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)
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
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
Please note: It appears that China has disabled the links to servers coming through the USA. We’ll will try to re-route some of those links.
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.