**Martin R. Bridson**, Mathematics Institute, Andrew Wiles Building

Oxford University, Oxford, England

Clay Mathematics Institute, Peterborough, New Hampshire and Denver, Colorado

**ArXiv** (81): *On the profinite rigidity of triangle groups* (April 2020), *The homology of groups*… (2019), *Subgroup separability in residually free groups* (2007)

Martin R. Bridson, Cornelia Druţu Badea, Linus Kramer, Bertrand Rémy, Petra Schwer, Geometric Structures in Group Theory. Oberwolfach Rep. 17 (2020), no. 2/3, pp. 877–918DOI 10.4171/OWR/2020/16

Book: *Metric Spaces of Non-positive Curvature*, published by Springer-Verlag, 1999**Homepage** (s): CV (PDF), Google Scholar, Wikipedia**YouTube**: International Centre for Theoretical Sciences, Part II

References within this website:

A new place to explore: From the Planck scale to wave-particle duality.*

Fifth email: 17 December 2022 at 10:01 AM

Dear Prof. Dr. Martin Bridson:

I just sent a note off to a scholar’s scholar within mathematics. He appreciated that I had given John Conway and Frank Wilczek models of our tetrahedron and octahedron. I mentioned your name regarding the Lagarias-Zong history and their 2015 AMS Conant Award for their article, *“Mysteries in Packing Regular Tetrahedra *(PDF)” and how it was already fading away (fewer and fewer citations).

Three points might help shape and focus on the key parts of those equations:

1. **The visualization of sphere-to-tetrahedron-octahedron dynamics**: I am, of course, referring to close-cubic packing: https://81018.com/ccp/ Let’s ignore packing densities to focus on the process by which tetrahedrons and octahedrons are created. Let’s focus on structures within both. Let’s focus on the gaps. Unfinished business, our best scholars need to grapple with these first principles.

2. **Interpreting the gaps**. The sequel should focus on the place and purpose of that gap. Is the gap related to quantum fluctuations? That’s a major discussion.

3. **The very nature of pi (π)**: I am no scholar but the mathematics of infinity has to be the penultimate challenge. There are so few discussions of the place and importance of pi (π) and infinitesimal spheres. I believe that the open questions about the very nature of homogeneity and isotropy of the universe are keys and that Planck units or Stoney units (or the ISO’s equivalent units) will define an approximate rate so tredecillions of infinitesimal spheres per second fill the universe. These three points are beyond a simple one-off engagement. It is all so idiosyncratic, it’ll take time to engage it, absorb it, and use it.

It is all very approachable with high school students. We even had our AP class of sixth grade savants get immersed in it. But, once our graduates started circling back, we realized that it was too disruptive within the current curriculums. The only hope is within the special integrity of people like you.

Might you have any advice for me and how to proceed? Thank you.

Most sincerely,

Bruce

Fourth email: June 28, 2022 @ 3:45 PM

( And, I agree that studies of packing densities is old news and over-studied.)

Dear Prof. Dr. Martin Bridson:

I thought you might be interested in seeing that earlier note to Alisa Bokulich. I had also copied it to Lagarias and Zong.

Those links and my reflections seemed like a natural sequel to my prior notes to you. The page that raises the questions is here: https://81018.com/geometries/

I think we are getting closer.

Also, I hope you are well and doing fine.

Warmly,

Bruce

Third email: July 12, 2021 at 6 PM RE: Lighting fires

Dear Prof. Dr. Martin Bridson:

My last note in July 2020 has been summarized as a web page: https://81018.com/conference/

Also, my notes to you and my references to your work are here: https://81018.com/bridson/

Again I make a case to use base-2 to parse time from the most infinitesimal scale (conceptually defined by calculations like those of Max Planck and George Johnston Stoney) to the current time. We know there are 202 notations. We know that quantum fluctuations and the wave-particle duality begin after Notation-64 (between Notations 65-and-67). First, what is going on between Notation 1-and-64? Is it a domain for M-theory, Langlands programs, and all the infinitesimal, mathematically-defined, theoretical particles? …what else? …geometric group theory?

