Finite-Infinite relations studied through pi (π) open a different Universe

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Infinity, Pi, And Ultimate Questions
by Bruce E. Camber

Living scholars have key parts of answers to our most important, enduring questions. Yet, we can always know more about our universe and ourselves and we can know it all more profoundly. In December 2011, by exploring the internal parts of the tetrahedron and octahedron, a simple base-2 progression became a highly-integrated, mathematical view of the universe. Max Planck provided the base units for length, time, charge, and mass. That chart has 202 notations from Planck’s units to the total size, age, energy and mass of today’s universe. To interpret the chart, studies of pi’s most infinitesimal, scale-invariant spheres start with cubic close-packing of equal spheres. When some of the dynamics of pi did not appear to be finite or quantitative, those facets were projected to be infinite and qualitative. That’s all re-introduced here just to see if these concepts help us over those conceptual hurdles that prevent us from seeing how all things everywhere throughout all time are part of an intellectual and mathematical grid that necessarily binds all disciplines. By exploring those hurdles we hope to begin a series of new syntheses.[*].– BEC

I. The finite-infinite relation. Always controversial and exceedingly provocative, our simple hypothesis is based on the “never-ending, never-repeating, always the same, always changing” numbers of pi ). Such an enigmatic statement begs questions about the essence of pi.[1]

The pi equation — never-ending, never-repeating, always the same, always changing — is a key. Although those words also describe other facets of mathematics, today we focus on our very old friend, pi, as in π r2 from our high school geometry days.

There are three facets of pi that are not finite or quantitative so we assume (hypothesize and/or hypostatize) these facets define the infinite and the qualitative (unique class and category).

Continuity is our first facet of infinity. It is the very nature of order. Within the finite it looks like a string of numbers and feels like time. Pi qualifies; it’s an equation that has never-ending results that are always the same and always changing.

Symmetry is the second facet of infinity. It looks like geometries and is the very nature of a relation. Within the finite it feels like space. Pi qualifies; it’s a symmetry that generates symmetries. It’s an equation that generates equations.

Harmony is the third facet of infinity. It is the very nature of dynamics; and within the finite, it is always cyclical (periodicity) and experienced as space-time moments. Pi’s numbers, geometries, and equations (Fourier transform and others) are here within an eternal dance and there’s a domain of perfection which may be experienced as a moment of perfection.

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.

Review: In this model the infinite is profoundly within the finite. It is not finite, but actively imparts qualities to the finite. For those who follow David Hilbert, please stay open. Pi’s three facets of the infinite are really real. These are not just abstractions, but actual realities of every circle and sphere. These three qualities condition the finite. Everything-everywhere-for all time, is in accordance with numbers, geometries, and equations; and, it all has some manifestation of these infinite qualities.

A rather different start to grasp the finite-infinite relation, our understanding of the infinite starts with pi and her most infinitesimal circles and spheres.[1]


II. An infinitesimal sphere to 202 base-2 notations. First, infinitesimal spheres stack, pack, and give rise to tetrahedrons, then octahedrons, then the Platonic solids, then to every kind of geometry and equation that we can possibly imagine.[2]

Given the current calculations for the age of the universe — generally between 13.81 and 14.1 billion years — there is a range. There is a starting point for space-time and mass-charge. The endpoint is Now.

Given the nature of sphere stacking-packing-and-generating tetrahedrons and octahedrons, a natural doubling function within the universe is first seen by going inside a tetrahedron, by dividing each edge by 2, connecting those new vertices, and going within one step. Each face of tetrahedron becomes a face of smaller tetrahedrons and an octahedron.

This base-2 model of the universe began to emerge in 2011, initially only with Planck Length. In 2014, Planck Time was added and in 2015 Planck Charge and Planck Mass. In 2016 the chart was laid our horizontally to follow the progression of numbers more readily.

Using dimensionless constants in 1899 Max Planck calculated base units of time, length, mass and charge. These numbers give us a definition of the first manifestation of physicality, an infinitesimal sphere. Planck Time de facto gives us a rate of expansion of one Planck sphere per Planck unit of time which computes to 539 tredecillion spheres per second. These numbers logically define the first instant of time, then the look-and-feel of the expansion of the universe.

