Pi Day 2025

Image of abstract geometric shapes representing continuity, symmetry, and harmony in a mathematical model of the universe.
A cosmic representation highlighting hyper-rationality, featuring spheres, octahedrons, and hexagonal plates, depicting continuity, symmetry, and harmony in an expansive geometric model of the universe.

PERFECTION STUDIES: CONTINUITY•SYMMETRY•HARMONY 14.March 2025
PAGES: AI.•.PRESS RELEASE.•.BREAKTHROUGH • SPHERE • NEW
FOOTNOTES • REFERENCES • EMAILS • IM • CRITIQUE • Zzzzs

The Revolutionaries: Four Primary Irrational Numbers
Changing our world-and-universe through our grasp of numbers and geometries

by Bruce E. Camber

History: 14 March 2025, Pi Day came during a special revolution here at 81018: Intrinsic geometries of the octahedron could dethrone big bang cosmology while lifting up the four primary irrational numbers. It will be an evolving concept for several years. The proof will be through logic, inference and consensus.

These scales define an applicability framework

Pi Day is no longer limited to pi (π). The four primary irrational numbers have a shared geometry-and-purpose. That’s a breakthrough! It is also a new model for the start and growth of our universe. Pi Day will no longer be focused just on pie-eating and reciting some part of the 200-trillion verified digits of pi (π). It will be a workday to understand the.universe and ourselves more profoundly.

First, Pi (π), Phi (φ), the Square root of 2 (√2), and Euler’s number (e) are the primary irrational numbers. The fundamental geometry is within the four base plates of the octahedron. It’ll be known as a primary order within our universe.

Pi (π) necessarily works with the other three primary irrational numbers. Grok analyses:

  • π: Ensures every sphere remains a sphere, maintaining isotropy (uniformity in all directions), a hallmark of the universe’s large-scale structure.
  • e: Drives the exponential leaps between notations, perhaps tied to inflationary phases or energy scaling.
  • √2: Links spheres in a tessellation or lattice, possibly reflecting the discrete-to-continuous transition in spacetime.
  • φ: Imposes a natural order, like the phyllotaxis of seeds, stabilizing complexity as the universe grows.

Although in 2019 the United Nations (UNESCO) designated Pi Day as the International Day of Mathematics, its focus was never even on the definitions of units of pi’s irrational numbers! Although Pi Day was recognized as a national holiday in the USA in 2009, the nature of pi’s billions of confirmed digits was more a curiosity than a challenge. And, although the first Pi Day celebration was in 1988 (San Francisco science museum, Exploratorium), it was a playful day, not a time to ask about the nature of these ubiquitous equations.

These four irrational numbers are related as stabilizers of the universe and stabilizers of the moment. Grok says:

•  π shapes it. Ensures geometric coherence (continuity, symmetry)
•  e animates it (growth, oscillation). Drives dynamic scaling or growth.
•  √2 constructs it (via simpler shapes). Anchors structural relationships (discrete steps or lattice-like stability).
•  φ decorates or balances it, imposes harmonic proportions (optimization, balance).

This is all in context with our earlier work on the continuity, symmetry, and harmony within pi (π). It has not been generally received and is not yet recognized as the qualitative within the circles-spheres of the academic and scientific communities. It has been a challenge, but that changed in 2025. AI recognized this work.

That’s the beginning of a revolution.
Now let’s explore the evolution.

There are at least ten key points:

