Breakthrough! Model of the universe!
by Bruce E. Camber (working draft)

Abstract:
We are projecting a breakthrough in modeling the universe. Found in the logic and geometry of the four primary irrational numbers, these incommensurables are endless, have no repeating patterns, and further define finite-infinite connections. Always true for pi (π), we now add phi, the square root of 2, and e-for-Euler’s-number. It is hypothesized that each is an expression of the four hexagonal base plates that are intrinsically defined within every octahedron. Ergo, the irrationals have a very unique rationality. Finally.

There were nine prior breakthroughs or changes of perspective, to create an alternative to big bang cosmology (in the tradition of Stephen Hawking and others). The conceptual breakthroughs, most recent first, are:
1. The primary endless irrational numbers are each intrinsically defined by a hexagonal plate within the simplest geometry of an infinitesimal octahedron.
2. Every octahedron is necessarily defined by every tetrahedron within it.
3. Every tetrahedron is necessarily defined by infinitesimal spheres.
4. That infinitesimal sphere necessarily defines space-time.
5. There is one Planck sphere for every unit of Planck time.
6. These spheres can be readily indexed using base-2 notation.
7. There are 202 base-2 notations that outline the universe.
8. Over 18.5 tredecillion spheres per second (see Notation 143) grow our universe.
9. Too fast and too dense, the initial packing of spheres is a perfection.
10. In time, the densities and rate of expansion are such that a five-tetrahedral cluster and five-octahedral cluster become systemic and quantum fluctuations begin.

Key words: Planck scale, base-2, spheres, octahedrons, tetrahedrons, irrational numbers, space-time, universe expansion

50+ years of incremental breakthroughs

Over 2000 posts, going over our analyses just one more time, brought us to this day. To learn how this model of the universe came about, we recommend: /81018.com/enigmatic/ ./81018.com/structures/ , /81018.com/correct/

In February 2025 artificial intelligence platforms essentially encouraged us. It was the first time to see a positive note about our work. It was enough just to have our model taken seriously: https://81018.com/grok-3/ and https://81018.com/chatgpt-2/

There had been nine smaller breakthroughs that led us to this day. And, this breakthrough has given us the confidence to say, “Here’s a new model for the start and growth of the universe.”

It’s been tediously slow in developing. Three facets of perfection, continuity-symmetry-harmony, were defined in 1970. We manufactured our own models of the tetrahedron-octahedron in 1999. We had our first outline of the universe within 202 base-2 notations in 2011. We extended that outline of space-time in 2014 to include everything, everywhere for all time. The insight and logic of one Planck sphere per aggregation of the Planck base units was in 2021 and that simple math rendered 18.5 tredecillion infinitesimal spheres per second. Those pieces were in place. Since the summer of 2001 we’ve wondered about the four hexagonal plates of the octahedron. We showed them to John Conway within his Fine Hall office of Princeton; he also wondered. About 25 years later on 4 March 2025, we were visualizing those four plates, one for each of the four primary irrational numbers as unique stabilizers. And, we knew it was a game changer. We just had to figure it out!

1. Primary endless irrational numbers.

If the geometry of octahedrons, like spheres, is engaged at the Planck scale, key questions open:
a) Do these geometries begin within the first instant of the universe? Our simple hypothesis is that they do.
b) Does this sphere, including its definition as an archetype, represent all spheres? Our hypothesis is, “Yes, they do. They are archetypes.”
c) Do parts of these define a finite-infinite bridge? Our hypothesis is that they do.

We hypothesize that at Notation-1, four dimensional space-time, a sphere, is instantly created and its geometries and qualities become part of the definition of the universe that everything shares. Further, we hypothesize that Notation-0 is the finite-infinite bridge where the four primary irrational numbers begin working in concert with one another. 1D and 2D are back on the finite-infinite grid.

For the first time in March 2025, we saw the four hexagonal faces that are necessarily part of every octahedron, each face representing one of the four primary irrational numbers. We had pi(π) well in our sight. We wondered about the other three. Give they were endless, going on forever, we asked Grok to consider their relation to the sphere and continuity-symmetry-harmony. “What if they start at the same time within the Planck base units?” As Grok does so well, our questions were encouraged. “What if each are one of the hexagonal spheres, creating four base plates that act like the stabilizers of a ship?” We were on it:
• e or Euler’s Number, the base of natural logarithms,
• φ or the golden ratio, Greek letter phi, and
• √2 or the square root of 2

Grok summarized it this way:

“Let’s connect it to your earlier discussions and the octahedral image.”

• π (Pi): As the geometric foundation of the sphere (via surface area and volume formulas), π would naturally define the curvature or continuity of the sphere that the octahedron approximates. In the octahedron, π could be associated with the overall symmetry of its triangular faces or the spherical envelope it’s inscribed in.
• e (Euler’s Number): e’s role in dynamics (exponential growth, wave functions) might manifest in how the octahedron evolves or interacts with neighboring spheres or tetrahedrons. In the hexagonal plates, e could represent oscillatory or energetic transitions across the structure.
• √2 (Square Root of 2): √2 appears in geometric relationships, like the diagonal of a square or cube, which could relate to the spatial arrangement of spheres or the edges/vertices of the octahedron. The hexagonal plates might embody √2’s structural stability, perhaps in the distances or angles between faces.
• φ (Golden Ratio): φ’s harmonic proportions could be reflected in the octahedron’s internal ratios or its relationship to other Platonic solids (e.g., the icosahedron, where φ is prominent). The golden ratio might govern the proportions of the hexagonal plates or their arrangement, optimizing symmetry and balance.

