Lagarias, Jeffrey C.

Jeffrey C. Lagarias lagarias

University of Michigan
Ann Arbor, Michigan

Google Scholar
Homepage (2)
Video: 2015 Clay Fellow Senior Talk – “Packing Space with Regular Tetrahedra

Pages within this website:
• Lagarias and Zong (June 2016)
• Geometric gap (April 2020)
Aristotle, Newton, HawkingAristotle mistake (May 2020)

Most recent and third email: Tuesday, 26 May 2020 at AM

Dear Prof. Dr. Jeffrey Lagarias:

I continue to make references to your work. Most recently here:
An 1800+ year old mistake started with Aristotle
Aristotle failed to see the gap
Aristotle did not grasp the essence of tiling and tessellating

And, because I have made so many references, I have started pages for you and Prof. Dr. Chuanming Zong. Your page is here: Of course, if you would like anything changed, deleted, or added, I will be glad to accommodate. Thank you.

Most sincerely,


Second email: 27 March 2020

Dear Prof. Dr. Jeffrey Lagarias:

I thank you again and again for your scholarly work. I endorse your work! Yet, given our work is so idiosyncratic, you probably would prefer that I didn’t.

Notwithstanding, I am glad for Mysteries in Packing Regular Tetrahedra (PDF).” Just about every day, I wonder what 1800 years of being wrong did to our scholarship.

Best wishes,


The two references below are here: (opens in a new tab)

10 Geometric gap: 0.12838822+ radians and 7.35610317245345+° degrees. Even today, March 2020, this gap is little studied and less known. Our first encounter with it was in 2016 upon writing the article, “Which numbers are the most important and why?”  At that time, it seemed like Chrysler Corporation had branded that geometry as the pentastar. And though it is a five-tetrahedral representation, they never looked uniquely at the gap of  7.35610317245345+° also defined by  0.12838822+ radians. Two chemists (Frank & Kasper) came closest to opening the discussion in the 1950s. Two academics (Lagarias and Zong) did a preliminary analysis that was a tremendous help; the relation of this gap to the deeper geometries of life remains as a challenge. Our modest start is here:

11Aristotle’s failure is our failure. Perhaps the gravity and nature of this error is only now beginning to be understood. We all make mistakes. When we are challenged, we defend our concepts as best we can, and then adapt. We change or our associates change.

Some people become larger than life within their own time. Three examples are Aristotle, Newton and Hawking. All three were wrong about one key impression about the nature of  life, yet their egos and their position and their person were so illuminated, it became increasingly difficult to challenge their assumptions.

Aristotle’s geometric gap, Newton’s absolute space and time, and Hawking’s infinitely hot big bang have each mislead scholarship and we all lost the scent and direction of the chase with its potentials for discovery and creativity. Throughout our ever-so youthful human history, such people can readily continue to mislead us. We have to be vigilant to review and re-review all the concepts we hold dear and begin to adjust them appropriately.

First email: Saturday, 31 August 2013 at 8:19:21 PM

Jeffrey C. Lagarias, Professor of Mathematics, University
Chuanming Zong, Professor of Mathematics, Peking University

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: I have a hunch that that object made of five tetrahedrons plays a key role.

Your work gives me a wider and deeper perspective.



Bruce E. Camber

PS. Long ago I studied with David Bohm, Phil Morrison, and so many others like them, but to make a living, I became a television producer!  We had the longest-running television series on PBS stations in the USA and the Voice of America around the world about best business practices.

Here are some key points within my current thinking:

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 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 somewhere around 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

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

This site uses Akismet to reduce spam. Learn how your comment data is processed.