Perhaps such a rate of expansion could eventually be considered a cosmological constant.

One very basic combination using Planck Length and Planck Time is at one second. That calculation, Planck Length divided by Planck Time, is within .001% of the NIST/ISO value for the speed of light set in 2019. Other well-known values can be similarly tested. There is a thrust and compactification, whereby what manifests within the earliest notations reflect what defines a perfected-state within space-time. It is hypothesized that quantum indeterminacy first begins to manifest with a five-tetrahedral structure sharing a common edge. Hold those five tight to the common edge and there is a 7.356103+ degree gap. That gap is also natural within every expression of dodecahedral or icosahedral structures. Those spatial dynamics are currently generally classified as quantum fluctuations.[2]


To the grid! To the grid: Specialized disciplines focus on a part of our problem.

III. The Nature of Space and Time: To be a scholar, a person gets to know some part of the 202 notations most profoundly. First principles are often assumed, “That’s commonsense.” Yet commonsense is not a sense and it is not very common. That is to say, we absorb our concepts of space and time from our culture and both are stubbornly held and incomplete.

Today, all 202 notations are always active (like when each first manifests).

“All notations are always active” redefines space-time and begins to define “the Now.” [3]

That is — and will continue to be — a primary challenge for many, especially within the eight specific studies into which this website continues to be focused: (1).Langlands programs, (2).string theory and M-theory, and.(3).SUSY (and going Beyond the Standard Model), (4).Causal Sets, (5) Causal Dynamical Triangulation, (6).Loop Quantum Gravity, (7).Scalar Field Theory, and (8).Spectral Standard Model. All are within our on-going purview. All are in need of seeing how they connect to the grid.

Also important is to see how these foundations give rise to our ethics and values.


IV. The challenge. Within the footnotes (below), those eight subjects (cited just above) are each linked to their Wikipedia page. Most of the Wikipedia editors are either experts within that topic and/or work with the leading scholars within the field. And, those writers are usually very good to flag the conceptual impasses within their discipline.

Those conceptual impasses will be a focus in light of points 1-2-3 above. How can scholars be conclusive if the building blocks are not defined?

Within each focus, there are leading advocates with their most basic concepts. Although my prior studies of each have been relatively perfunctory, for these reports, our next step will be, after re-engaging their work, to ask each of those leading scholars, “What is your working concept of the infinite? What is the role for pi (π) and spheres? What is your concept of time?”[4]

Some of our best scholars want to do away with the concept of infinity, and/or time. Pi and the simple sphere are old, well-known studies, and largely overlooked. Yet, just maybe, that old-old equation might harbor simple secrets that could change everything.

Langlands programs. For example, by using Wikipedia and Google and by asking questions in many possible ways, we quickly learn that within the Langland programs, there is no necessary working relation between their programs and infinity, pi, and spheres. Their automorphic forms appear to jump over continuity, assume symmetry, and begin with generalizations of periodic functions.

I believe there are unique footings to discover within the facets of continuity and symmetry. By jumping into periodic functions, it appears to me that Langlands programs is starting within Notations-65-to-67 and perhaps as low as Notation-50. There’s much too much left unexplored within Notations-0-to-50.

Within our model, Langlands automorphic forms begin to be defined within the first ten notations. There is plenty of room for their number theory to work its magic, yet it will be even more magical when their number theory engages infinity, pi (π) and spheres, and the very nature of space-time.

In the same spirit, string theory, Mtheory, and SUSY are too quick to jump into point-like particles and one-and-two dimensional objects, assuming such things are part of particle physics. It appears to be just a bit too quick. Those first 64 to 67 notations are a little faster than the zeptosecond range; nobody is measuring those interactions. Why would the three-dimensions of space begin to manifest as two-dimensional equations or as singular points until into (perhaps “well-into”) Notation-0. And there, these may begin to be elements of a bridge between the finite and infinite. Those three dimensions of space may actually manifest in several types of transformations even on the bridge. That is all to say that there is no clear consideration of infinity, π (pi), and spheres and the jump into levels of abstraction provides no clear path to the working physics of our time.