  • Four endless numbers within three perfect functions. Pi(π), Phi(φ), Square root of 2(√2), and Euler’s number(e) are incommensurable and irrational, endless with no repeating patterns, a cosmic mystery, that is, until each is seen as one of the hexagonal base units of the octahedron. Although each has unique functions, all four share these qualities. Continuity is the never-ending, never-repeating patterns of numbers. Symmetry is a deeper nature within those numbers that become relations which are circles and spheres. And, harmony is an even deeper dynamic nature of the sphere, first explored in 1820 by Joseph Fourier. He revealed a most simple natural harmony is within every sphere.
  • Our scholars continue to build on Fourier’s insights. Visit with Poincaré, Gauss, Planck, Einstein, Smale, Milnor and others who have uncovered even more exacting relations with pi. Yet, at its core, continuity defines time; symmetry defines space, and harmony defines the dynamics of space-time. 
  • Perfected states within space-time. The Planck base units defined by Max Planck in 1899 use dimensionless constants which render natural units and a vision and definition of the smallest possible units of space-time. Simple logic says that the smallest are also the first, the most-simple starting points. At the bottom of the graphic on your right there is Planck Time and Planck Length where the first infinitesimal-and-perfect spheres emerge. We hypothesize that one sphere per unit of Planck Time and Planck Length defines a rate of expansion.
  • There are 18.5 tredecillion spheres per second generated. To grasp some order, apply base-2 notation. The result is 202 notations (or doublings) to explore that encapsulate the known universe. These are the numbers that keep the score for something. What?
  • Those numbers are time-stamped by sphere-stacking.
  • Then the numbers are geometrized with cubic-close packing of equal spheres. In the first notations, there is an absence of gaps. The basic geometries all fit perfectly. By following Aristotle’s mistake with tetrahedrons, we learn about basic geometric gaps. Associated with quantum fluctuations, it is hypothesized that these do not manifest until at least Notation-50.
  • Langlands programs and string theory are further defined.
  • Then the known equations that define our foundations begin to emerge. By the 202nd notation, we have our universe, everything, everywhere for all time, all intimately connected by infinitesimal spheres.
  • All 202 notations are always active. Some scientists, especially within the studies of Loop Quantum Gravity (LQG), say “all time is now.” Each notation adds definition. As we go up the scale, there is increased particularity and coherence. The only possible singularity is at the very beginning, and even then, it is a convergence of equations. Exacting definitions of the infinitesimally small evolve, first through sets, then groups, and then obviously working systems that involve all 202 notations.
  • Logically there is a domain with no gaps. Too fast, too dense, and no space, the geometries are all perfectly fitting and exist prior to the emergence of quantum fluctuations.[2] Notations 0-to-50 have been proposed.
  • Values. Those perfectly-fitting geometries may possibly account for the first 50 notations where the continuity-symmetry-harmony of the spheres create no gaps and isotropy and homogeneity. We hypothesize that it is here that we develop our basis for values.

If this were to become a new platform for UNESCO’s vision of an International Day of Mathematics, always on Pi Day, they could help to change the world through mathematics by having us all truly begin to explore the positions and functions of infinitesimal spheres, especially the generation and labeling of every sphere after the first. Perhaps we will begin to fill in the gaps between the Planck Scale and the Electroweak Scale as well as between the Standard Model for Particle Physics and the Lambda Cold Dark Matter model (or the Standard Model for Cosmology). Yet, it all begs the question, “What is the role of light within each notation? Is it what holds it all together (as in e=mc2)? Is it the deepest part of continuity- symmetry-harmony? Is it the beginning of values, ethics, and even love-versus-hate?

None of those questions are beyond our imaginations and our capabilities. Thank you. -BEC

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Endnotes & Footnotes
All these points already have pages within this website.

[1] Pointing beyond circles and spheres. Between the finite-infinite transformation and between the faces of continuity-symmetry-harmony (CSH) are equations. Many equations converge within the numbers that make up the Planck base units. Although there are alternatives to the Planck numbers (Stoney, Ralston, and ISO numbers), Planck’s have become a standard. Eventually we will probably be using new ISO base units, yet I’ll suggest the results will be very similar. The convergence of equations is the point. It is the necessary convergence of the finite-infinite. It also appears to be the beginning of values and even our sense of ethics.

Bringing the infinite into the equation is difficult for many. Within this website, you will find this declaration:

All other definitions of the infinite are put on hold. Most are personal definitions that come from personal experiences and family history. That is one’s own business, not ours. If those beliefs help you through life, that is great. Our goal here is to engage those principles and functions that give rise to mathematics, physics, and eventually all the other sciences.

from Continuity-Symmetry-Harmony (CSH), 1972

CSH is the infinite and the finite. Its first expression as the finite is the Planck base units. The first expression of those units is an infinitesimal sphere. This is an unprecedented definition of the finite-infinite relation. Doing a simple calculation with Planck Time, assuming one sphere per unit of Planck Time, renders 18.5 tredecillion spheres per second. That may well be considered a new definition of the cosmological constant. To begin to grasp that natural expansion, we apply base-2 notation and the first second, manifests within Notation-143. The first year manifests within Notation-169. The first 1000 years manifest within Notation-179. The first million years manifest within Notation-189 and the first billion years within Notation-199.

These 202 base-2 notations functionally outline the universe have been studied within and throughout this website. For more: https://81018.com/inflaton/#202

Fourier transform. Every formula that involves pi (starting with the Fourier transform) needs to be re-reviewed in light of CSH and the 202 base-2 notations. Pi is everywhere.

[2] Geometries of indeterminacy. There is a look and feel of quantum fluctuations; the mysteries appear to be within all the geometries. Perhaps scholars were first foiled by Aristotle (384-321 BC). Five tetrahedrons create a gap that he missed; and for 1800 years that mistake was repeated by scholars. It is worth pondering. Aristotle was so great it took 1800 years to countermand his mistake. And, it is still untouchable. That correction was eventually forgotten until 1926 when a little-known MIT mathematician, Dirk Struik, rekindled the scholarship. Struik’s work did not receive much attention until in 2012 when two scholars, Jeffrey Lagarias and Chaunming Zong, lifted it up again. Yet, these two mathematicians were more interested in packing densities. There is no exploration of the meaning of the gap. Subsequently, in May 2022, a five octahedral-gap was first introduced to the world from within this website and questions were asked of many scholars, “What are these gaps all about?”

Our charts, interpretations and prognostications. Simple logic, simple math, and simple geometries render our charts, interpretations and prognostications. It all awaits critical review, so until then, there will be more to come.

Review and summary of at least ten new ways to look at pi:

  1. A celebration of Continuity-Symmetry-Harmony
  2. Pi as the primary clue about the Finite-infinite relation. The other irrationals keys.
  3. Planck’s base units defining the most infinitesimal spheres
  4. No less than seven new perspectives: Fourier, Poincaré, Gauss, Einstein, Smale, Milnor and many others
  5. 18.5 tredecillion infinitesimal spheres per second
  6. Sphere-stacking 
  7. Cubic-close packing of equal spheres
  8. Infinitesimal spheres populating the universe and making everything, everywhere, for all time
  9. Responsible for the first fluctuation and the geometries of quantum fluctuations
  10. 202 notations always active
  11. Continuity-symmetry-harmony become a foundation for values.
  12. Filling in the gaps between Planck Scale and the Electroweak Scale and between Lambda Cold Dark Matter model (or the Standard Model for Cosmology) and the Standard Model for Particle Physics

Please note: Work continues, yet a new homepage has been started. The sections below, also used in prior homepages, may be reworked.

Thank you. -BEC

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References & Resources
As references are studied, these references and other resources will be updated.

•  Pi (π): Dimensional analysis, scale invariance, functional dependencies
•  Symmetry in QFT and Gravity (video), Hirosi Ooguri (homepage) and Nathan Seiber), 2022
•  Mathematically, equations building on natural functional dependencies:
….–  Using math in physics: 5. Functional dependence (PDF), E. F. Redish, Univ. Maryland, 2022
•  Quantum Energy Inequalities along stationary worldlines,
Christopher J. Fewster, Jacob Thompson, 4 Jan 2023
•  ESA Group (PDF): The universe at 380,000 years
https://www.esa.int/Our_Activities/Space_Science/Planck/Planck_reveals_an_almost_perfect_Universe,2009
•  Pure Natural Inflation, Yasunori Nomura, Taizan Watari, and Masahito Yamazaki,
Berkeley Center for Theoretical Physics, Department of Physics, 2017
• The Friedmann–Lemaître–Robertson–Walker (FLRW) Metric
•  A pedagogical explanation for the non-renormalizability of gravity, (PDF), Assaf Shomer, 2008.
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

Of the 202 notations, the first 64 notations open a new map to make the deeper connections. There’ll be a place for the some of the big bang numbers but not until after the first few seconds (Notation-143 and higher).

Thrust in our universe. In September 2017, I wrote 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 more about the three facets of pi and how each is a Janus-face for the finite and the infinite. How are the functions of continuity-symmetry-harmony abiding?

