Defining Forms from Plato to Langlands


Homepages: Just Prior: Intro Langlands | 1 | 2 | 3 | 4| 5 | 6 | 7 | 8 | 9 | 10|11|12|13|14|15|16|17|18|19|20|Original

Number generation, the perfections of circles-and-spheres, formulas, and the nature of forms

by Bruce Camber   A First draft.   Your comments are most welcomed.

A partial of first column of our Universe Table
Austin, Texas:  In light of our horizontally-scrolled chart (image just above) and my intuitions about the Langlands programs, let’s open these subjects in the headline for further discussion and even debate. I envision no less than four postings or articles. For me these concepts have been enriched in light of base-2 notation starting at the Planck base units and going to the Age of the Universe. This chart encapsulates our universe within 202 exponential notations. [1] It creates a mathematically-integrated outline, a grid or matrix, within which to consider how the simple becomes the complex. The first ten notations out of the 202 should be keys and the Langlands programs just might give us the needed structure and logic.

Perhaps simple math and logic will serve us (links back to prior article). 

 “… nature is supposed to be simple and elegant, not complex and ugly.” – Nobel Prize website

Question: What generates numbers?

Propaedeutics: Numbers – functions – geometries – logic. Why? How?

Answer: Regarding just numbers, perhaps a natural place to begin is with the transcendental, irrational and incommensurable numbers that are never-ending and never-repeating.[2] Though a constant (the sequence of numbers doesn’t change), these are random number generators. Known through formulas, these numbers are homogeneous and isotropic. Pi is a perfect example.

Pi is pi is pi no matter where you are and what position you are in. It is isotropic and homogeneous. It defines continuity/order and symmetry/relations as well as asymmetry and randomness. It has a built-in function and geometry!`

Next come the fundamental physical constants and mathematical constants. These simple numbers are all ratios; they have a certain kind of continuity (the number is the number) yet each is uniquely dynamic. Each has a specific function.

1Discussion: From the first moment, the array of numbers is dense and intense, yet it is not infinitely dense and infinitely hot (as the big bang theory would have it). And, it may be a singularity [3] in the technical definition of the word, however, there is nothing singular about it. With just these numbers mentioned, it is more like a nexus for transformations, an  amalgamation of equations and ratios. The most simple and most perfect use pi to create geometries and symmetries and these become our first forms.

Our first forms, right here, are with real numbers (Planck base units)
and real formulas.  One might guess that it all begins with pi and
complexity begins immediately!

Question: How do we know each number is the right number?


Answer:  The initial sequence of numbers, 1-10,  was established and that order is taken as a given.  The formulas that use pi define basic relations. Within pi there is an unchanging sequence of randomness, a flow of trillions (upon trillions) of non-ending, non-repeating numbers, yet the actual number that defines pi does not change. That fact is worthy of our deepest reflections.

Discussion: Pi defines (1) an infinite-finite relation, (2) a constant-and-universal sequence of non-ending, non-repeating numbers, (3) a geometry of circles-and-spheres, (4) the centerpoints of all circles
and spheres, and  (5) straight lines between centerpoints.

All quite remarkable. That’s five stars in the pi crown!

Question: How are these circles and spheres being generated?

3eAnswer: It is math — a special math to be sure — yet at the Planck scale the sequence includes a length, a time (which we might consider to be a transaction time intimately tied to a length and light), a mass, and a charge. The numbers and dynamics are infinite. The circles and spheres are finite.

Discussion: Here there is a very real mathematically-defined substance, so very small, I do not believe it has been studied as such among the scholarly communities. My first analysis in the early summer of 2017 was very preliminary and quite chunky.
Though necessarily starting at the Planck scale, I quickly
— too quickly — jumped to the 31st notation.

Are the most substantial secrets within these first ten notations!?!

Question: How do we visualize that first notation?

: Circles and spheres are being generated “faster than the eye can see.” There are no simple doublings as I have written and imagined from December 2011 until November 2017.

