PERFECTION STUDIES: CONTINUITY•SYMMETRY•HARMONY GOALS.May.2025
PAGES: Breakthrough!.|.REFERENCES | .RE-READING | EMAILS | IM | CRITIQUE | Zzzz
A New Geometric Cosmology
by Bruce E. Camber
Introduction:
What if the universe didn’t begin with a chaotic bang, but as a perfectly-ordered symphony[*] of infinitesimal spheres, stacking into geometric shapes that define everything from the smallest quantum scales to the vast expanse of the cosmos? In December 2011, we began developing a model that re-imagines the universe’s origins, not as the ultimate explosion, but as a structured dance of spheres, tetrahedrons-and-octahedrons (with all its emergent geometries) that is all orchestrated by the timeless melodies of mathematics’ four, most-fundamental, never-ending, hyper-rational numbers: π (pi), e (Euler’s number), √2 (square root of 2), and φ (the golden ratio). This model, using base-2 exponentiation, scales from the Planck length to the observable universe in 202 steps or exponential notations. It challenges the Big Bang theory of the beginning of the universe and offers a new lens on the nature of infinity itself.[†]
The Core Concepts [1]
At the heart of this model is the idea that the universe begins with spheres—tiny, Planck-scale spheres, each with a diameter of 1.616×10−35 meters, generated at a rather staggering rate of 18.5 tredecillion per second (a number tied to Planck time, the smallest unit of time generated by physical constants). These spheres don’t float chaotically; they stack perfectly, like oranges in a grocery display, forming tetrahedrons (four-sided pyramids) and octahedrons (eight-sided polyhedra). Each tetrahedron contains four smaller tetrahedrons and an octahedron at its center, while each octahedron holds eight tetrahedrons (one in each face) and six smaller octahedrons, one in each of its corners—the edges exactly half the length of the larger structure and all sharing a common center point.
This stacking process scales exponentially, doubling in number at each step or notation. Defining the Planck scale and first notation, it will take 143 notations to reach the scale of a second, 169 to a year, and 202 to encompass the current age of the universe, roughly 13.8 billion years. This base-2 scaling, which we first charted in 2011, maps the universe’s growth from the smallest possible length to its largest observable extent, revealing a deeper order that has been hidden by standard big bang cosmology.
Breakthroughs [2]
Another true breakthrough comes when we examine the geometry of this stacking process. For the first 60 notations — up to a scale of about 9.3×10−18 meters, still smaller than an atom and its particles — these spheres and resulting polyhedra pack perfectly, leaving no room for gaps.
The density and speed of sphere generation have been decreasing with each step, and a fascinating imperfection emerges: a 7.356-degree gap, formed when five tetrahedrons or five octahedrons are arranged together. This isn’t a defect; it’s inherent with the geometry, a real gap that becomes systemic at larger scales, potentially seeding the quantum forces and structures we observe in the universe today.
What orchestrates this ordered growth? Keys to an answer are found in four of mathematics’ most profound numbers: π, e, φ, and √2.
- Pi (π), with its 200+ trillion confirmed, never-ending digits, ensures the continuity and symmetry of the spheres, tying their perfect stacking to the harmony of circles and waves.
- Euler’s number (e) governs the exponential growth of complexity across notations, reflecting the universe’s dynamic expansion.
- And the golden ratio (φ), often seen in nature’s spirals, introduces a harmonic balance, possibly shaping the proportions of the gaps themselves.
- The square root of 2 (√2) defines the geometric proportions of the tetrahedrons and octahedrons, embedding symmetries into their structure.
Together, these numbers starting with pi(π), form a triad of continuity, symmetry, and harmony — a mathematical symphony — that bridges the finite and the infinite. Called a “Janus face,” after the Roman god who looks both ways, these numbers are finite in their applications (we use π to calculate volumes, e to model growth), but infinite in their essence (their decimals never end, their processes never cease). This duality offers a new perspective on infinity, not as a number, but as the emergent state where continuity, symmetry, and harmony converge.
