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
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:
January 14, 2021: https://81018.com/precis/#Zong
May 26, 2020: https://81018.com/duped/#R3-2
May 5, 2020: https://81018.com/duped/#Aristotle
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.
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
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.
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/
Your work gives me a wider and deeper perspective.
Books: Continuous Symmetry: From Euclid to Klein (AMA, 2007)
(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.
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.
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,
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.
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.
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.
- Goldstone’s theorem
- Infinitesimal transformation
- Noether’s theorem
- Sophus Lie
- Motion (geometry)
- Circular symmetry
References: William H. Barker, Roger Howe, Continuous Symmetry: from Euclid to Klein (2007)
- Edward Anderson, formerly of DAMPT of Cambridge and OTC, Paris
- Anousheh Ansari, CEO, Prodea Systems and sponsor of the Ansari X Prize
- Nima Arkani-Hamed, theoretical physicists, Institute for Advanced Studies, Pricneton
- Peter Diamandis, founder/chairman, X Prize Foundation and Singularity University
- Gil Elbaz, founder and CEO, Factual, American entrepreneur, investor, and philanthropist
- Brian Greene, professor, Columbia University; co-founder, World Science Festival
- Justin Khoury, Professor of Physics, University of Pennsylvania
- George Musser, contributing editor for Scientific American magazine
- Thanu Padmanabhan, theoretical physicist and cosmologist
- Tony Rothman, American theoretical physicist, academic and writer
Within this website: https://81018.com/2020/01/21/communications/
First email: 21 January 2020
Dear Dr. Kristina Starklofft:
Congratulations on all that you doing to preserve the best among our scholars.
I have a deep and abiding affection for Max Planck and his base units.
Is it your office where the Vetter printing of Max Planck hangs today? Does your museum/library gift shop sell a replica?
So entirely delightful. That it was painted in my birth year, 1947, is sweet. That it is an artistic image, probably the last prior to his death, gives it even more vibrancy.
How very special.
Thank you so very much.
Max Planck, 1947, by Ewald Vetter, painter, Max-Planck Gesellschaft
PS. With every use, the picture would have the two line description
and the picture itself link to your website. -BEC
Born: March 18, 1937; died, July 3rd, 2019
Most recent email: Thursday, April 29, 2019
Though Freeman Dyson and I go back to work in 1979, I remember writing to you in and around 1970! I am getting too forgetful and these postings are a way for me to check on my most recent communications with a person. If you would like anything changed, updated or deleted on this page, just let me know and I will be as expeditious as possible. Thanks.
Third email: Thursday, June 28, 2018, 9:21 PM
Hi Elizabeth –
You might find today’s homepage to be of some interest:
In a few weeks it will be easily accessed at this URL:
We are in South San Francisco and Sacramento for the next two months!
Might you be coming into the area?
Second email: Monday, April 30, 2018, 3:46 PM
Hi Elizabeth –
I just came upon the work of Nassim Haramein where I learned about his
Unified Field And Sacred Geometry that you were attempting to write
a scaling law essentially to encapsulate the universe. How is that going?
Might the 202 doublings of the Planck base units to the Age of the Universe
and the size of the universe be a simple solution? Perhaps too simple, at least
it is a start: https://81018.com/chart
First email: Sunday, July 17, 2016, 9:05 AM
We corresponded back in the ’70s. Noyes, Bastin, Bohm were all mutual friends. My interim story is too arduous, perhaps for another time. You have been prodigious. What a vitae!
In 2011 in a high school I had the five geometry classes go inside the tetrahedron and octahedron. Dividing by 2, this perfect, interior tessellation brought them face-to-face with the proton in just 40 steps, and then face-to-face with the Planck base units in another 67 steps. A sweet journey it was. In our next time together, we multiplied by 2 and in about 90 steps we were out to the edge of the universe, well beyond Kees Boeke.* 3.33 times more granular with imputed geometries and the Planck base units, what was not to love about it?
The project got away from us and has it own life:
That link goes to a rather large, horizontally-scrolled file.
Is it all wet? …too idiosyncratic? …too simple?
I thought you would find it of some interest, if just as a novelty. A penny for your thoughts? Thanks.
*Of course, Kees Boeke’s base-10 is great fun, but it doesn’t mimic
cellular reproduction and bifurcation theory, nor does it engage
cellular automaton, or the automorphic forms of the Langlands
References within this website:
This page is https://81018.com/2019/04/18/engquist/
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.