Discussion: All the Planck numbers have multiple “jobs” built into them. There are so many actual and potential formulas to consider! But trying to keep it simple, consider literally quntillions of spheres and circles, edges touching, radius to radius, diameter to diameter.

Though an infinitesimally small length, the second, third and fourth doublings are virtually predefined, and virtually immediate and constant. It is a dynamic pile-on with actual numbers for length, time, mass, and charge. Even before the first doubling these spheres align in such a way to create the possibility to create tetrahedrons-and-octahedrons. This discussion may well become a book some day!

Edward Frankel [4] believes Robert Langlands conjecture is opening the path to a Standard Model of Mathematics whereby numbers, geometries, and functions are united and then begin to individuate with unique flavors, textures, and dynamics. It was in the spirit of Langlands-Frankel that this page was engaged.

This is our second posting on a journey to learn from Edward Frankel and his colleague and friend, Robert Langlands.[5] So, I will be looking through their writings for their statements about boundary conditions, spheres, pi, never-ending and never-repeating numbers, and finite-infinite to develop the next report on the fabric of the universe and notations 1-4. – BEC

Our simple first principles

  1. Everything starts simple before it can before complex.
  2. Numbers create continuity and define order.
  3. The circle and sphere are primary forms that define symmetry and create relations.
  4. Never-ending, never-repeating numbers are the beginning of uniqueness and all dynamics, and these numbers begin to define infinite-to-finite relations.


If you click on 81018 in the top navigation bar, you can go to the chart of 202 notations with all the figures doubling from the Planck Scale to the Age of the Universe.  The chart looks rather static. The doublings look static.  I don’t think anything is static. As I view it today, within that first notation, there is an endless stream of circles and sphere, edges touching, radius to radius, diameter to diameter, all their centerpoints somehow in the nexus of transformation between the finite and infinite —  all active, never-ending, creating a first layer of the fabric of the universe.

Every notation may well be a layer. Yes, I can now see this exquisitely fine layer of notation 1 surrounding all other layers and from which all layers emerge.

Sphere to tetrahedron-octahedron couplet
Imagine this dynamic from the beginning of time where every sphere is pushing out the next.

Here is the initial push of inflation coupled with Planck charge, constantly expanding. Yet, even as they emerge, each has the potential to be uniquely defined with one or more of the many inherent mathematical definitions within this nexus of transformation. The second doubling emerges from the first as clusters of spheres begin to create tetrahedrons. Remember this dynamic image that has been on many prior pages? [6]

There is more to come and updates of that which is already here!

Here are a few of the pages preceding this page:

  1. November 15:  Before we can understand the complex…We need to understand the simple things. An introduction to our study of the Langlands programs.
  2. November 12:  Seven reasons to look more deeply at our chart (at the top). It is still a  largely-unexplored  model of the Universe
  3. November 9:  Over 1000 Simple Calculations Chart A Highly-Integrated Universe
  4. November 8: We live in an exponential universe.


[1] All links to outside resources will be first discussed with these endnotes. Frequently used links to pages inside this account will often go go directly to those pages without discussion. All other links will be first discussed within these endnotes. The chart is

[2] Never-ending (also called non-terminating) and never-repeating numbers define a nature of infinity.  These are known through their relations and equations. We are just beginning this study and will look up all the resources we can find within academic circles and beyond.

For further study: (Many more to come)
Priya SubramanianResearch Fellow Applied Mathematics, University of Leeds

[3] Physics and mathematics each have their own definition of a singularity. Neither is a commonsense definition of “just one” or a point; their singularities open to an infinity.    The singularity defies a simple logic. Just look at all the activity in this transformation nexus between that which is defined as infinite and that which is defined as finite.

[4] The page on the emails to Frankel has numerous references to his excellent book, Love & Math:

[5] The page of emails to Robert Langlands has many references to his work since 1965:  And, Wikipedia has a rather in-depth study of Langlands; begin with

[6] is another page, the most recent, where this dynamic image was used. Another document is: