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Is pi (π) both finite and infinite?

by Bruce E. Camber (first draft)

**James Webb Space Telescope**^{1} (JWST): The JWST is challenging everyone who studies his/her/its findings. I predict that it will push the Standard Model of Cosmology^{2} well-beyond the formulations and justifications for big bang theories. Lemaître, Hawking, Guth^{3} and followers have had to ignore the most basic formula of science, pi (π), as well as the key dimensionless constants in order to make big bang cosmology appear to work.

An article, *Standard Model of Cosmology Survives a Telescope’s Findings*, by Rebecca Boyle^{4} was published on January 20, 2023 in Quanta Magazine.^{5} It focuses on the people and issues shaping this Standard Model. However, if we place all those issues within our 202 base-2 notations^{6} from Planck Time to this moment in time, our predictive, mathematical model would readily begin to absorb and transform the sense of logic within big bang thinking. The continuity equation from Planck Time to the current day is perhaps the most-basic continuity equation there is. The progression of numbers from the very-first moment of the universe to the very-first second^{7} of the universe is within our Notation-143^{[8]} out of the 202. It is a “must-study” progression. It is a look at the earliest universe unlike any proposed. It has Planck units (numbers), geometries, logic, a built-in thrust^{9}, and a host of studies^{10} yearning to be on that grid.

**Penultimate grid**.^{11} In this model key symmetry-and-harmonic functions, the essence of the finite-infinite transformation, give rise to a real cosmological constant that within Notation-0 emerges as an infinitesimal sphere. Assuming one Planck sphere per unit of Planck Time and Planck Length, within the first year )which is within Notation-169), the mathematics of this universe is already showing signs of greatness. Starting with Planck Mass at 2.176.470(51)×10^{-8} (kg), within just over one year our universe is already 1.628×10^{42} kg. As a comparison, our sun is estimated to be *1.989×10 ^{30} kg*. Even with Jupiter, the sun is estimated to be 99.5% of the total weight of the Solar System. This expansion is clearly inflation. The entire Milky Way has been estimated (Sloan Digital Sky Survey). Currently it is thought to have a total mass of around 6×10

^{42}kilograms. And, within the first year, the universe has densities in the range of neutron stars and blackholes. With just the first year, we can begin to understand why and how the universe is quite so large in 13.8 billion years.

This trajectory for the universe has a commonsense logic based on a simple mathematical progression that is emulated by nature everywhere.

From that first infinitesimal sphere, tetrahedrons and octahedrons naturally emerge. Fourier kicks in. Inherent harmonic functions of the Fourier transform should readily inspire because, yes, here is a place for Langlands programs and for string and M-theory, and loop quantum gravity and all the hypothetical particles. Here is a place of all those disciplines that are not on the grid. There’s even a place for Smale and Milnor with their attractors and repellers.

Here is a model of the earliest universe where our thinking is least developed. And, it offers a new challenge and opportunity. It’s a new opening of possibilities.

When the Boyle/Quanta article came out, I had been reflecting on the Scientific Method and how currently we all *de facto* assume the big bang. Of course, *de jure* is the counterpoint and I was particularly looking at pi and the dimensionless constants to provide foundations for natural law. It was increasingly clear that pi’s continuity-symmetry-harmony were descriptions of both the finite and infinite and *de facto* was finite and *de jure* was infinite. Although the Boyle article gives a passing reference to the big bang, that theory has nothing to do with the results of the JWST and it really doesn’t change any of the conclusions made by Boyle. She has given us an excellent introduction to some of the key challenges the JWST is making. It is not about the Standard Model. It is about the first microseconds and minutes and years and how the universe began so smoothly.

It is an article to which I will return and watch as a wonderful encapsulation of this moment in time.