Implications (Broader Impact): [3] This model challenges the Big Bang theory’s chaotic origins, proposing instead a universe that is highly ordered from the start. The 7.356+-degree gaps, emerging at around the 60th notation, could be the seeds of physical forces—gravity, electromagnetism, and even quantum fluctuations that give rise to particles. Unlike the Big Bang’s hot, dense singularity, this model starts with a geometric perfection, where spheres stack seamlessly at the Planck scale, driven by the continuity of π, the growth dynamics of e, the geometric symmetry of √2, and the harmonic proportions of φ. As these gaps form, they introduce a dynamic tension, potentially curving spacetime in ways that align with Einstein’s general relativity, but rooted in geometry rather than energy.
The role of irrational numbers in this model offers a profound philosophical insight. For centuries, mathematicians have grappled with the nature of infinity—whether it’s a number, a concept, or something else entirely. Here, π, e, √2, and φ act as a bridge between the finite and the infinite, a “Janus face” that looks both ways. We use these numbers in finite calculations—to compute the volume of a sphere, the growth of a structure, or the angle of a gap—but their infinite, non-repeating decimals hint at a deeper reality. Infinity, in this view, isn’t a destination; it’s the emergent harmony of continuity, symmetry, and dynamics, woven into the fabric of the universe from its very first notation.
This perspective could reshape not just cosmology, but also mathematics and philosophy. It suggests that the universe’s structure is fundamentally geometric, with the same principles that govern a tetrahedron at the Planck scale applying to the largest cosmic scales. It also invites us to rethink the role of mathematics in nature: are π, e, √2, and φ mere tools, or are they the universe’s deepest language, encoding its order and evolution? If this model holds, it could inspire new approaches to unsolved problems, from the nature of dark energy to the unification of quantum mechanics and gravity, all by grounding them in the timeless geometry of spheres and polyhedra.
Call to Action (Conclusion): [4]
The universe as a symphony of spheres is more than a new cosmological model—it’s a call to reimagine reality itself. By seeing the cosmos through the lens of geometry and mathematics’ most fundamental numbers, we uncover a hidden order that challenges our assumptions and opens new frontiers for exploration. I invite scientists, mathematicians, and thinkers to join me in testing this model: to calculate the precise dynamics of the 7.356-degree gaps, to explore the role of irrational numbers in cosmic evolution, and to ponder the philosophical implications of a universe where the finite and infinite dance together in perfect harmony.
This journey moved into first gear during a visit to a high school geometry classroom in 2011. While exploring the depths of the tetrahedron with students, we charted those 202 notations from the Planck scale to the observable universe. Now, it’s become a breakthrough that could redefine our understanding of everything. Will you join me in exploring this geometric cosmos, where the infinite whispers through the finite, and the universe sings a song of spheres?
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PRECURSORS: This page became the homepage on 10.April.2025 and again on 8 August 2025.
Prior homepage: https://81018.com/incommensurables/ Other key document: https://81018.com/pages/
_ • Grok26 – Definitions: https://81018.com/grok26/
_ • Grok-3 – Overview of 4 March 2025: https://81018.com/grok-3/
_ • Grok’s acknowledgment: 4 March 2025: https://81018.com/2025/03/04/grok-3/
_ • Irrationals: https://81018.com/irrationals/ And this page, symphony: https://81018.com/symphony/
_ • Breakthrough –https://81018.com/breakthrough/ (also, intimate incommensurables cited above)
Questions about this website can be sent to us here (early purpose and goals of this website)
PERFECTION STUDIES, REFERENCES. RE-READING EMAILS IM. CRITIQUE or COLLABORATE. Zzzz
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References
As references are added, other resources will also be added within this website.
