Five emails to Thomas Callister Hales

Thomas C. Hales, University of Pittsburgh, Pittsburgh, Pennsylvania

Articles: Computer related
________•  A formal proof of the Kepler conjecture (Jan. 2015) (PDF) (PDF-Princeton)
ArXiv (27)The Kepler conjecture (Nov 1998) (PDF)
BooksDense Sphere Packings: A Blueprint for Formal Proofs (2012)Review
Blog: Jiggerwit  Sloppiness of NIST
Homepage(s):   Another, Google Scholar, Wikipedia, YouTube

References to Thomas Hales work within this website: (July 2020) (May 2020)

Most recent email: 31 December 2022 at 11:18 PM (revised)

Dear Prof. Dr. Thomas Hales:

  1. I am sure you are aware of the tetrahedral gap that Aristotle missed and whose mistake stood for over 1800 years unchallenged and even affirmed. The results of work in the 15th century were so little discussed, knowledge of Aristotle’s error had to be rekindled in the 1920’s by mathematics professor, D.J. Struik, who would eventually join the MIT faculty. It had to be rekindled again in 2012 by Jeffrey Lagarias (Michigan) and Chuanming Zong (Tianjin) yet even with their AMS Conant Award in 2015 for that work, it is quietly fading from view again. Scholars seem to be afraid to ask, “What are these gaps? Do they have anything to do with quantum fluctuations? If the gaps begin with fluctuations, are there places without gaps, where space is being perfectly filled?”
  2. Are you aware that the five octahedrons sharing an edge create the same gap? I don’t think you will yet find it in any books or articles. Even the online geometric construction tools are not able to replicate it automatically. Physical construction sets like the Zometool cannot as well. It is a striking object unto itself but even more so when combined with the tetrahedral gap.
  3. The most-squishy quantum geometry is created with twenty tetrahedrons sharing a centerpoint to create an icosahedron.

You are bold. You are willing to push the equations. Would you begin to look into these gaps so we can get beyond sphere-stacking and into the cubic-close packing’s methodologies for creating tetrahedrons and octahedrons? Thank you.

Warmest regards,


Fourth email: 8 April 2020

Dear Prof. Dr. Thomas Hales,

Reviewing some of your work, I noticed problems with my reference/resource page for you, so just updated it. Just so you have the last word, here is that reference: (this page). Essentially because I am older now — part of the high risk pool — and my acuity is not so acute, I have had to resort to such postings just to remind me, keep me on track, and be polite. Any changes? Your requests will be quickly honored.

Again, I thank you for all your most brilliant work,


Third email: Aug 1, 2019, 1:18 PM

Dear Prof. Dr. Thomas Hales,

Because it was called idiosyncratic and it has not been encouraged, and I did not want to be responsible for misleading our high school students with crackpot concepts, I’ve pulled our base-2 exploration out of the classroom. Some of our old web pages are still up including but my current work is here:

Not to taint your work in any way, because it is so important and spheres, Kepler, conjectures, and cubic close packing of equal spheres, is all so important, I continue to study your work and still encourage people to visit your site. See:

I thought you might appreciate hearing from me after three years.
If not, I’ll hold back further notes!


Second email: Thursday, July 7, 2016, 10:53 AM


Dear Prof. Dr. Thomas C. Hales,

In 2011 we divided the edges of the tetrahedron, connected the new vertices to find the half-sized tetrahedrons in each of the four corners and the octahedron in the middle. We did the same with the octahedron and found the half-sized octahedrons in each of the six corners and the eight tetrahedrons, one in each face, all sharing the common center point.

We continued dividing by 2 and in 45 steps we were down in range of the proton and fermion. In another 67 steps we were in the range of the Planck base units. To normalize the process we used the base-2 exponential notation from the Planck base units and went out to the Age of the Universe and the Observable Universe in just 90 additional notations for a total of 202 notations. Kees Boeke’s base-10 work is interesting; this simple model was fascinating.

Just high school people, we thought we had a great STEM tool.

As we studied it, it became more. The first 67 notations challenged us to see this very small universe in new ways. In time, we began to think it could be a possible paradigm shift in the way we look at our universe. Our chart of numbers is here:

We have done a little work with your proof of Kepler’s conjecture.
The first reference is here:
Might you give us a little feedback? We would be most grateful.

Most sincerely,


First email: 17 January 2016 @ 9:56 PM

A proof of the Kepler conjecture – Annals of Mathematics

Dear Prof. Dr. Thomas Hales,

We are high school folks wrestling with issues that are way beyond our training. I need a really good mathematician to help with what we consider to be a possible paradigm shift in the way we look at our universe.

In my very rough-draft paper, working title, “On Building The Universe From Scratch,” I have made several references to your proof of Kepler’s conjecture. It is my second-most important number (after pi).

The first reference is here:

“The Feigenbaum constants, Buckingham Pi theorem, the fine-structure constant, dimensionless quantities and physical constants were cited less often. We have added two numbers not cited at all: mathematician Thomas Hales‘ number from his proof of the Kepler Conjecture and what we call the Pentastar 7.38 degree gap.”

The primary reference is here:

I know you are a busy person with the highest credentials; I humbly ask if you could you take a look and give us a little feedback? We would be most grateful.

Most sincerely,