Thank you. –BEC

**Editor’s Note**: Obviously this article was inspired by Rebecca Boyle and her article in Quanta Magazine. This homepage evolved from my note to her. -BEC

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Endnotes & Footnotes

All these points already have pages within this website. Another 14 footnotes are being added…

**[*] PI (π)**. In this website, pi (π) always has a face of the infinite and a face of the finite. When focused too much in either direction, we lose our bearings. For this posting, we had first named it, pi-pi, but quickly discovered there was a hip-hop recording named, Pi-Pi, by Milli Music, director, Shane Creative. Their recording of pi-pi opens with the lyrics, “Only you can bring the demon home.” So, here’s a viewer warning: drugs, sex, and darkness and nothing to do with circles or spheres was our first encounter with Pi-Pi. Our next, another recording named, Pi-Pi-Pi, is humorous and it is all about Pi Day and the circumference of the circle with a very light touch.

My work with pi (π) started in 1961 in high school. But it took a much later high school geometry class to begin the progression from the Planck units using base-2 that resulted in 202 notations. Those notations truly opened the discussion around the question, “What is the first thing to manifest in this universe?” After a false start, we’ve settle on an infinitesimal sphere defined by those Planck units. However, we are also open to using the Stoney units or new ISO units. Arguably, studies of pi began between 2500 to 5000 years ago. Euclid’s *Elements* was published around 300 BC. So with so much focused study over such a long period of time, of course, we think we have milked it dry. The opposite is true. We’re making slow progress to grasp its deepest, broadest, highest, most-comprehensive meaning.

**[1] JWST**. The James Webb Space Telescope is technically named for the head of NASA from 1961 to 1968. Huge progress was made during the Mercury, Gemini, and Apollo programs under his leadership. Yet, it could be argued that the array of sixteen hexagonal mirrors of this telescope is more like a *she* than a *he*. In the spirit of the time, I think the pronouns, *he/she/it*, are probably suitable. Wikipedia‘s summary of this work is an ongoing venture that is often updated with new information.

**[2] Standard Model of Cosmology doesn’t care how the universe started**. The more comprehensive it is the better. The more mathematical it is, the more compelling. If the model includes some of the key concepts of the big bang, yet not its time line or its place of importance, so much the better. The 202 base-2 notations do all that and so much more; so of course, we’ll come back to this footnote “for more” as we attempt to build a connection between the it and the Standard Model for Particle Physics.

**[3] Lemaître, Hawking, Guth** were the most pivotal thinkers to promulgate the big bang. Of course, Lemaître is long dead, and the very few who knew him are now close to the end of life. Hawking died on Pi Day, March 14, 2018. I can well-imagine he had had enough. Photo-op after photo-op, it is hard to be a celebrity and even consider doing serious science. Our infinitesimal sphere just may be a very good definition of Guth’s inflaton. It is creating the laws of physics as it populates the universe which based on either Planck‘s or Stoney’s base units could anywhere from 539-to-4605 tredecillion infinitesimal spheres per second… More to come.

**[4] Standard Model of Cosmology Survives a Telescope’s Findings**, R.Boyle, Quanta, Jan. 2013 Rebecca Boyle has written an excellent article. She gives the big bang passing acknowledgement yet focuses on the current tensions in cosmology created by the results of the JWST. The focus has to be on the results of real research. And, the focus is to answer the question, “How can the universe look like these images 330 million years from the start?” Although the base-2 expansion from a single infinitesimal sphere is dramatic, it is orderly and entirely geometrical and mathematical. It is consistent with the JWST’s findings… More to come.

*Standard Model of Cosmology Survives a Telescope’s Findings*, by Rebecca Boyle

**[5] Quanta Magazine** Thomas Lin started the publication in 2012 and *Quanta Magazine *has already won the 2022 Pulitzer Prize in Explanatory Reporting and the 2020 National Magazine Award for General Excellence. The magazine is primarily sponsored by the Simons Foundation which is also responsible for the Flatiron Institute in NYC (Wiki), Simons Center for Geometry and Physics at Stony Brook, and hundreds of programs related to the advancement of scientific knowledge. James Simons used the power of mathematics to understand the markets and trading and made billions. His investment in Quanta Magazine is richly paying back.