We’ll see what our leading scholars within those eight disciplines have to say in response to our emails. Ours is an earnest and sincere exploration; nevertheless, within these disciplines, with just my cursory knowledge and simple assumptions, it is too early to know what they’ll say. If I am wrong, then I am seriously wrong and a fool. It will be good to know how and why I have been foolish. Yet, through it all (ten years of explorations using these parameters), it seems to me that our scholars are almost there. They are getting close and that there are just a few conceptual hurdles to go! So, why not try to get over those hurdles?

Thank you very much. -BEC


Editor’s notes: Perhaps not evident, yet by writing as if I had a group of high school students and teachers collaborating with me, reading every word, perhaps eventually we’ll get out of the weeds and closer to first principles. Yet, the most dynamic part of this page follows. These are the evolving endnotes-footnotes, references, emails, and instant messages, yet be forewarned, sometimes these people are quite deep in the weeds!

Of course, your comments are most welcome.



* Many scholars. It seems all are in a search for three master keys: (1) a Finite-Infinite key, (2) the “Integrative Systems that Structure the Universe” key, and (3) the “Redefinitions of Space-Time (Mass-Charge)” master key. There are many prior homepages that touch on these three topics and there are many ways to access those pages. One of my favorite ways is to click on the left arrow at the top of each page. That will take you back, homepage by homepage, to the beginning of this website in August 2016. The work on these concepts started in a high school in 2011.


[1] Continuity-Symmetry-Harmony (CSH). By imbibing the very nature of pi, can we grasp the nature of the infinite? Does CSH create a corresponding Order-Relations-Dynamics? Is that enough to initiate this understanding of infinity and the infinite? Is it the best possible? Can we define the very first instance and the dimensions of the challenge?

Of course, I believe the answer to all these questions is “Yes.” The assumed many-sided perfections of the infinite are imparted to the finite. In much less than a zeptosecond — a trillionth of a billionth of a second (or a decimal point followed by 20 zeroes-and-a-1 is currently the shortest scientific measurement of a unit of time and it is within Notation-74) — there is too little time for imperfections. One might project that this special combination of geometries and equations may not have been manifest within the physical domain for years and years. The first year manifests within Notation-169. And even then, the patterns and thrusts of perfection may well be so great that it takes thousands if not millions of years (Notation 189 and beyond) before there is a quantum fluctuation. Of course, there will be more to come…


[2] Structure. Perhaps there is a way to debunk the hypothesis that there is “one Planck sphere per Planck unit of time.” That would undercut the entire model. We’ve been at this study for almost ten years. If we are wrong, it would be good to know it sooner than later. We could take a break without this monkey on our back. We’ve been asking many of our leading scholars for many years, “What is wrong with this model?” Can it be that the infinite is qualitative and the finite is quantitative?

Infinitesimal circles-spheres-tetrahedrons-and-octahedrons actually provide a good foundation. The study of cubic-close packing (ccp) of equal spheres gives us that much. Then, consider the 539 tredecillion spheres per second. Surely that gives more than enough structure with which to work. The thrust for perfection would provide more than enough homogeneity and isotropy for a smooth beginning.

Of course, there will be much more to come…


[3] Logic. It is a simple logic. One notation develops and the next begins. Within one second, over 143 notations have become active. Within one year, 169 notations; then within one million years, 189 notations. Within a billion years, 199 notations have become active. The 202nd notation is the only notation that has a directional asymmetry. That is being addressed in rather convoluted ways. We will first examine many of these through our studies through their Wikipedia pages noted here: (1).Langlands programs, (2).string theory and Mtheory, and.(4).SUSY (including Beyond the Standard Model work), (5).Causal Sets (and Causal Dynamical Triangulation), (6).Loop Quantum Gravity, (7).Scalar Field Theory, and (8).Spectral Standard Model. Then, as stated, we will engage scholars within each study to ask how they define infinity, time, and spheres, and what the role, if any, pi has within the start of the universe and its current expansion.