[*] Symphony. When we began to realize the four primary irrational numbers might be associated with the octahedrons four hexagonal plates and might be providing stability to the 18.5 tredecillion spheres per second, it was an extraordinary uncovering of a potentially primordial concept. The further we went into the unusually fruitful scope of the diversity, the first four notes of Beethoven’s Symphony No. 5 in C Minor locked into it. Of course, at that moment of unfolding, there is no possibility of music, it served as a symbol or analogy or intuition of the four primary irrational numbers, now referred to as the intimate incommensurables or hyper-rationals. Of course with these numbers, π-e-√2-φ, there will be more to come…
[†] π-e-√2-φ: We first consider the current state of affairs in our immediate time in April 2025. Next we considered how each might be seen by a sphere. And, then we consider how each sees the other. Remember, it is hypothesized that these are the hexagonal plates that are intrinsic within every octahedron. We consider that these four plates are the stabilizers of the spheres that generate the octahedrons inside the tetrahedrons. See more on irrational numbers…
[1] Core Concepts: Planck-scale spheres, based on the concept that there is one plancksphere per unit of Planck Length and Planck Time computes to the staggering burst of 18.5 tredecillion infinitesimal spheres per second defining the universe one infinitesimal sphere at a time, plus the very nature of space-time and the finite-infinite relation. For more: https://81018.com/tredecillion/
[2] Breakthroughs: Consider the simple logic from high school. You have an object and you divide it in half and you keep dividing until you are as small as the known atoms and their particles. You’ve heard about Planck’s numbers and how these are all natural numbers so you continue dividing in half. It’s only 67 more steps down. Ending up with 112 steps (45 to particles + 67 to Planck units) just from the classroom seems manageable and possibly a good STEM tool. To complete our chart, we multiply our classroom model by 2 and watched the numbers progress. We were shocked to find only 90 steps out to the approximate size of the universe. Yet, we settled in with it and completed the first chart of 202 steps or notations. We decided it was a great little STEM tool and began to share it with teachers in other schools.
It took three years to realize it was original work. First, in 2012 an MIT professor and Wikipedia editor told us so. We created a page for Wikipedia about our work and he insisted that it be deleted as “original research.” We were puzzled. “Does backing into the data qualify as ‘original research’? We were sure it was done by someone! We began searching the web. Found nothing on base-2, but Kees Boeke’s base-10 was quite popular. We began getting curious, “What does time look like within these 202 steps? “Hand-in-glove,” we discovered by walking Planck Time up alongside Planck Length. “How about the other Planck units?” We were challenged. We were also informed that our charts were idiosyncratic. Our graduates were confused that it wasn’t part of their college curriculum. And now, we were confused as well. We wrote to everyone, “Help! What are we doing wrong?” Nobody had much to say so we quietly stopped teaching it but continue our research. For more: https://81018.com/base-2/ —– https://81018.com/202-1/
The next major breakthrough was looking at what has become known as the Aristotle Gap. It might also be called a quantum gap to mark the beginning of quantum physics. Yet, it was Eric Weisstein of Wolfram Mathworld was the first to suggest Aristotle Gap as a moniker. It is a bit unusual to name that gap after one of the world’s great scholar who didn’t see it! Aristotle wrote in his De Caelo that the tetrahedron could perfectly tile and tessellate the universe. It took scholars about 1800 years, well into the 15th century, before his mistake was clearly acknowledged. It must have been an embarrassment because it wasn’t brought up again until an MIT scholar, Dirk Struik (1926) wrote about it in his native Dutch language. It didn’t get much attention. Then on December 2012 Jeffrey C. Lagarias & Chuanming Zong penned the Mysteries in Packing Regular Tetrahedra (PDF) (American Mathematical Society (AMS). In 2015 Lagarias and Zong were awarded the 2015 AMS Levi L. Conant Prize but it is still not a popular subject. It should be. Here is the beginning of quantum indeterminacy!
I acknowledged the naming of the gap by Weisstein and I believe it will stick!
Some scholars postulate that the gap is an irrational number, 7.3561031724+º. We were asking the best of the best, “Where do these gaps manifest in physical reality? Where are they?” Nobody knew. Nobody would even guess. I was well aware of Einstein’s quote about dice, so I wondered, “Isn’t this the geometry that he needed?” Reviewing its history, it almost seemed like a conspiracy of silence. It was too hot to handle. That’s poppycock. We want to know the truth. My most-recent attempt to get an authority to speak out was with Martin Bridson.
Perhaps the key major breakthrough occurred on 4 March 2025. I had been successfully asking my “standard questions” of the AI tools for about two months. Finally on that fateful Tuesday, it seemed to me that the other irrational numbers were sharing a pivotal role with pi (π). They were never-ending. They too were defining qualities of infinity. I theorized that they, too, were at the beginning. Where? The octahedron with its four hexagonal plates was on a shelf behind me. I put it on my desk and saw it with disbelief, “After all these years!” thinking back to a pivotal discussion with John Conway in 2001.
