The first second of numbers-geometries-equations hold the keys that define our universe.

Image of abstract geometric shapes representing continuity, symmetry, and harmony in a mathematical model of the universe.
A cosmic representation highlighting hyper-rationality, featuring spheres, octahedrons, and hexagonal plates, depicting continuity, symmetry, and harmony in an expansive geometric model of the universe.

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Hyper-Rational: 202 Base-2 Notations
by Bruce. E. Camber

Abstract:
What if the universe is a highly-ordered, mathematical system? Our model goes from the smallest scale (Planck units, ~1.616×10-35 meters) to the largest (the observable universe, ~1026 meters) in just 202 base-2 steps. The geometries of each step double the previous, creating a hyper-rational structure that is first defined by pi’s continuity, symmetry, and harmony.*

The very first second of the universe is within Notation-143.
Unlike Big Bang cosmology, this model suggests a smooth, exponential progression up to and beyond Notation-143. Compare our charts. In 2011 our most simple chart emerged. Then, in 2016 work began on our huge horizontally-scrolled chart: https://81018.com/chart/ Started in July 2025, you can see our numbers in action here: https://81018.com/chart-downloads/

The Power of 202 Notations
Starting at Notation 0-1 (Planck scale), our model doubles through to Notation 67 (particle scale) and then to Notation 202 (cosmic scale). If this scale is always active and all time is Now, this exponential growth could readily explain phenomena like dark energy without external forces. Being on the edge of knowledge and understanding, one asks, “Can AI simulate these transitions?”

Hyper-Rationality Examined
Hyper-rationality refers to a universe governed by precise mathematical relationships. Each notation, in some measure, preserves the continuities and symmetries imparted through the four primary irrational numbers. Order emerges naturally. Thinking about the power of AI, we ask, “How can design visualize this harmony and the qualitative?”†

Challenging Cosmology
Unlike Big Bang cosmology’s singular event, our model starts with finite Planck units, scaling smoothly. It is a different model and we need help to test it against observational data. Planck units defined: https://www.nist.gov/pml/planck-constant Big Board-little universe numbers: https://81018.com/home/ Open Data Science Conference (ODSC) AI East 2025 encouragement: https://81018.com/odsc/

Core Concept: Hyper-rationality constructs a mathematical model of the universe, starting at the Planck scale (Planck Time-Length-Mass-Charge). It uses base-2 exponentiation to map its expansion across 202 notations to the present. Visualize it as a universal grid. Metaphorically, it is like the old aether theories, or poetically, a cosmic symphony of interconnected spheres.

Model Structure:

  1. Begin with the Planck base units. The key hypothesis is that there is one Planck sphere per unit of Planck length and Planck time. This becomes a foundational grid of infinitesimal spheres. Poetically, each sphere is a pulsating note in the universe’s melody.
  2. Sphere Density: Approximately 185 tredecillion spheres emerge per second . Again, please note that the first second is within Notation 143.
  3. Early notations (1–50/60) are hypothesized to define a perfection with spheres and their emergent octahedrons. Hexagonal plates form a flawless lattice, driven by π, e, √2, φ and the continuity, symmetry and harmony emergent from each of these irrational numbers, a finite-infinite relation that is quantitative-qualitative.
  4. Natural geometric gaps of 7.356+ degrees introduce a dissonance around Notations 60–64, enabling complexity currently understood as quantum fluctuations and particle archetypes.
  5. Scales via base-2 notation from the Planck scale (10-35 meters, 10-44 seconds) to the observable universe (1026 meters, 13.8 billion years) across 202 doublings, each notation a harmonic layer.

An animated 3D model of a geometric shape, featuring a cube with triangular faces in various colors, rotating to illustrate mathematical concepts.
First debut: 8 July2025 – history

Key Features:

  • Continuity, Symmetry, Harmony: Rooted in π within the octahedron’s four hexagonal plates, these principles, with e, √2, and φ, create a qualitative scaffold, replacing physical constants with ratios, resonating infinitely across all notations.

  • Octahedron’s Four Hexagonal Plates initially dubbed, the Planck Polyhedral Core: The speed of light (c) bridges finite measurements and infinite qualities which shape the universe’s relational grid.