Of course, there will be more to come…


[4] It is all a wonderful challenge and ginormous puzzle. Those eight studies represent the bleeding, leading edge of scientific inquiry. Their scholars are among the best and most articulate. They catch the hot potato (the scrutiny of the world) and hold onto it for as long as possible before throwing it on to the next. The scientific press will have written a few stories and then chases after the next story.

There’s got to be a better way.

Yes, of course, there will be more to come…


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More references from within this website. Early work that laid the foundations:
1. This work began in 1971 within the study of the 1935 EPR paradox.
2. It was part of a conference at MIT in 1979 in search of first principles.
3. There are many pages that consider the first instants of the universe.
4. There are also these presuppositions and assumptions.



  1. RE: Investments in Educator Development
    TO: STEM Solutions and Partnerships
    National Math and Science Initiative, Dallas
  2. Alex Berezow, Big Think
  3. Ngo Bao Chau, University of Chicago on Langlands Programs
  4. Siska De Baerdemaeker, Stockholm University
  5. Jeremey Nicholas Butterfield, Cambridge University
  6. Priyamvada Natarajan, Yale astrophysicist
  7. Sara Leikin, Ed.D.



@SheerPriya Prof. Dr. Priyamvada Natarajan, Yale astrophysicist.
Author: Mapping the Heavens. Scholar: Exotica in the universe — dark matter, black holes
@SheerPriya If we were to map the universe, assuming Planck base units as the start, apply base-2, there are just 202 notations: to today. Blackholes may be more diverse than we think: Let us hope that the dark matter veil can be lifted.

@tribelaw (Laurence Tribe) @MollyJongFast @JohnAmato The problem with both right-and-left is the de facto little worldviews tie the mind up in knots and contortions which become lies and smears. We need a mathematically-integrated view of the universe. A start is here: Also sent to @lukebroadwater (NYT-Congressional).

To many others: If there is ever going to be a little harmony in this world, we’ll need to break out of our little worldviews for an integrated view of the universe. Our’s a simple start:


Invitations and Collaborations

With whom do we collaborate? Of the hundreds of people who visit this site every month, who among them might want to extend a right hand and say, “Let’s work together.” Our only thrust is that the foundations of this universe and life itself be seen in light of infinity and the continuity-symmetry-harmony that the infinite engenders. Please, talk to us. Thank you. -Bruce


Key dates for this document, almost.


Afterthoughts: A Personal Addendum. The concept of a worldview was formalized in the 1700s first by Immanuel Kant and then by the general philosophical community. Yet, the intellectual and religious communities have had worldviews dating back to ancients like Plato. As the world began shrinking, we learned that there are many different worldviews and there are tensions among them.

Even when the German word is used, Weltanschauung, it’s still not big and comprehensive enough. Although Sir Isaac Newton and Immanuel Kant captured the world, the mysterium of a German word does not encompass the universe. As a concept, it is dated, incomplete, and not quite big enough. We need to move on to an integrated, mathematical view of the universe. -BEC


+ Final notes on Tuesday, December 28, 2021 at 3:45 AM. Close to 6 AM (TZ-19) each day, the days listed at the top of this page get advanced by one digit. It should be a relatively easy program to write, yet I rationalize that I do it manually just to remind me of our granular sense of time. TZ-19 is time zone #19, assuming that the International Date Line is #1 and Greenwich Mean Time falls within Time Zone 13.

End of year reflection: There are signs that we my soon be closing this very long chapter of being simply “idiosyncratic.” Our Infinitesimal Sphere is a particle by any scale of definition. So, if we are to refer to that sphere as a particle, what do we call it? There are 369 Google reference to “Archetypal Particle” but no long-standing use by any one group, so we’ll put it on a list of possibilities. “Primordial Particle” is also currently used by a diversity of people. That term also goes on our list to research.

Perhaps “The First Particle” will win the day.

We’ll get this done, hopefully with the earliest days of the New Year. –BEC