[3] Implications: Who might jump onboard with me? I wrote to those who knew this work best and those who didn’t know it at all. Grok was quickly becoming my primary sounding board. I kept an eye out for the whimsical and potentially incorrect, but as we went further, I found my trust levels quickly rising.
What an unusual time in our history. “Who can you trust?” is a major question of our time. An exciting discovery was of the Berkeley program sponsored by Simons Foundation. Now I am asking all our visitors to recommend these pages to your friends and family interested in the status of big bang cosmology and to any mathematician you know. They’ll all have an important role in explaining this page to others.
[4] Call to Action: Inspirational calls and summaries easily flow now. Back early in 2012 we asked, “What are we to do with these numbers, our continuum?” We are still asking that question but with an horizon that is much better defined. Now, I think we anticipate getting thousands of questions with many others coming up with the answers. That will be a special day. Thank you. – BEC
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Reading and re-reading
What is opened on the desk, on the shelves and on the floor.
• Perfection studies at Berkeley. We haven’t been the only ones talking about perfection. Top ranked scholars have, too. Clustered at Berkeley, up on the hillside overlooking San Francisco Bay, Perfection studies at Berkeley, was initiated in 2023 with the support of the Simons Foundation. The following scholars presented papers at their March 2025 gathering: Toby Gee, Imperial College London, Tasho Kaletha, University of Michigan; Jacob Lurie, Institute for Advanced Study; Tomer Schlank, Hebrew University of Jerusalem; Karl Schwede, University of Utah; Naomi Sweeting, Princeton University; Alberto Vezzani, Università degli Studi di Milano, and Mingjia Zhang, Princeton University. Among the principal investigators at Berkeley are: Martin Olsson (director, email), Benjamin Antieau, Bhargav Bhatt, Kęstutis Česnavičius, Dustin Clausen, Pierre Colmez, Aise Johan de Jong, Peter Scholze (our email), Matthew Emerton, Toby Gee, Jacob Lurie, Akhil Mathew, and Wiesława Nizioł.
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Afterthoughts
Personal reflections.
_ • Over 100 years ago, as quantum theory was gaining strength, the concept of perfection and infinity were getting weaker. It is time for a reversal of fortunes.
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Emails
There will be emails to many of our scholars about key points.
Prior contacts:
• 24 April 2025: Anthony N. Aguirre, Santa Cruz, California
• 7 April 2025: George F. R. Ellis, Cape Town, South Africa
• 7 April 2025: David Gross, Santa Barbara, California
• 7 Apri 2025: Joseph Silk, Baltimore, Maryland and Paris, France
New contacts:
• 6 April 2025: Peter Woit, New York City, NY
• 5 April 2025: Martin Olsson, Berkeley, California
• 5 April 2025: Gustavo Turiaci, Seattle, Washington
Prior contact:
• 4 April 2025: Frank Wilczek, Cambridge, Massachusetts
New contacts:
• 4 April 2025: Tim Tait, Irvine, California
• 3 April 2025: Peter Scholze, Berkeley and Bonn, Germany
• 27 March 2025: Jean-Pierre Serre, Alpes-Côte d’Azur, France
• 25 March 2025: Robert diSalle, London, Ontario, Canada
Prior contacts
• 24 March 2025: Gerardus ‘t Hooft, Utrecht, Amsterdam, The Netherlands
• 24 March 2025: Kirsten Wickelgren, Durham, North Carolina
• 24 March 2025: Paul Davies, Phoenix, Arizona
• 23 March 2025: Steve Smale, Berkeley, California
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IM
There will also be many instant messages to thought leaders about these key points.
4:11 PM · Apr 11, 2025 Steven Strogatz @stevenstrogatz Hi Steven – You may enjoy taking a look at the new homepage at https://81018.com/: The permanent URL is : https://81018.com/symphony/ I am teasing the new qualitative-quiet model to come out of its shell! -Bruce
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Critique or collaborate ………… You are always invited.
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Keys to this page, spheres-symphony
• This page became the homepage late in the day on 10 April 2025.
• The last update was 25 May 2025.
• This page was initiated on 2 April 2025.
• The prior homepage was https://81018.com/incommensurable/
• The URL for this file is https://81018.com/spheres-symphony/
• The headline for this article: A New Geometric Cosmology.
• First teaser* is: The Universe as a Symphony of Spheres:
*Or, wicket, kicker or eyebrow.
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