**[6] 202 base-2 notations**. The horizontally-scrolled chart of 202 base-2 notations started in December 2011 with just Planck Length. That chart had somewhere around 202 notations. Not until we mapped it with Planck Time did we have a more definitive stopping point: 13.79 to 13.81 billion years. The 201st doubling has taken Planck Time, 5.391 16(13)×10^{-44} seconds, around 173,272,944,073,600,000 seconds or 5.4908 billion years. If we add up each notation up to the 201st notation, we are one unit of Planck Time shy of 10.98 billion years. Simple math tells us that just 2.8284 billion years has passed since the beginning of the 202nd notation (Calculation: 13.81 minus 10.9816 ≈ 2.8284). It is an important, hard-earned perspective. Our calculation for the UniverseClock helped us along that path. We are now challenged to exegete each notation! We quickly discovered how difficult that notational analysis can be: 0, 31, 64, 67, 101, 137, 143, 167, 197, 199 and 202.

There is a place for the big bang numbers but not until after the first few seconds.

**[7] The very-first second**. One might think that the very-first second of the universe is well understood. It’s not. Steven Weinberg within his 1977 book, The First Three Minutes, says on page 5, that “…one-hundredth of a second (is) the earliest we can speak with any confidence” and then adds with great aplomb that “the universe was about a hundred thousand million (10^{11}) degrees Centigrade” as if he had observed it in the laboratory. Then there is a group of 27 scholars from around the world who titled their article, The First Three-Seconds. They didn’t get anywhere close to the first second. It is such a blindspot.

**[8] Notation-143**. Then, there is Notation-143, or the 143rd doubling of Planck Time, and we are well into the last third of our chart. Notations 1-67 are virtually unexplored and, to the best of our knowledge, had never been cited in our scholastic literature. We’ve called it the *small-scale universe* when perhaps The *Infinitesimal Universe* would be more appropriate. Notations 67-to-134 have been cited as the *human-scale universe* while Notations 135-to-202 have been cited as the *large-scale universe*. By that 143rd notation, on the first pass through, the core geometries, mathematics, and physics have been shaped by efficiencies and densities. There is no time or space for indeterminacy. In that first pass there is only perfection. It is the most efficient and the most simple. I can well-imagine those efficiencies become precedents and that perfection, currently called smoothness, readily defines the first 330 million years right up to and within Notation-197.

**[9] Thrust in our universe**. In September 2017, I took a stab at my first real article about the thrust in our universe. So now, over five years later, it is time to revisit that article and update it as much as possible. The major update would involve our understanding that the three facets of pi are each a Janus-face for the finite and the infinite. Another major update will include the continuity-symmetry-harmony functions that are abiding.

**[10] Major studies not on the grid**. I consider nine major studies not on the grid. I have written to Robert Langlands, Ed Frenkel, and others within Langlands programs. They have not yet acknowledged the 202 mathematical notations. Why not? It’s just math and logic. There is no philosophy. There are no agendas. It is either correct or not.

I have also written to people within string theory. None have acknowledged the 202 notations. Why not?

I believe people are naturally incrementalists. It is more comfortable. The Planck units were ostensibly ignored until 2001 and by that time Hawking-Guth-and-family had a choke hold on the theory about the start of the universe. With Hawking’s death, the choke hold has become somewhat more relaxed. With the JWST it’s time to breathe again. Of course, conformal-quantum-and-scalar field theories (CFT, QFT) have holds on all the old-timers within the industry. You can imagine that each night they are hoping that they may see a breakthrough before they die. Yet, although John Wheeler’s sense of simplicity was a good idea, for most of the nonagenarians, this base-2 model of 202 notations is just too simple. It is too obvious. Yet, prior to 2001 and Frank Wilczek’s three articles about the Planck scale, Planck’s numbers were aloofly small in a similar way that Dirac’s were aloofly big.

Perhaps a little more to come…

**[11] Penultimate grid**. The first infinitesimal sphere has been likened to Lemaître’s primeval atom and Guth’s inflaton. Both are hypothetical. Our very first infinitesimal sphere is a little less hypothetical because it has a geometry, a mathematics (an algebra), and a deep-and-abiding logic. The universe has to start with something to create space-time. Of course, our postulation that our universe is totally populated by such infinitesimal spheres is also hypothetical. Notwithstanding, this hypothetical penultimate grid warrants inspection. Although the finite-infinite transformations between the faces of continuity-symmetry-harmony (CSH) are assumed, our focus is on the finite. The finite is first defined by CSH, then defined by the Fourier transform, and the Poincaré sphere, and then the Poincaré homology sphere. Any and all types of spheres are included as potential spheres waiting for their mathematics to evolve (be possible, come out). This, of course, would include Smale and Milnor’s spheres, attractors and repellers.

