Zong, Chuanming

Chuanming Zong

Tianjin Center for Applied Mathematics (TCAM)
Tianjin, China

Articles: Mysteries in Packing Regular Tetrahedra (PDF)
• “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
ArXiv (19): On Lattice Coverings by Simplices, 2015 (PDF)
Award: 2015 AMS Levi L. Conant Prize
Books: Sphere packings, Springer, 1999
The Cube-A Window to Convex and Discrete Geometry, 2009
Mathematics Genealogy Project
Wikipedia: Keller’s conjecture, H. F. Blichfeldt, Kissing Number

References within this website to your work:
May 26, 2020: https://81018.com/duped/#R3-2
May 5, 2020: https://81018.com/duped/#Aristotle
_______________ https://81018.com/duped/#1b
April 2020: https://81018.com/fqxi-aristotle/
March 2020: https://81018.com/imperfection/
October 2018: https://81018.com/realization6/
January 2016: https://81018.com/number/#En7

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/2020/05/28/zong/

In these days and times, my most important conclusion is here about all our work, collectively and individually: https://81018.com/duped/#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,


Second email: Wednesday, January 8, 2014

Your paper is sensational.
It is exactly what I needed to be assured that Frank-Kaspers
and many others were not leading us astray. 

Your mathematics and analysis are spot on.

Let me share my reasons for my enthusiasm below this note to you. Thanks.


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 somewhere over 202.34
and under 205.11 notations, steps or doublings.  NASA’s Joe Kolecki
helped us with the first calculation and JP Luminet (Paris Observatory)
with the second. Our  work began in our high school geometry
classes when we started with a tetrahedron and divided the edges
by 2 finding the octahedron in the  middle  and four tetrahedrons
in each corner.  Then dividing the octahedron we found
the eight tetrahedron in each face and the six octahedron
in each corner.  We kept going inside until we found the Planck Length.
We then multiplied by 2 out to the Observable Universe.  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 with its  20 tetrahedrons and call 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

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.




Georgi, Howard

GeorgiHoward Georgi

Mallinckrodt Professor of Physics
Harvard University
Leverett House, 28 DeWolfe St.
Cambridge, Massachusetts

Articles: Why Unify? (Nature, v.288, pages 649–651, 1980)
ArXiv (51): Unparticle Physics (May 2007) Wiki
Books: Lie Algebras In Particle Physics (Westview, 1999) (CRC Taylor & Francis, 2018) (PDF)
Wikipedia: Unparticle Physics

Most recent email: Tuesday, March 17, 2020 (Rewrite: 19 May 2016)

Dear Prof. Dr. Howard Georgi:

Your work on Unparticle Physics has finally come to my attention so my studies of your work are still quite young. I apologize in advance for my lack depth.

In and around 1979 John Wheeler sent me a copy of his booklet, The Frontiers of Time (PDF). Unfortunately, soon thereafter, I went back into a business that I had started nine years earlier.

I recently revisited Wheeler’s writings about quantum foam and simplicity. I would ask him today, “What about the Planck base units?” Might we consider Planck Time the first unit of time? Might we consider today, the Now, to be to be an endpoint that gives us the current estimated age of the universe between 13.81-to-14.1 billion years?

If we apply base-2 notation to that continuum, there are just 202 notations that encapsulate the universe. At one second (between Notation 143 and Notation 144) the Planck Length is within .001% of the distance light travels in a vacuum.

Throughout those 202 notations, there are many places to check the validity of the numbers, including the Planck Charge and Planck Mass doublings. There is a deep logic to it all. The first 64 notations are too small to be measured. The first doubling of the Planck Length that can be measured is within Notation-67. The first measurement of a unit of time that can be measured is the attosecond; it is within Notation-84.

Here is a domain, 1-64, for your unparticle physics, including Langlands programs, string theory, and loop quantum theory. If real, it has dimensionality and physicality that cannot be measured directly. Indirectly, it just may become part of the definition of dark energy and dark matter.

When we considered the look and feel of these unparticles, might an infinitesimal sphere at the Planck level be defined by the Fourier transform, Poincaré spheres, and cubic close packing of equal spheres? What are  our limitations within mathematics and physics?

All notations appear to be active, so time is surely redefined. It would appear that there is symmetry across all but the current notation.

I hope you will comment.  Thank you.

