TO: Erica Klarreich, American Scientist, Nature, Quanta Magazine, Scientific American, journalist and scientific reporter with PhD in Mathematics
FM: Bruce E. Camber
RE: Your many articles: What is the geometry of the universe? Quanta, March 16, 2020 and Foams and Honeycombs, American Scientist (PDF); your arXiv article, The Boundary at Infinity of the Curve Complex…, March 2018, along with your homepage, bsky, Twitter, Wikipedia, YouTube: Hedetniemi’s conjecture. Also, Semiconjugacies between Kleinian group actions on the Riemann sphere (1999); The boundary at infinity of the curve complex and the relative Teichmüller space (2018); and A Number Theorist Who Solves the Hardest Easy Problems (James Maynard)(2020).
URL (this page): https://81018.com/klarreich Other: 81018.com/essence/#Readings
Third email: 18 November 2025
Hi Erica,
I sent this note through bsky but usually go deeper in a personal message. Pictured here is a natural gap created by five tetrahedrons on top; five octahedrons in the middle and five tetrahedrons on. I manufactured the molds to make it. It is one of a kind but a natural gap. It was only with an article published by the AMS did the five-tetrahedral natural gap get some attention. The five-octahedral natural gap is a mystery. I haven’t read anything (except what I have have written) about it. It’s a natural, primitive geometry that got some notoriety because Aristotle wrote that one could tile-and-tessellate the universe with just the tetrahedron. He was wrong and it stood for 1800 years!
In August 2001 I visited with John Conway in Princeton for a day of discussions. I asked him about the octahedron and the four natural hexagonal plates within it. It was his first exposure to it! Salvatore Torquato and I have kind of carried on that conversation. There seems to be no literature about those plates. Isn’t that odd? In March 2025 in a conversation with Grok and ChatGPT, it seemed to come alive as geometric stabilizers of spheres. I know that I am reaching around in the dark here: https://81018.com/planck-polyhedral-core/ I thought with your background and expertise, you might advise me. How crazy have I been since March? Thanks, Erica. -Bruce
“@ericaklarreich.bsky.social Have you ever seen these geometries:
1) Five tetrahedrons, on top of five octahedron on top of five tetrahedrons with a natural gap online here: https://81018.com/15-2/
2) A dynamic gif you can see at https://81018.com/planck-polyhedral-core/
It is all quite fascinating. Thanks. -Bruce
Second email: November 4, 2021
Dear Erica:
Thank you for all your excellent work over the years. With so many issues you have captured, it appears that your work on the Mathematicians Chase Moonshine’s Shadow (Quanta, 2015), prompted this note to you. Wow. Thank you.
-Bruce
First email: 26 February 2021
Dear Erica,
Just to let you know that your March 2015 article, referencing the work of John McKay (1978), Don Zagier, Richard Borcherds, Shamit Kachru, John Duncan (Case Western Reserve), Miranda Cheng, Jeffrey Harvey, Andrew Wiles, Igor Frenkel, James Lepowsky, Arne Meurman, David Kaplan, Tohru Eguchi, Hirosi Ooguri, Yuji Tachikawa, Sander Zwegers, Ken Ono, Michael Griffin, and Terry Gannon.
Of course, we all know the work of John Conway and Edward Witten. What a fascinating group!
Thank you for your introductions. Fabulous.
Now, let’s find the deeper wisdom among all the pies on Pi Day. Here are my two feeble attempts: (1).https://81018.com/challenge/ and (2).https://81018.com/instance/
Also, at one time I thought Natalie Wolchover was my favorite Quanta writer. Now that I am on your Twitter overview and excellent webpages, and with all those introductions above, I most profoundly thank you. You are first class.
Warmly,
Bruce
PS I had already engaged Miranda Cheng, Hirosi Ooguri, and Don Zagier. -BEC