In 1980 in Paris at the Institut Henri Poincaré, Jean-Pierre Vigier discussed (and I listened) the EPR paradox in light of the work of Alain Aspect in d’Orsay. Instead of infinitesimal spheres, Vigier had suggested using the metaphor of the dominos which was not instantaneous. Infinitesimal spheres with the packing densities suggested by the Planck-or-Stoney-or-ISO numbers, would be instantaneous. Mathematics and physics begin here.

**Please note**: Today, more of the linked words or expressions may still become a footnote. Today is indeed February 3, 2023.

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References & Resources

As these references are studied, key references and resources will be added.

• The Friedmann–Lemaître–Robertson–Walker (FLRW) Metric

• Path integrals and Gaussian fixed point. See Assaf Shomer’s on page 7: “The derivation of the path integral formula in quantum mechanics of a massive particle involves chopping up the quantum evolution into very short time intervals and inserting complete sets of states between them.”

• Doplicher S, Fredenhagen K, Roberts JE (1995) The quantum structure of spacetime at the Planck scale and quantum fields. Communications in Mathematical Physics 172(1):187–220

• Scale invariance and conformal symmetries

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Emails

There will be emails to many of our scholars about the key points.

31 January 2023, **Thomas Sumner**, Simons Foundation

30 January 2023, **Basil Hiley**, University of London (UCL)

29 January 2023, **Thomas Lin**, Quanta Magazine

27 January 2023, **Drew Harrell**, Washington Post

26 January 2023, **Carl Zimmer**, New York Times

25 January 2023, **James Sethna**, Cornell

24 January 2023, **Rebecca Boyle**, Quanta Magazine

23 January 2023, **Rohan Naidu**, MIT Pappalardo Fellow

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IM

There will also be many instant messages to thought leaders about the following key questions:

- Is it possible that the first instance of the universe is defined by Planck’s base units?
- Is it possible that the first manifestation of those base units is an infinitesimal sphere?
- Might the characteristics of pi describe those spheres?
- Might the Fourier Transform impart either electromagnetism or gravitation to each sphere?
- Is it possible that one sphere manifests per unit of length and time?
- If so, doesn’t that compute to 539 tredecillion spheres per second using Planck units and 4605 tredecillion units per second using Stoney time?
- Is it possible that the densities within the earliest notations are on the order of a blackhole?
- To create some sense of order with the generation of infinitesimal spheres, may we use base-2 notation?
- Using base-2 notation, are there 202 base-2 notations from Planck Time to the current time?
- Is the calculation significant at one second where that Planck Length multiple is a very close approximation of the distance light travels in that second?
- Is it significant that quantum fluctuations are measured within Notation-67? Notation-72 appears to be the limit of our abilities to measure a duration of time.
- Would these notations, 1-64, provide 64 possible redefinitions of a point-particle? (And, I would add a vertex.)

**11:14 AM · Jan 31, 2023 @RBReich** Everyone should find their creative thing that makes them happy and brings them joy and ask, “Is there a business in there?” Millions have. We call it small business and it enriches the soul and satisfies the heart and inspires the mind. https://smallbusinessschool.org

**8:45 PM · Feb 1, 2023 @DrOsamaSiddique @Harvard_Law @UniofOxford @IGLP_HarvardLaw** There is a type of natural law within pi (π) (https://81018.com/starts-2/) that also creates a mathematically-integrated view of the universe where value comes from its continuity-symmetry-harmony. https://81018.com/values/

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Participate **You are always invited.**

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• This page became the homepage on January 27, 2023 at about 8:42 PM.

• The last update was February 3, 2023 in the morning.

• This page was initiated on January 23, 2023 at 8:42 AM

• The URL for this file is https://81018.com/pi-π/

• The headline for this article: Pi Defines the Finite and Infinite.

• First byline is: Filename changed to “pi-π” because pi-pi was already engaged.

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