Diagram illustrating the geometric relationship between five tetrahedrons sharing an edge, including a highlighted angle labeled as the 7.356-degree gap.
Physics of gaps
  • Geometric Gaps as Creative Tension: When five tetrahedrons or five octahedrons share a common edge, a 7.356-degree gap emerges, a subtle dissonance in the cosmic symphony. These gaps, too small for direct measurement below Notation-64, introduce a potential for quantum fluctuations, seeding complexity and individuality within the universe’s otherwise perfect structure.

  • Finite Grid: The universe is a finite, ever-expanding tapestry (not infinite) with all 202 notations active, weaving past-present-and-future into a single, luminous composition.

  • Comprehensive Scope: Captures all time, space, and matter-energy, from the Planck scale’s first sphere to the galaxies’ radiant dance, each notation a movement in the universe’s unfolding score, and with time, an active agent of emergence.

Visual Imagery:

  • Envision a cosmic orchestra: Planck-scale spheres with emergent tetrahedrons and octahedrons, structured by hexagonal plates, pulse like notes. The infinite numbers of these plates — π, e, √2, φ — sing of continuity, growth, scaling, and proportion, in syncopated rhythms, harmonizing light, gravity, and life across the universe, all forever, infinite equations.

Implications:

  1. Challenges big bang cosmology by grounding the universe in numbers and geometry, not an infinitely hot singularity, offering a much “cooler-start” model where spheres sing the universe into being.
  2. Critiques academic physics for overlooking dimensionless constants and holistic grids, proposing instead a model where every sphere’s vibration contributes to cosmic unity.
  3. Suggests gaps in academic models (e.g., big bang theory) due to overemphasis on particles and neglect of a holistic grid.
  4. Provides an educational framework, teaching exponential growth through simple doublings, as if learning the scales of the universe’s music.

Next Steps:

  • Investigate the model’s application to mysteries like dark matter, quantum gravity, CMBR, the finite-infinite divide, and the quantitative-qualitative divide especially focusing on the exploration of how geometric gaps amplify cosmic creativity.

Critique of Alternatives:

  1. Big bang cosmology ignores mathematical continuities witchin the Planck scale and the irrationals.
  2. The four most prominent irrational numbers have not been seen within the intrinsic geometries of the octahedron which are part of the intrinsic geometry of the sphere at the Planck scale.
  3. Academic physics often ignores dimensionless constants and ratios, focusing on complex particle interactions.

In the world:

Submitted to PASCOS 2025. Our hyper-rational model challenges big bang with 202 notations, Planck base units, perfect geometries, imperfect geometries, and the four most prominent irrational numbers within the four intrinsic hexagonal plates within every scale-invariant octahedron. That link is: https://81018.com/pascos/

PASCOS 2025, An Abstract by Bruce Camber

“Big bang cosmology, defined by and reliant on an infinitely hot singularity, struggles to reconcile Planck-scale dynamics with observable cosmic structures. We propose a hyper-rational model that redefines the universe as a geometric and mathematical start and expansion from the Planck scale (10-35 m, 10-44 s) to the observable cosmos (1026 m, 13.8 billion years) across 202 base-2 notations, all simultaneously active. Initiated by infinitesimal spheres at one per Planck Time (~1043 spheres/second by Notation 143), this lattice evolves through sphere stacking into tetrahedrons and octahedrons, anchored by four irrational numbers (π, e, √2, φ) within the octahedron’s four hexagonal plates. These plates encode continuity-symmetry-harmony, replacing particle-centric models with a qualitative scaffold of dimensionless constants. Geometric gaps of 7.356°, arising from five tetrahedral or five-octahedral configurations above Notation 60, seed quantum fluctuations, potentially manifesting as dark matter halos, CMBR power spectra, and quantum gravity’s spacetime foam. Testable predictions include gap-induced density variations aligning with N-body simulations for dark matter and Planck satellite data for CMBR, achievable through AI-driven computational geometry (e.g., topological deep learning). Unlike big bang frameworks, this model posits time as a dynamic agent, driving continuous emergence from Planck spheres to galaxies, implicating humans as participatory co-creators. We invite PASCOS 2025 attendees to collaborate on simulating these geometric structures, exploring their implications for cosmology and particle physics, and advancing a unified, mathematically coherent view of the universe.”