Most sincerely,


PS. Another recent attempt to describe all this ideation was for FQXi (Aguirre and Tegmark group):  https://81018.com/3u/

Long, long ago… I was a member of Harvard SDS ’64 (local high school student – recruited from an all-night teach-in at Memorial Hall), also a member of the Harvard Philomorphs with Arthur Loeb and Bucky Fuller, 1970-1973, and one of nine (1977) with Arthur McGill (HDS) on Austin Farrer’s Finite and Infinite.

Barker, William H.

William H. Barker

Bowdoin College
Brunswick, Maine

Books:  Continuous Symmetry: From Euclid to Klein (AMA, 2007)

________ Harmonic Analysis on Reductive Groups

(NOTE: A conference on Harmonic Analysis on Reductive Groups was held at Bowdoin College in Brunswick, Maine from July 31 to August 11, 1989. The stated goal of the conference was to explore recent advances in harmonic analysis on both real and p-adic groups. It was the first conference since the AMS Summer Sym­posium on Harmonic Analysis on Homogeneous Spaces, held at Williamstown, Massachusetts in 1972, to cover local harmonic analysis on reductive groups in such detail and to such an extent. While the Williamstown conference was longer (three weeks) and somewhat broader (nilpotent groups, solvable groups, as well as semisimple and reductive groups), the structure and timeliness of the two meetings was remarkably similar. The program of the Bowdoin Conference consisted of two parts. First, there were six major lecture series, each consisting of several talks addressing those topics in harmonic analysis on real and p-adic groups which were the focus of intensive research during the previous decade. These lectures began at an introductory level and advanced to the current state of research. Sec­ond, there was a series of single lectures in which the speakers presented an overview of their latest research.

_____ Lp harmonic analysis on SL(2,R)


Most recent email: Friday, 7 February 2020

Dear Prof. Dr. William H. Barker:

My work in 1972 focused on continuity, symmetry, and harmony. I was attempting to define what I thought would entail “a moment of perfection” within our quantum universe. By 1980, after working with an array of distinguished scholars in Boston, Cambridge (USA), and Paris, I went back to work within a business that I had started in 1971. From a little service bureau, we soon had a software business with well over 100 employees. My first opportunity to attempt to dig back into it all back was in 2011. I was helping a nephew with his high school geometry classes when we went inside the tetrahedron — https://81018.com/tot/ — and then its octahedron, step-by-step, deeper and deeper by dividing all the edges by 2 and connecting those new vertices. Within 45 steps we were within particle physics. In 67 additional steps, we were within the Planck scale. By multiplying those classroom objects by 2, in 90 steps we were out to the approximate age and size of the universe. Instead of base-10 like Kees Boeke (1957), we used base-2, we had an inherent geometry, and we went from the Planck units to the current time.

It was an unusual, albeit, rather idiosyncratic chart of 202 notations: https://81018.com/chart/

Prima facie, do you see any merit to such a chart?

I will continue my readings of your work, Continuous Symmetry: From Euclid to Klein (AMA, 2007) and Harmonic Analysis on Reductive Groups (Springer, 1991) in hopes that you might have some guiding thoughts for  this rather idiosyncratic chart of the universe.  Thank you.

Most sincerely,


PS. In 1746 our family settled in Bremen, Maine. Bowdoin had always been on my list of schools to consider, but in 1965 the call for voter registration in the South won the day.  I always think of you all on my way out of Freeport and as we go through Brunswick. -BEC

First email: August 2, 2016 3:20 PM

Dear Prof. Dr. William H. Barker:

My grandmother lived up the road a ways (Bremen…Damariscotta, then out to 1A and the coast). Often Dad would stop at Valerie’s in Ogunquit, my sister’s favorite restaurant; they shared the name. We’d fall quickly back to sleep as children for the final long slog up from Cambridge. For some magic reason, I would awake just as we were passing by Bowdoin. Bathed in the soft summer lights, I would secretly dream, “That’ll be my school.”

1965 came quickly and I marched off to the south to register voters, but Bowdoin always held that special place.

Today, I am delighted to find your book on continuous symmetries and remember my childhood once more. Images imprint the soul and make us who we are.

When and why is there spontaneous symmetry breaking?
Have you given it much thought?

So, I have discovered your work and I am grateful to now be taking a de facto course with you through your writing. And so I say, “Thank you!”

With warm regards,
Most sincerely,



How does one find your work:  https://en.wikipedia.org/wiki/Continuous_symmetry
In mathematics, continuous symmetry is an intuitive idea corresponding to the concept of viewing some symmetries as motions, as opposed to discrete symmetry, e.g. reflection symmetry, which is invariant under a kind of flip from one state to another.

