PERFECTION STUDIES: CONTINUITY • SYMMETRY • HARMONY . GOALS . May.2025
PAGES: Breakthrough! | REFERENCES | RE-READING | EMAILS | IM | CRITIQUE | Zzzz
Creating a new category of numbers
by Bruce. E. Camber
First draft
Abstract
The four primary irrational numbers — π, e, √2, φ — are hyper-rational.* We propose creating a new category of numbers to be called hyper-rationals. These numbers are keys to logic, meaning, and rationality within a notation and across all notations. Without them, the universe would be irrational and could not support life as we know it. Those four equations deserve to begin to be recognized for what they do. -BEC
Proposal. We propose that the numbers, π, e, √2, φ, should herein be known as the primary hyper-rational numbers and these establish the baseline of never-ending continuity equations between the finite and the infinite. Additional deep-seated values include symmetry and harmony which are facets of pi (π). All define the infinite and qualitative. The Janus face defines the finite and the quantitative as order (numbers), relations (space), and dynamics (spacetime). All the characteristics of these four numbers are deeper definitions of the infinite and it is all qualitative.
These numbers are the key elements of our Qualitative Expansion Model.
Other numbers that might be considered hyper-rational are the Planck base units. Max Planck calculated these natural units between 1899 and 1905. First published as Zur Theorie der Wärmestrahlung in 1910 and in English as The Theory of Heat Radiation in 1914. These numbers provide our starting points for spacetime.
A third possible category of hyper-rational numbers might include the mathematical and physical constants. Still studied and debated by scholars, John Baez; Frank Wilczek, Anthony Aguirre, Martin Reiss, and Max Tegmark; and the US National Institute of Standards and Technology (NIST), just to name a few. There are many such constants; they vary greatly in scope and application, yet a few are more important than most.
A Natural Geometry Intrinsic Within the Tetrahedron and Octahedron. In the summer of 2001 when I visited one of the foremost geometers in the world, John Conway, a professor at Princeton. I had to discuss the interiority of the tetrahedron and octahedron. When I gifted him with our model of the octahedron, I asked him what he knew how those four hexagonal plates manifest within every octahedron, “Where is that manifest in nature?” He did not know. We had outlined each of the four hexagonals with colorful tape within our clear-plastic octahedral models. We had made a game out of it so our students would become familiar with its intrinsic eight tetrahedrons and six octahedrons within it. All shared a common center point. We did not compute that we were looking at an intrinsic geometry and that the best possible candidates were the four most-fundamental geometries in our universe: π, e, √2, φ.
We had been studying the sphere and pi (π) for years, but it wasn’t until 4 March 2025 did I ask Grok to consider that what would happen if we started with all four irrational numbers at the first moment of time and each with a hexagonal plate within the octahedron? How does each irrational number view the circle and sphere? How about symmetry and harmony? I started learning about these incommensurable numbers. They each had uniquely definitive things to say. It became an exploratory discussion that went on for many weeks. Finally coming up for air to explore the very nature of artificial intelligence, I returned to ChatGPT to report those results.ChatGPT’s assessments was affirming and encouraging.
Basic numbers, basic geometries, basic mathematics, and basic equations became the foundations of the Qualitative Expansion Model. We had turned to those studies that were not on the grid but with functional dependencies that caught our attention. There were no less than nine such studies and it included some of the most brilliant scholars alive today. We believe there is a logic to support a particular prime number and each discipline would emerge on the base-2 grid. We’ve challenged the leaders within these studies to begin to consider it.
Now we are turning to AI specialists to see if we could get some help to validate each notation within itself, with those notations around it, and as a strategic part of the entire model.
Here is a relational network defined by the following: continuities, symmetries and harmonies, the other three hyper-rational numbers (including base-2 notation), the Planck base units, and mathematical-and-physical constants. It is a mathematically-integrated model like none other. Thank you.
–BEC
____
References
As references are added, other resources will also be added within this website.
[*] The four primary irrational numbers — π, e, √2, φ: The initial analysis by Grok on 3 March 2025 is very telling: https://81018.com/grok-3/ (retrieved 23 May 2025). Grok set the basis for the conclusions the following day: https://81018.com/irrationals/ (Retrieved 23 May 2025) We began an earnest exploration of how π, e, √2, φ — all retired on 23 May 2025 — might be the four hexagonal plates intrinsic to the octahedron.
_____
ChatGPT: chttps://81018.com/chatgpt/
_____
_____
IM
There will also be many instant messages to thought leaders about these key points.
24 May 2025, YouTube: We are not limited to one theory. The Big Bang starts with a so-called singularity of everything-everywhere (Hawking). Instead, if we were to hypothesize the most-simple start with the most-simple three-dimensional object, the sphere, and we were to define that sphere by the Planck natural units, the results would be 18.5 tredecillion infinitesimal spheres per second manifesting within Notation-143: https://81018.com/tredecillion/ If we continue to follow those notations, within Notation-169 is a year. Within Notation-179 is 1000 years, in Notation-189, a million years, and within Notation-199, one billion years. The JWST is looking beyond Notation-197 and its 343.15 million years. Also, if within Notation-1, where it is most simple, if we might recognize the four irrational numbers and their influences. We hypothesize an intrinsic octahedral geometry that supports them, and we have all its rich qualities to analyze and speculate about the logic of the first notations. A first draft of the numbers: https://81018.com/chart/ Now we all can begin speculating about the infinite diversity of our universe with isotropy and homogeneity. https://81018.com
_____
Critique ____ You are always invited.
_____
Keys to this page, hyper-rationals
• This page will become a homepage.
• The last update was 24 May 2025.
• This page was initiated on 11 May 2025.
• The URL for this file is https://81018.com/hyper-rational/
• The headline for this article: Creating a new category of numbers.
• First teaser* is: Irrational numbers redefined as hyper-rationals….
*Or, wicket, kicker or eyebrow.
_____