References within this website:
Dear Prof. Dr. Weinan E:
I don’t think we did a very good job teaching geometry. In 2001, after spending a few hours with John Conway there in Fine Hall, he asked me, “Why are you so hung up on the octahedron?” I answered, “Because nobody knows what is perfectly enclosed within it.” I continued, “…we don’t know its most simple interior parts. …we don’t know about its four hexagonal plates. …we fail to recognize its necessary relation with the tetrahedron. Shall I go on?”
Ten years later with a high school geometry class we went deep inside that tetrahedral-octahedral complex. In 45 base-2 notations going within, we were well within particle physics. Within another 67 notations we were within the Planck Scale. We went back out; and from the desktop, it was only 90 additional doublings and we were at the approximate age and size of the universe. We then discovered Kees Boeke’s base-10. It had no inherent geometry. It had no Planck scale doublings. It was an empty shell while we had the dynamics of cubic-close packing.
We later learned that we had the penultimate multiscale model. Would you classify it as part of your heterogeneous multiscale method (HMM)?
Now, regarding all this data, there are three pages about which I would enjoy your harshest judgments:
Can you help? Thank you.
• The Origins of Dark Matter (Symmetry, November 2018)
• The quest to test quantum entanglement (Symmetry, November 2018)
• Already beyond the Standard Model (Symmetry, October 2018)
• Five mysteries the Standard Model can’t explain (Symmetry, October 2018
• Will We Recognize Alien Life When We See It? (Mosaic / digg, October 2015)
• Quantum and Consciousness Often Mean Nonsense (Slate, May 2014)
Websites: http://bowlerhatscience.org/ (Personal Website)
• http://GalileosPendulum.org (Blog)
First email: 15 November 2018
Dear Prof. Dr. Matthew R. Francis,
Is simplicity good?
We took the Planck base units of Length, Time, Mass and Charge and
applied base-2 exponentiation. In 202 doublings, the chart is out to the
Age of the Universe. It seems straight forward, however, the results are
First, it is a simple, logical, mathematical map of the universe.
I am not sure… are there any others?
By studying the numbers associated with each doubling, we see
that most of the 202 doublings are about the early universe.
Notation 143 contains the first second.
Notation 197 contains the beginning of large structure-formation.
All the numbers are here: https://81018.com/chart/
Though entirely idiosyncratic, I think there is something here.
1. Of course, the Planck Length doubling at one second, divided
by the Planck Time doubling at one second is very close to the value
of the speed of light in a vacuum. It is consistent with Planck’s initial
equation for Planck Time. Yet, it is naturally also consistent within
each of the other 202 notations.
We started this project in a high school geometry class in 2011
( https://81018.com/home/ ) with just the Planck Length. We
did not introduce Planck Time until 2014 and Planck Mass and
Planck Charge until 2015. So, really we have just begun to study
and attempt to understand these numbers in light of current theories
within cosmology and physics. It is entirely provocative!
Even though it is idiosyncratic, is there any hope for it?
The current homepage is my latest attempt to spotlight key ideas
and problems. Thank you.
After three years now, a new email is in process.
First email: 19 October 2018
Dear Prof. Dr. Jenann Ismael:
I started seeing references to you and your work regarding the structure of space and time. Then, came FQXi, then symmetry, Rovelli’s work, a conference about time at Perimeter… so I started to investigate. When I do, I create this little reference page along with a copy of my notes as I struggle to see how it all fits together.
Since December 2011 we have been studying an application of base-2 notation from the Planck base units to the age and size of the universe. We know well that it falls outside the normal work within physics-philosophy-mathematics today. But, such a simple concept renders rather unusual-even surprising results:
• There are just over 202 doublings. Our working numbers: https://81018.com/chart/
• Too small to measure, the first 64 notations: https://81018.com/64-notations/
• Does it address the derivative structure of space-time? https://81018.com/c/
• The doublings create a natural inflation: https://81018.com/ni/
• Perhaps it is just too simple. The first second emerges within the 143rd notation.
• The 202nd notation has a processing speed of 10.9816 billion years and I am not sure what it means to be just 2.8 billion years into it!
I thought you might find it all of interest. I don’t think it’s just poppycock… If it is, it seems we’ll have to re-examine the foundations of logic, mathematics, and integrity, and the concepts of continuity and symmetry!
On leave from Columbia, I hope your work is going very well and you are making very special progress. I would be delighted to hear from you either way,
poppy cock or not poppycock!