Following the work of Gerardus ‘t Hooft…

Gerardus ‘t Hooft, Institute for Theoretical Physics & EMMEΦ
Centre for Extreme Matter and Emergent Phenomena, Utrecht University
Utrecht, The Netherlands

ArXivAn unorthodox view on quantum mechanics (PDF), 2021
Ontology in quantum mechanics, 2021
Free Will in the Theory of Everything, 2017
Books: Time in Powers of Ten: Natural Phenomena and Their Timescales, 2015
 In Search of the Ultimate Building Blocks, Cambridge UP, 1996
Homepage(s): Nobel Prize (1999), Wikipedia
YouTubeLemaitre Workshop, May 9-12, Vatican Observatory (INAF and INFN), 2017

Most recent (third) email: 1 October 2022 at 11:59 PM

Dear Prof. Dr. Gerardus ‘t Hooft:

Most paradoxes are a red flag for incompleteness. Obfuscation is not uncommon. And, I suspect that we will not find anything among all Einstein’s writings about the Theory of Relativity that will satisfy you or me with an insight about how “space-time itself does expand faster than the speed of light” and “how it is possible that there may be galaxies that are almost 100 billion light years apart.” Of course, both of the quotes are from the Forward of Time in Powers of Ten, page xiv, second paragraph.

So, what is already known that might accommodate such conclusions? Before inventing the new, might we consider Planck’s 1899 calculation of natural units? George Stoney made similar calculations in 1874 (published in 1881). The density of those natural units are on the order of a neutron star. Quick calculations using base-2 place the first minute within Notation-143, the first year within Notation-169, and today within Notation 202. It’s a natural inflation that absorbs the big bang without extralogic and new theories. It opens new possibilities for Langlands, strings, and SUSY, to name just a few.

We have the makings for a Wheeler-like simple beginning (see quote under image). Thank you.

Warm regards,


PS. There is a geometry associated with the powers of 2. All notations are always building on each other. It’s a basic outline, but here our grasp of the very nature of time becomes penultimate. -BEC

Second email: 30 September 2022

RE: The Paradox described on page xiv in the Forward of Time in Powers of Ten

References:  Time in Powers of Ten and your most recent ArXiv articles
Lemaitre Workshop May 9-12 at the Vatican Observatory, 2017

Dear Prof. Dr. Gerardus ‘t Hooft:

The words in your book’s Forward are: “…space-time itself does expand faster than the speed of light.” Does this paradox have a formal name? Have there been any peer-reviewed papers about it? Thank you.

Warm regards,


Updated: 8 February 2018   First email: Sunday, June 16, 2013

References: How to become a GOOD theoretical physicist
One of your pages that the students enjoy: Tilings

Dear Prof. Dr. Gerardus ‘t Hooft:

Our geometry classes worked with clear, plastic tetrahedrons and octahedrons. We created the twenty-tetrahedral icosahedron, and the sixty-tetrahedral dodecahedron (Pentakis dodecahedron). We’ve built models and looked inside each (more pictures here). When asked if there were any questions, one of the students conjured up Zeno and sweetly asked, “How far within can we go?”

Using just the perfectly-fitting octahedron and tetrahedron, on paper we went inside each object about 112 times, dividing by 2 to discover that we were in the range of the Planck Length. We also discovered how each group of models was growing exponentially as we went further inside.

Next class we went the other way. We multiplied by 2 until we were in the range of the Observable Universe. There were just 90 steps. We thought it was very cool until we couldn’t find any references to it on the web. Of course, we found Kees Boeke’s work from the 1957. Back in days long gone by, occasionally I had dinners with Phyllis & Phil Morrison; they helped to popularize Boeke’s work with their classic coffee table book, The Powers of Ten.

Yet, now many of us are asking, “Where do the powers of 2 and base-2 exponential notation fit into the larger picture? Do we live in an exponential universe? Does Euler’s equation capture the universe as well?” We initially thought it was a very good STEM tool, but now we think it just might be more.

Six years later, we are still asking the experts, “What are we doing wrong?” Would you tell us? Thanks.



PS. Since our first email (a first-draft of this email), we followed your work with Stefan Vandoren on Time in Powers of Ten: Natural Phenomena and Their Timescales and we have begun to engage your work within ArXiv. Long, long ago I studied the foundations of physics with Abner Shimony and Robert Cohen at BU (1973-1980), and then with David Bohm, Costa de Beauregard, and J. P. Vigier (1977 & 1980). But, I was idiosyncratic.  Siding more with Leibniz than Newton, I began my graduate studies thinking that the essence of the infinite was continuity, symmetry and harmony, not absolute time. Though I enjoy Frank Wilczek’s work (2012 to now), I am even too idiosyncratic for him. Notwithstanding, if you’ll look at our current chart you will see that only two relatively brief epochs — the Grand Unification and Inflationary Epochs — are defined a little differently.  –BEC


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