The first formula changes the way we see every formula.

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Pages: Agreements | Hypostasis | Mistakes | Pi (π) | Questions | Spheres | STEM | UP

Pi (π) opens our universe & infinity
through the concrescence of continuity, symmetry and harmony.

by Bruce E. Camber

Background: No other mathematical symbol is recognized with its very own celebration and holiday. In 1988 the Exploratorium, a San Francisco museum, celebrated “Pi Day” and set a precedent for eating pies and pizza decorated with pi’s symbol and endless numbers. An historic moment in 2009, the day, March 14, was recognized as a national holiday in the USA. In November 2019 Pi Day was adopted by the United Nations (UNESCO) as the International Day of Mathematics.* As the day becomes a general celebration of the extraordinary power of mathematics in our lives, more detailed studies of pi|(π) take a backseat. This posting is a simple precaution… Pi (π) is more than we think. Within the ratio of the circumference of a circle to its diameter is the keyway to the universe and infinity..BEC

Key words: Pi (π), Pi Day, International Day of Mathematics, continuity-symmetry-harmony, sphere-stacking, cubic-close packing of equal spheres, 539 tredecillion spheres per second, 202 base-2 notations from the Planck scale, quantum fluctuations, and everything-everywhere-for-all-time

We should not lose focus of our studies of the very nature of pi (π) especially her perfections that reveal a deeper essence of this ubiquitous equation.

Pi(π) is the concrescence of continuity, symmetry, and harmony from which all circles and spheres evolve. That opens key points to explore further:

Perfect functions of the circle and sphere: Continuity is the never-ending, never-repeating numbers of pi. Symmetry is a seemingly magical, deeper nature within those numbers that become relations which become perfect circles and spheres. And, harmony is an even deeper dynamic nature of the sphere. First explored in 1820 by Joseph Fourier, his work reveals a basic harmony of every sphere.

Taken as a whole, continuity-symmetry-harmony are both finite-infinite, a.perfection and a transformation nexus.1 Our ongoing studies of the finite-infinite relation are here.

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. Continuity defines time; symmetry defines space, and harmony defines the dynamics of space-time.

Perfected states within space-time. The Planck base units were defined by Max Planck in 1899 using dimensionless constants which render natural units and a vision and definition of the smallest possible units of space and time. Logic tells us 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. An expansion rate, one sphere per unit of Planck Time and Planck Length, is hypothesized.

If so, 539 tredecillion spheres per second are generated. Base-2 notation is applied to investigate basic relations. The result is 202 notations (or doublings) to explore that encapsulate the universe.

These numbers keep the score.

The numbers are time stamped with sphere-stacking.

Cubic-close packing of equal spheres further geometrizes each number. In the first notations, there is an absence of gaps. The time interval requires efficiencies; and, densities render perfectly-fitting geometries.

By studying 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, string theory, and others are further defined.

Then the known equations that define our foundations begin to emerge. By the 202nd notation, the universe as we know it — everything, everywhere for all time — is encapsulated and all intimately connected by infinitesimal spheres that are many orders of magnitude smaller than the elusive neutrino (of which about 100 trillion pass through our bodies every second according to the IceCube Neutrino Observatory, the University of Wisconsin–Madison and the National Science Foundation (2017).

 All 202 notations are always active. Some scholar-scientists say “all time is now.” Each notation adds definition. As we go down 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. Perfectly-fitting geometries are hypothesized to come prior to the possible emergence of quantum fluctuations.[2]

 Values. These perfectly-fitting geometries might account for the first 50 notations where the continuity-symmetry-harmony of the spheres (and the extreme densities and absolute efficiencies) create no gaps and this becomes the basis for value.

If this were to become a new platform for mathematics, the world would change because mathematics would be pushing us to explore the meaning of those values as expressions of continuity, symmetry and harmony.

The positions and functions of infinitesimal spheres, especially the generation and labelling of every sphere after the first, would be the beginning of a new paradigm for the sciences. We would begin to recognize the tools to fill in the gaps between the Planck Scale and the Electroweak Scale (and between the Lambda Cold Dark Matter model or the Standard Model for Cosmology and the Standard Model for Particle Physics). 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 and ethics?

None of those questions are beyond our imaginations and our capabilities to answer. Thank you.


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

[*] Einstein’s birthday and the day of Hawking’s death. Many historians mention that the day, March 14, has unusual symbolic meaning. Two of the most influential people of our time seem to be pointing at pi (π) as if saying, “Pi (π) is a key.”

[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 likely be using new ISO base units, yet 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), 2022. Originated in 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 539 tredecillion spheres per second. That may well be considered a new definition of the cosmological constant. To begin to grasp that natural expansion, apply base-2 notation and the first second, manifests within Notation-143. The first year manifests within Notation-169. The first 1000 years manifest are within Notation-179. The first million years manifest within Notation-189 and the first billion years within Notation-199. All notations are always active.

These 202 base-2 notations functionally outline the universe; having been studied within and throughout this website, there’s more:

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 in 1926, 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 has been no exploration of the meaning of the gap. Subsequently, in May 2022, a five octahedral-gap was 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 closer-look at Continuity-Symmetry-Harmony
  2. Pi, the primary clue about the Finite-infinite relation
  3. Planck’s base units defining the most infinitesimal spheres
  4. No less than seven new perspectives: Fourier, PoincaréGaussEinsteinSmaleMilnor and many others
  5. 539 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

This homepage follows


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
•  ESA Group (PDF): The universe at 380,000 years,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
•  Scale invariance, conformal symmetries, and asymptotic freedom.

Of the 202 notations, the first 64 notations are a novel map to examine. 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 dominos 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?


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

3 April 2023: Dudley Herschbach, Harvard
3 April 2023: Cora Dvorkin, Harvard
31 March 2023: Arieh Ben-Naim, Jerusalem, Israel
29 March 2023: C.S. Unnikrishnan, DIAT (Pune)
26 March 2023: Christoph Sorger, IMU, University of Nantes
25 March 2023: Michael Spannowsky, Durham
25 March 2023: James Warren, NIST, DC
21 March 2023: CIMPA’s educators throughout the world
14 March 2023 Audrey Azoulay, UNESCO, Paris*
3 March 2023: David F. Robinson, Kings College (London)
2 March 2023: Steven Strogatz, Cornell
2 March 2023: Kevin Dykema, National Council of Teachers of Mathematics
2 March 2023, Pi Day Puzzle to many of those already contacted
1 March 2023, Edward F. Redish, Univ. of Maryland
25 February 2023, Anna Ijjas, NYU
25 February 2023, Katherine Freese, University of Texas, Austin


IM Many instant messages to thought leaders.

3:49 PM · Mar 9, 2023 @TheAtlantic @marinakoren You won’t believe this history of the universe. It’s too simple. Base-2 notation from the Planck scale: Pi day celebrations shed some light, too.

7:46 AM · Mar 9, 2023 @TonomusNEOM Three levels of cognition: continuity, symmetry, harmony work together within pi and define the finite-infinite relation: opens that door. “Let’s pi… work together.”

9:22 AM · Mar 4, 2023 @DrOsamaSiddique @Harvard_Law @UniofOxford @IGLP_HarvardLaw Natural law can be found within the fundamentals of pi (π): The relation becomes the primary real. Editor’s note, also see:

4:00 PM · Mar 2, 2023 @RushHolt Pi Day is coming up on March 14. I think we’ve underestimated the importance and place of pi (π):

  1. Is it possible that the first instance of the universe is defined by Planck’s base units?
  2. Is it possible that the first manifestation of those base units 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 539 tredecillion spheres per second using Planck units and 4605 tredecillion per second using Stoney Time?
  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.)


Participate       You are always invited.

Editor’s Note: 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. He is ill and facing death; he just may be receptive to their words about education,


Keys to this page, continuity, symmetry, harmony

• This page became a working homepage at 8 AM on March 2, 2023.
• The last update was March 21, 2023.
• This page was initiated on February 24, 2023 at 7 AM.
• The URL for this file:
• The initial headline for this article: Here’s the point!
• A headline: A Global Celebration of Mathematics
. . Sub-header and link: Our Call to Expand Our Understanding of Pi (π) for Pi Day 2023
• First byline: Pi Day points us beyond circles and spheres.


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