call it, squishy geometry. We also call it
quantum geometry (in our high school).
Perhaps here is the opening to randomness.
5. The Planck Length as the next big thing.
Within computational automata we might just find the early rules
that generate the infrastructures for things. The fermion and proton
do not show up until the 66th notation or doubling.
I could go on, but let’s see if these statements are interesting
to you in any sense of the word. -BEC
First email: Fri, Aug 30, 2013, 7:19 PM
RE: Mysteries in Packing Regular Tetrahedra, Jeffrey C. Lagarias and Chuanming Zong, AMS Volume 59, Number 11, Dec. 2012
Just a terrific job. A wonderful read.
Thank you.
Coming up on two years now, we still do not know what to do with a simple little construct: https://81018.com/2014/05/21/propaedeutics/
That five-tetrahedral construct plays a key role.
Your work gives me a wider and deeper perspective. Thanks.
Warmly,
Bruce
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