References

___ * Continuity, symmetry, harmony. Retrieved 25 June 2025, https://81018.com/csh/

___ † Harmony and the qualitative: Retrieved 26 June 2025, https://81018.com/good/

  • Planck Units: NIST Planck Constant (https://www.nist.gov/pml/planck-constant, retrieved 2 July 2025). Supplemented by Misner, C. W., Thorne, K. S., & Wheeler, J. A. (1973). Gravitation, Chapter 43. Defines Planck units rigorously in a cosmological context, grounding the model’s starting point.
  • Planck Scale: Misner, C. W., Thorne, K. S., & Wheeler, J. A. (1973). Gravitation, Chapter 43 (same as above, consolidated for efficiency).
  • Base-2 Exponentiation: https://81018.com/chart/ (retrieved 2 July 2025). Consolidated link on the 202 notations. Supplemented by Barrow, J. D., & Tipler, F. J. (1986). The Anthropic Cosmological Principle, Chapter 3. Discusses exponential scales in cosmology, aligning with the model’s 202 notations.
  • Four Irrational Numbers and Octahedral Plates: https://81018.com/hyper-rational-universe/ (retrieved 2 July 2025). Analysis linking π, e, √2, φ to the octahedron’s four hexagonal plates. Supplemented by Coxeter, H. S. M. (1973). Regular Polytopes, Chapter 7, on octahedral geometry and hexagonal structures.
  • Archetypes: https://81018.com/archetype/ (retrieved 2 July 2025). On formative structures. Supplemented by Lachièze-Rey, M. (1995). Cosmology: A First Course, on geometric archetypes.
  • Rational Analysis and logic. Jammer, Max, The Philosophy of Quantum Mechanics, Wiley Imterscience (1974), on Abner Shimony’s infinitely many components (page 355) and Hilary Putnam’s many-valued logic (page 373).
  • Exponentiation in Physics: Penrose, R. (2004). The Road to Reality, Chapter 5. Covers exponential functions in physical contexts, supporting the model’s mathematical framework.
  • Big Bang Critique: Steinhardt, P. J., & Turok, N. (2007). Endless Universe: Beyond the Big Bang. Critiques big bang cosmology, aligning with the model’s alternative approach.
  • Big Board-little universe Project: https://81018.com/home/ (retrieved 2 July 2025). Supplemented by Pi (π): Barrow, J. D. (2002). The Constants of Nature, Chapter 4. Explores pi and φ’s role in geometry and physics, supporting the model’s spherical foundation.
  • Continuity-Symmetry-Harmony: http://81018.com/continuity-symmetry-harmony. (retrieved 2 July 2025). Internal page defining these principles in the model. Supplemented by Weinberg, S. (1992). Dreams of a Final Theory, Chapter 5, on symmetry in physics.
  • Numbers and Ratios: https://81018.com/numbers/ (retrieved 2 July 2025). Internal page on dimensionless constants. Supplemented by Wigner, E. P. (1960). The unreasonable effectiveness of mathematics in the natural sciences. Communications on Pure and Applied Mathematics, 13(1), 1–14.
  • Finite-Infinite: http://81018.com/finite-infinite/ (retrieved 2 July 2025). Internal page on finite universe concepts. Supplemented by Ellis, G. F. R. (1993). Before the Beginning: Cosmology Explained, Chapter 2, on finite cosmologies.
  • First Second: https://81018.com/the-firsts/ (retrieved 2 July 2025). Details Notation 143. No external source needed unless particle emergence is emphasized.
  • 185 Tredecillion Spheres: https://81018.com/tredecillion/ (retrieved 2 July 2025). On sphere-based grid. Supplemented by Conway, J. H., & Sloane, N. J. A. (1998). Sphere Packings, Lattices and Groups, on sphere packing.
  • Qualitative and the perception of value from pi. https://81018.com/good/ (retrieved 29 June 2025).
  • Summaries: https://81018.com/handout/ and https://81018.com/1/

More References

Privacy Policy. A result of PASCOS!

Three files about numbers and number theory:

More to come…

Keys to this document, /mathematical-model/ . (Initiated 9 June 2025)
The last update of this page was 8 August 2025.
The URL for this page is https://81018.com/mathematical-model/

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The first use of these models was on television in 1999. We were doing a weekly, half-hour television series called Small Business School (originally called Small BusinessToday (1994-1995) and then Small Business 2000 (1995-2001) and then Small Business School (2001-2012). The series aired on PBS-stations around the USA, on the PBS-Adult Learning Satellite to all the colleges and universities in the country, and then by the Voice of America nine global satellites to every other country in the world in Spanish, French and Arabic. In China it was broadcast in English on CCTV-9 and had an estimated 45 million viewers because case studies guides and transcripts were all online.

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