If the first manifestation within space and time is a sphere, the Lagarias-Zong work has value to begin to discern how this universe took shape. Packing densities are a fact, a detail. The dynamics of cubic close packing of equal spheres and the generation of the five platonic solids is key. The dynamics of the sphere with the Fourier and integral transforms also seems key. The grasp of autonomic forms may be a key.

With the advent of tools for relatively large virtual meetings, theoretically a subject could be announced and a “flash mob” spontaneous emerge within virtual meetings that focus on key open questions. If 100 are invited, would five show up? Perhaps 100? More? Might that group make decisions or stimulate new insights? Why not try?

Just a few questions for a Monday late afternoon here in Nashville. Thank you.

Warmly,

Bruce

Second email: Jun 1, 9:27 AMUpdated/resent:July 20, 2020

RE: Continuity-symmetry-harmony becomes a sphere becomes geometry becomes finite

Dear Prof. Dr. Martin Bridson:

Again, I thank you for your support of the work by Jeffrey Lagarias and Chuanming Chong. I believe their work has been under-valued within academic studies. I am not referring to their interests in packing densities; that has been addressed by many since Kepler. The more important question, it seems, is the very nature of the infinitesimal scale and that 7.35+ degree gap that is created by the geometry of five tetrahedrons sharing a common edge.

Does this gap have anything to do with quantum fluctuations? Is this the geometric gap that Aristotle did not see and scholars missed for over 1800 years? Does it hold key insights about the indeterminate and the nature of chaos, unpredictability, undecidability and uncomputability?

Beyond the origin of fluctuations, could these geometries within the infinitesimal between the Planck scale and particle-and-wave duality, also create the conditions for the unique identity of everything-everywhere, including consciousness and creativity?

To create space for such an analysis is a straight-forward exercise if we assume that Planck Time and Planck Length are the first units of time. If we apply base-2 or simple doublings of these numbers, we create a modest but fascinating grid from the first moment of time to this very day. There are 202 notations and the first 64 notations are below our current thresholds of measurement. Yet, logically and mathematically, those 64 notations have potential to provide new insights and answers to persistent questions.

Perhaps a seminar of leading scholars addressing this configuration and the question about the relation of geometries to fluctuations would gather some attention and may render fascinating results that could substantially impact both mathematics and physics.

**I’ve begun mapping an overview of such a seminar**. Even today, David J. Gross, Stephen Weinberg, Lee Glashow, Anthony Zichichi and a few others of their caliber and age might like to attend and/or contribute.

Thanks again for taking time with me.

Most sincerely,

Bruce

First email: May 31, 2020 at 5:28 PM

Dear Prof. Dr. Martin Bridson:

In 2012 Jeffrey Lagarias and Chuanming Zong wrote, *Mysteries in Packing Regular Tetrahedra* (PDF). In April 2015 his Clay Fellow Senior Talk, *Packing Space with Regular Tetrahedra*, was a natural sequel.* *There are elements of those analyses that need a deeper review; i.e. what is that gap created by the five tetrahedrons? Where does it first manifest? Does it have anything to do with quantum fluctuations within the Planck scale?”

Perhaps CMI might entertain such a sequel. I have written to Brendan Hassett, Clifford Taubes, Simon Donaldson and Arthur Jaffe to suggest further examinations of this gap. I also believe that Robert Langlands and Edward Witten would have insights.

Might these questions be worth pursuing? Thank you.

Most sincerely,

Bruce

PS. A link to that 2015 Clay ICERM lecture is here.

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PhD advisor to Dani Wise(McGill), Claas Roever (NUI Galway), Tim Riley (Cornell), Adam Piggott (ANU–Canbera), Henry Wilton (Cambridge), Michael Tweedale (Bristol), Will Dison (Bank of England), Owen Cotton-Barratt (Future of Humanity Institute, Oxford), Dawid Kielak (Hertford College, Oxford), Ric Wade, Royal Society URF, Oxford; Benno Kuckuck, Muenster; Robert Kropholler, Muenster; Claudio Llosa Isenrich, Karlsruhe; Giles Gardam, Muenster; Nici Heuer, Cambridge; Sam Shepherd, Oxford; Jonathan Fruchter, Oxford; Dario Ascari, Oxford; and Monika Kudlinska, Oxford.

Clay board: Simon Donaldson, Michael Hopkins, Carlos Kenig, Andrei Okounkov, and Andrew Wiles

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