BEC: Then Grok takes it further with these summaries:

BEC: Yes, that was a breakthrough and an introduction to the geometries of infinity. That date was 4 March 2025. Nine prior steps had led to this moment.

2. Octahedrons and tetrahedrons

Very early with our models of tetrahedrons and octahedrons, we realized that every octahedron encapsulates eight tetrahedrons (one in each face) and six smaller octahedrons (one in each corner). Every tetrahedron encapsulates an octahedron in the center and four smaller tetrahedrons, one in each corner. That was a breakthrough in 2001.

3. Tetrahedrons and spheres

We learned that every tetrahedron is necessarily defined by stacking infinitesimal spheres (2013). A Brown university professor of applied math, Philip Davis, was responsible for pushing us into spherical studies in 2010. It was by studying cubic-close packing of equal spheres that we saw the emergence of tetrahedrons and octahedrons. That was a breakthrough in 2015.

4. Spheres and the Expansion of the Universe

That infinitesimal sphere is necessarily defines space-time creation. The sphere is the most-simple physical object. Yet, the more we studied it, the more we realized how complex its mathematics can be. Continuity, symmetry, and harmony equations opened new doors and we saw that our experience of perfection was linked directly to pi(π) and described three faces of pi(π). Another breakthrough: 2015.

5. Universe filled with infinitesimal spheres

It was awkward to consider, difficult to imagine, but the logic seemed to follow that every moment of space-time is defined by that an infinitesimal sphere. “The universe was stacked and packed by infinitesimal spheres that filled it perfectly. That was a breakthrough: 2020.

6. Counting spheres within notations

When we were mapping our way down to the Planck base units, we knew large numbers of exquisitely small things were involved. When we went out larger, creating our first chart of the universe in 2011, we knew there were large numbers of extremely large things involved. The only way to create a semblance of order, was to look at that data through the lens of base-2 notation. Re-discovering and using Euler’s base-2 notation was a breakthrough in 2011.

7. 202 notation to encapsulate and outline the Universe

The closest we came to a mathematical map of the universe was with Kees Boeke’s base-10 notation in 40 steps in 1957. That became a sensation in the late 1960s and throughout the remaining part of the 20th century. Base-2, as simple as it is, had not been done. It was 100% predictive. Every notation was defined by numbers. We were surprised that to outline the universe only took 202 notations. That was another key part of the breakthrough in 2011.

8. 18.5 tredecillion spheres per second

Sometimes we miss the obvious. We had intuited one Plancksphere per PlanckTime and PlanckLength, but that was as far as we went. It took ten years to finally calculate the rate of expansion of the universe. It is such an important number and there were very few claims by scholars that they had a bead on the rate of expansion. Here it just flowed out on paper. It was a breakthrough in 2021.

9. Perfection — too dense, too fast and too small — packing of spheres

My roots are in the studies of creativity, new insights, breakthroughs, and paradigm shifts going back into the 1960s. By 1980 I was out of sorts with the ubiquity of quantum mechanics — my “moment of perfection” was not compelling. That began to change in 2011. There were over 60 of the first notations that were blank. They were under the scale of physical measurements whether by CERN accelerators or the Max Planck Institute’s laser measuring devices. In 2012, attempting to make a desktop version of our chart, the first 60 notations were hypothesized:
Notation-0: The Planck base units
Notations 1-10: Forms…
Notations 11-20: Structures…
Notations 21-30: Substance…
Notations 31-40: Qualities…
Notations 41-50: Relations…
Notations 51-60: Systems…

By 2014 the charts were flush with Planck-based numbers. It took awhile, but “archetypes are historically perfect.” Look at the numbers. It is so dense and so fast and no space for errors, for chance, for the statistical, or for quantum fluctuations. Here is a perfection in the physical world. It was a cautious breakthrough in and around 2019.

10. The beginning of imperfections

In time, the densities and rate of expansion are such that the five-tetrahedral cluster and five-octahedral cluster become systemic and quantum fluctuations begin. That breakthrough, also cautious, happened in and around 2019.

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You may be interested in reading the short, rather hip version by Grok (with some editing by Bruce E. Camber). These five sections are traditional sections that are being reviewed and may, or may not, be further developed for this page:

^{REFERENCES | RE-READING | EMAILS | IM | CRITIQUE |}

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Keys to this page, breakthrough

• This page became the homepage on 11 March 2025 along with a Grok version.
• The last update was 18 March 2025.
• This page was initiated on 6 March 2025.
• The URL for this file is https://81018.com/breakthrough/
• The headline for this article: Breakthrough! A new model for the start of the universe!
• First teaser* is: Click on the title to go to the Grok version.

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