TO: Jacob Bourjaily, Pennsylvania State University, University Park, Pennsylvania; once an Assistant Professor of Physics, Niels Bohr Institute, Copenhagen University, Copenhagen, Denmark
FM: Bruce E. Camber
RE: Your Articles, Google scholar citations, ArXiv articles especially Amplitudes at Infinity (2018), Prescriptive Unitarity, even your homepage(s) — Copenhagen, CV-PDF, Twitter, Wikipedia, and YouTube: Vernacular of the S-Matrix, 2018; The Surprising Simplicity of Scattering Amplitudes, 2019
Most-recent fourth email: 14 March 2025
Dear Prof. Dr. Jacob Bourjaily:
I am still caught by the header of your Mathematical Structures group; I don’t think there could be more crisp mathematical structures than the four primary irrational numbers, Pi (π), Phi (φ), the Square root of 2 (√2), and Euler’s number (e). I asked Grok about them: https://81018.com/irrationals/
Might the four hexagonal plates of the octahedron be an appropriate representation of that geometry?
I have discussed it in the last four homepages:
• Pi Day 2025: https://81018.com/pi-day-2025/
• Today’s homepage: https://81018.com/incommensurable/
• Breakthrough: https://81018.com/breakthrough/ https://81018.com/breakthrough-indeed/
Thank you.
Most sincerely,
Bruce
PS. If you have any updates-changes-deletions within our page about your work, just say the word. That page is: https://81018.com/bourjaily/
An evolving page: https://81018.com/penn-state/ -BEC
Third email: 24 June 2023
Dear Prof. Dr. Jacob Bourjaily:
The statement at the top of your listings for the Mathematical Structures group is empowering: “Physics often advances when crisp mathematical structures are uncovered in a framework developed to describe observed phenomena.“
We have two quick questions:
1. Have you ever seen the five- tetrahedral gap (famously missed by Aristotle and documented by Lagarias and Zong, AMS-2015 Conant Award) over and under a five-octahedral gap? It’s a pretty object, but I don’t think any scholar has made a study of it.
Do you think it could be important?
2. Do you know if there are any scholars who have written about the nature of these gaps? How do such gaps manifest within space and time?
Thank you.
Most sincerely,
Bruce
Second email: Wednesday, 18 September 2019
Dear Dr. Prof. Jacob Bourjaily:
So much has transpired in the five years since my first letter (below). We are still asking questions about our simple construct from 2011 when we started at the Planck Length and went out to the size of the universe in somewhere just over 202 doublings. Of course, we have all appreciated exponential notation as an academic study; this chart suggests it is part of our very being.
The first 64 doublings, coming up to just a few steps below where CERN’s scale of measurement picks up, are problematic. So, to explore these doublings, we treat the chart as a little mathematical puzzle. If we take as a given that these 64 doublings exist, what do they tell us about the nature of our universe? What can we do with them? The answer appears to be, “Quite a lot.”
https://81018.com/bottom-up/ is a page where I’ve begun to explore those 64.
There’ll be more to come…
-BEC
First email: May 28, 2014 @ 4:32 PM
Dear Dr. Prof. Jacob Bourjaily,
Besides engaging your research publications listed within your vitae, I thoroughly enjoyed your walk with harmony and Kepler. By the way, you already have a role in our work with our high school geometry classes (I’ll post it below and provide a link if you want to see more.)
We would love your best guess as to where to place your amplituhedrons within our 202+ base-2 exponential notations from the Planck Length to the edges of the Observable Universe.
It all started when we made models of a tetrahedron with a smaller tetrahedrons in the each corner and the octahedron in the middle. We then divided the edges of that octahedron in half, connected the new vertices, and found the six smaller octahedrons in each corner and eight tetrahedrons in each face: http://81018.com/tot/ https://81018.com/octahedron/
We chased that progression first to the Planck Length and then out as far as we could. We then realized we just tiled the universe. Of course, it is all simple logic and an ideal construction. Yet, we found it useful, so for over two years now, our students have been using this range, we call it a Universe Table or the original, Big Board – little universe, and considered our STEM tool.
I’ll insert our current range and highlight our guess for the amplituhedrons.
Thank you.
Warmly,
Bruce
Notes:
Notation 66. ___ 1.19254509×10-15 m In the range of the diameter of a proton or a fermion
Notation 65. ___ 5.96272544×10-16m In the range of neutrinos, quarks
Notation 64. ___ 2.98136272×10-16 Possibly in the range of largest possible strings
Notation 63. ___ 1.49068136×10-16m Possible Amplituhedron
Notation 62. ___ 7.45340678×10-17 m
#12b. May 2014: Discovering Quanta Magazine, amplituhedrons and more
We have been discovering the writings of Natalie Wolchover within Quanta Magazine where she focuses on the work of Andrew Hodges (Oxford), Jacob Bourjaily (Harvard) and Jeremy England (MIT). We believe these young academics are opening important doors so our simple work that began in and around December 2011 has a larger, more current scientific context, not just simple mathematics. We are now pursuing all their primary references; our introductions are here:
• http://www.simonsfoundation.org/quanta/20130917-a-jewel-at-the-heart-of-quantum-physics/
• http://www.simonsfoundation.org/quanta/20140122-a-new-physics-theory-of-life/
• https://81018.com/2014/05/08/fifteenideas/