The notion of continuous symmetry has largely and successfully been formalised in the mathematical notions of topological group, Lie group and group action. For most practical purposes continuous symmetry is modeled by a group action of a topological group.

One-parameter subgroups
The simplest motions follow a one-parameter subgroup of a Lie group, such as the Euclidean group of three-dimensional space. For example translation parallel to the x-axis by u units, as u varies, is a one-parameter group of motions. Rotation around the z-axis is also a one-parameter group.

Noether’s theorem
Continuous symmetry has a basic role in Noether’s theorem in theoretical physics, in the derivation of conservation laws from symmetry principles, specifically for continuous symmetries. The search for continuous symmetries only intensified with the further developments of quantum field theory.

See also:

References:  William H. Barker, Roger Howe, Continuous Symmetry: from Euclid to Klein (2007)


Engquist, Björn

Björn Engquist

The University of Texas at Austin
201 E. 24th Street, 1 University Station, Austin, Texas 78712-1229

ArXivNumerical methods for multiscale inverse problems (January 2014)
YouTube: Basis in Information Theory

References within this website:
This page is https://81018.com/2019/04/18/engquist/

First email: 18 April 2019

Dear Prof. Dr. Björn Engquist:

Not being a scholar or expert, I am still fascinated with your work to define the development of absorbing boundary conditions.

We began our work in a high school geometry class where we were observing how an octahedron is in the center of a tetrahedron with half-sized tetrahedrons in each corner. Within the octahedron, there is a half-sized octahedron in each of the six corners and a tetrahedron in each of the eight faces all sharing the centerpoint.

We decided to do a Zeno-like progression and applied base-2 going back deeper and deeper inside. In 45 steps we were in the range of particle physics, and in another 67 steps we were in the range of Planck’s base units.

We then decided to multiply by 2 and in 90 steps we were in the range of the age and size of the universe.

For high school people, it was great fun. We encapsulated the universe in 202 steps. We only then found Kees Boeke’s work and began thinking of the differences between base-10 and base-2.

I suspect that you are one of the few people on earth who has thought very deeply about computational multi-scale methods. Might you advise us? Are we being illogical? Are we doing something wrong? Thank you.

Most sincerely,

Links above:

Current research:

Click to access 6all.pdf


Click to access Engquist77.pdf

Nima Arkani-Hamed says, “Spacetime is doomed.”

From an October 2017 homepage:

Nima Arkani-Hamed1 of the Institute for Advanced Studies in Princeton, with a sweeping naturalness, proclaims “Spacetime is doomed2 (go to 5:45 minutes of 1.22.44). Pacing throughout3, his really-real reality is within his ever-so-illusive amplituhedron4 with a string to a planar N.5 He declares that it’s equal to the perturbative topological B-model string theory in twistor space6 (an ever-so-positive Grassmannian7) (related ArXiv article).

To which I reply, “Yes, of course. We knew that.”

1 The first link goes to our correspondence (mostly emails) to Nima Arkani-Hamed. He has also received a few tweets as well.

2 Spacetime is doomed is a YouTube video. If you go to 5:45 minutes of his 1 hour 22 minute lecture, you will hear that indeed, spacetime is doomed. The entire lecture is his focus on this concept.

3 Pace-Man is our affectionate name for Nima, from his pacing back and forth when he gives a lecture.

4 The amplituhedron has a special place within our work. It was one of many basic structures and by no means is the most basic.  The transformations between circles and spheres to lines and tetrahedra and then octahedra are.

We knew that:

5  The N=4 supersymmetric Yang–Mills theory within Wikipedia is entirely readable, but not easily understood.

6 Twistor space

7 Grassmannian

Torquato, Salvatore

Salvatore Torquato

Lewis Bernard Professor of Natural Sciences
Princeton Institute for the Science and Technology of Materials
Princeton, New Jersey

Articles: ArXiv (149): The structure factor of primes (2018), Hyperuniform States of Matter (2018)
Google Scholar
Group Homepage
YouTube: Hyperuniformity in many-particle systems and its generalizations

First email: Mar 10, 2014, 8:54 PM
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 2002 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.



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:

Here is what I said about John Conway:

“An earlier history began with the study of perfected states in space time.
Sometime around 2002, at Princeton with geometer, John Conway, the discussion focused on the work of David Bohm, once a physicist from Birbeck 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.”