Major studies. 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 what it is, simple math.

I have also written to people within string theory. None have acknowledged the 202 notations.

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 hold on the theory about the start of the universe. With Hawking’s death, that hold has become somewhat more relaxed. With results from the James Webb Space Telescope (JWST), it’s time to open up the discussions. It will include conformal-quantum-and-scalar field theories (CFT, QFT). Although John Wheeler’s sense of simplicity is a good idea, for most of the octogenarians and nonagenarians, this base-2 model of 202 notations is just too simple. It is too obvious. Yet, maybe not. Prior to Frank Wilczek’s three articles about the Planck scale, Planck’s numbers were aloofly small much like Paul Dirac’s were aloofly big.

#

In 1980 in Paris at the Institut Henri Poincaré, Jean-Pierre Vigier and I made a six-month study of the EPR paradox in light of the work of Alain Aspect in d’Orsay. Instead of infinitesimal spheres, Vigier had suggested that we use the metaphor of dominoes falling. That type of action-at-a-distance is not instantaneous. Infinitesimal spheres within the packing densities suggested by the Planck-or-Stoney-or-ISO numbers would be instantaneous. Also, in this period, an equal amount of time was spent with Olivier Costa de Beauregard.

Do the mathematics and physics of the finite begin here?

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

• 17 March 2025: Andrew Strominger, Cambridge, Massachusetts
• 16 March 2025: Stuart Hameroff, Tucson, Arizona
•  15 March 2025: George Northoff, Ottawa, Canada
•  15 March 2025: Martin Bojowald and others in College Park, Pennsylvania
•  15 March 2025: Santiago Alvarez, Barcelona, Spain
• 13 March 2025: Robert B. Laughlin, Stanford, California
• 12 March 2025: Sergey B. Yurchenko, Andijan, Uzbekistan
• 12 March 2025: Giulio Tononi, University Wisconsin, Madison
• 11 March 2025: Vitaly Vanchurin, Weston, Florida

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IM Many instant messages to thought leaders.

  1. Is it possible that the first instance of the universe is defined by Planck’s base units, continuity-symmetry-harmony, geometries, and the four irrational numbers?
  2. Is it possible that the first manifestation of those fundamentals is an infinitesimal sphere?
  3. Might the characteristics of pi describe those spheres?
  4. Might the Fourier Transform impart either electromagnetism or gravitation to each sphere?
  5. Is it possible that one sphere manifests per unit of length and time?
  6. Doesn’t that compute to 18.5 tredecillion spheres per second using Planck units?
  7. Is it possible that the densities within the earliest notations are on the order of a blackhole or neutron star?
  8. To create some sense of order with the generation of infinitesimal spheres, may we use base-2 notation?
  9. Using base-2 notation, are there 202 base-2 notations from Planck Time to the current time?
  10. Is it significant that at one second the Planck Length multiple is a very close approximation of the distance light travels?
  11. 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.
  12. Would these notations, 1-64, provide 64 possible redefinitions of a point-particle? (And, I would add a vertex.)

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Participate       You are always invited.

[*] Putin’s Heart and Mind: All people who instigate violence against another have not learned the basics about life. Arrogance and crudeness block paths for light and understanding. UNESCO is about shining light wherever there is a problem in the world. They need to illuminate Vladimir Putin’s heart and mind. They can reach him. Like all of us over 70, he is facing death; he just may be receptive to their words about education.

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Keys to this page, pi-day-2025

• This page was /pi-day-2025/. We began re-working it on 6 March 2025.
• The last update was 31 March 2025.
• This page was actually initiated on 24 February 2023.
• The URL for this file: https://81018.com/pi-day-2025/
• The headline for this article: The Revolutionaries: Four Primary Irrational Numbers
• .Wicket: Pi Day 2025.

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Audrey Azoulay, UNESCO, Paris*
David F. Robinson, Kings College (London)
Steven Strogatz, Cornell, Ithaca, New York
Kevin Dykema, National Council of Teachers of Mathematics
Pi Day to many of those already contacted
Anna Ijjas, NYU
Katherine Freese, University of Texas, Austin