Initial focus: Packing, tiling, and covering with tetrahedra, J. H. Conway, S. Torquato, PNAS V.103 No. 28, 2006

**Salvatore Torquato**, Lewis Bernard Professor of Natural Sciences,

Princeton Institute for the Science and Technology of Materials, Princeton, New Jersey

Articles**ArXiv** (156): The structure factor of primes (2018), Hyperuniform States of Matter (2018)**Homepage**(s): Complex Materials Theory Group, Google Scholar, inSPIRE^{HEP}, Prabook, Wikipedia**Publications**: Resume, Books

• Packing, tiling, and covering with tetrahedra, J. H. Conway, S. Torquato, PNAS V.103 No.28, 2006

• New family of tilings of three-dimensional Euclidean space by tetrahedra and octahedra, PNAS, July 5, 2011 108 (27) 11009-11012; https://doi.org/10.1073/pnas.1105594108**Twitter**: ICERM, Physical Review, Quanta Magazine and many more

YouTube: Hyperuniformity in many-particle systems and its generalizations

Third email: 1 December 2022 at 1:11 PM

Dear Prof. Dr. Salvatore Torquato:

Yes, it is too simple. The five octahedral gap is overlooked. I have asked dozens of people now and everyone has been puzzled. All our computer graphics programs appear to ignore it or compensate for it. Even the construction kits like Zometool do not account for it. Isn’t that fascinating?

So, what’s next? Can you write about it and get a larger group discussing it?

If it is in any way related to quantum fluctuations — and with my models, there is a nervousness with those models — it’s significant. We can actually make those constructions dance and bounce all around!

Would you like to have a set of models made of the clear plastic that we use? I’d be glad to send a few models to you so you can see-and-experience that “nervousness” to which I am referring. Thanks!

Warmly,

Bruce

Second email: July 6, 2022, 5:56 PM (Updated: July 15)

Dear Prof. Dr. Salvatore Torquato:

I have not found references online to a five-octahedral gap much like the five-tetrahedral gap that Aristotle missed and, of course, you and John Conway did a major study of it. Have you studied or are you aware of any studies of the five-octahedral gap?

Here is a picture of both gaps together: https://81018.com/15-2/. It is much too simple, but for that reason perhaps it has been overlooked.

Thank you.

Warmly,

Bruce

PS. We’re making a study of that cluster of fifteen sharing a common centerpoint (with the hexagonals within each octahedron). It would make an interesting gate within circuitry of the infinitesimal. If we introduce the twenty-tetrahedral icosahedron in place of the five-tetrahedral cluster, its complexity and potential functionality increases exponentially. -BEC

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

###### REFERENCES:

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 August 2001 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.

Warmly,

Bruce

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.

**Here is what I said about John Conway:**

“**An earlier history began with the study of perfected states in space time**.

Sometime in the Spring of 2001, at Princeton with geometer, John Conway, the discussion focused on the work of David Bohm who was a physicist at 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… scolding myself, “You’re just too simple and easily get hung up on simple things.”

_____