A Discrete Geometric Toy Model to bridge and scale from Planck Units to Cosmological Expansion

Authors: Bruce E. Camber and AI-assisted editing-and-review by (alphabetically)
ChatGPT, Claude, DeepSeek, Gemini, Grok, Meta, Mistral, and Perplexity. (March 31, 2026)

Abstract

The 81018 Project presents a toy model, a discrete geometric framework that spans from Planck-scale structure to cosmological scales using a base-2 notation system. Beginning with stacking and packing of spheres defined by Planck base units, tetrahedral-octahedral geometric seeds become emergent. Successive edge bisections generate nested refinement layers indexed by an integer notation n, where n=0 anchors at Planck length/time and positive n correspond to binary doublings (formulas below). Extending this chain to n≈202 naturally produces length/time scales comparable to the observable universe’s radius (~10²⁶ m) and age (~ 13.8 billion years). The model functions as a combinatorial laboratory for exploring discrete scale hierarchies, making no claim to derive cosmic dynamics from first principles. We detail the geometric construction, scaling relations, and numerical mapping to cosmological milestones, positioning 81018 as a heuristic tool for relating Planck discreteness to large-scale expansion within a unified binary framework.

1. Introduction

Contemporary cosmology describes the large-scale universe with remarkable precision using general relativity and $\mathrm{\Lambda }$CDM, yet the Planck regime and the ultraviolet completion of gravity remain a major issue open questions. Discrete geometric frameworks offer one approach to exploring scale hierarchies that bridge quantum and cosmological domains. The 81018 model adopts this perspective through a simple base-2 construction rooted in tetrahedral-octahedral geometry.

This paper presents 81018 as a toy model, a combinatorial system, not a physical theory. Our French AI partner, Mistral, says, “In physics, a ‘toy model’ is a simplified framework used to explore ideas or test hypotheses without claiming to be a complete or realistic theory. The 81018 model is such a tool—it helps us visualize and organize scales, but it does not replace established physical theories like general relativity or quantum mechanics.”

Beginning at Planck scales ($n=0$), successive binary doublings generate a notation chain that reaches cosmological scales near $n=200$. The model serves as a discrete analog to logarithmic cosmological plotting, providing a geometric template for thinking about scale separation from Planck discreteness to large-scale homogeneity. Yet, unlike conventional logarithmic plots, this construction provides a discretized, integer-indexed template where each step corresponds to a doubling of geometric complexity, offering a new combinatorial perspective on the Planck-to-cosmos scale hierarchy.

Sections 2–4 develop the construction, scaling relations, and cosmological mapping. The construction makes no dynamical claims and is intended as a heuristic laboratory for scale hierarchy questions.

1.1 Model specifications (v1.0)

Four defining rules:

1. Geometric seed: Tetrahedron-octahedron pair, unit edge length (Fig. 1).
2. Notation index: $n\in \mathbb{Z}$, $n=0$ at Planck scale, $L_n=2^n{L}_{\mathrm{P}}$, $t_n=2^n{t}_{\mathrm{P}}$​.
3. Binary process: Edge bisection (inward), edge doubling (outward).
4. Domain: ~202 notations spanning Planck to cosmological scales.

Core parameters (CODATA 2018):

1.2 Toy Model: 

Again, we use ‘toy model’ in the combinatorial sense: a simplified, discrete framework for organizing scales and exploring logical consequences, not a candidate for fundamental dynamics. It does not propose new forces, particles, or modifications to general relativity.

2. Geometric construction

The 81018 model begins with a tetrahedral-octahedral geometric seed, chosen for its duality and space-filling properties. A regular tetrahedron and its dual octahedron form a natural pair: the face centers of a tetrahedron define vertices of another tetrahedron, while octahedron vertices lie at cube face centers.

Figure 1. Geometric seed: (a) unit-edge tetrahedron, (b) dual octahedron, (c) first edge-bisection step. Midpoints on each edge generate 4 smaller tetrahedra + 1 central octahedron.

The refinement process follows three explicit steps:

1. Start with a tetrahedron (or octahedron) of unit edge length $a_0=1$.
2. Edge bisection: Place vertices at midpoints of all edges. For a tetrahedron (6 edges), this yields 6 new vertices. Connect to form:
   • 4 corner tetrahedra (edge $a_1=1\mathrm{/}2$)
   • 1 central octahedron (edge $a_1=1\mathrm{/}2$)
3. Iterate: At step $k$, bisect all edges from step $k-1$. Edge lengths scale as $a_k=2^{-k}$generating nested layers indexed by notation $n=-k$ (inward).

Figure 2. Refinement sequence: four spheres, centers connect, first tetrahedron. Green tetrahedra actualized. Each edge is halved, four smaller tetrahedrons – one in corner, and an octahedron is in center (pictured above as a yellow triangle). One octahedron, each edge halved for eight tetrahedrons (one in each face) and six octahedrons, one in each corner.

Outward construction reverses the process: from $n=0$ (Planck scale), edge doubling ($a_{n+1}=2a_n$​) generates the cosmological sequence. This bidirectional refinement produces the full notation chain:

$$n\in \mathbb{Z},L_n=2^n{L}_{\mathrm{P}},\text{edge length scales with }{2}^n\mathrm{.}$$

The tetrahedron-octahedron pair provides both visual intuition and combinatorial structure for the base-2 hierarchy. Section 3 maps this geometric progression to physical scales and the math is seen in our 2016 vertically-scrolled chart of 202 notation.

3. Base-2 scaling and 202 notations

The 81018 model indexes refinement levels by an integer notation $n\in \mathbb{Z}$, where $n=0$ corresponds to Planck‑scale quantities and positive $n$ represent successive doublings of characteristic scales. If this nested bisection process is extended to its logical limit, the edge length of the smallest tetrahedron is determined by the Planck length (l_P). Setting this as the base unit (n=0), each outward doubling step (n=1,2,3,…) defines a discrete scale. The time associated with each scale follows identically from the Planck time (t_P), assuming the speed of light c = l_P/t_P. The length and time associated with notation $n$ are defined by:

(1) .,…….. .,……. L_n = 2ⁿl_P .,……. T_n = 2ⁿt_P

where l_P ≈ 1.616255(18)×10⁻³⁵ meters and t_P ≈ 5.39116(13)×10^-44 seconds are the Planck length and time, respectively. These basic equations, simple rules, provide a discrete logarithmic bridge between Planck-scale structure and much larger physical scales.

Table 1 lists selected notations depths, showing how $n\approx 202$ produces $T_{202}\approx 13.8$ billion years and $L_{202}$​ on the order of the observable universe radius ($\sim {10}^{26}$ m), up to factors of order unity.

Table 1. Selected notations in the 81018 base‑2 chain.

… … . .. …. …… .. Notes: Numerical values use CODATA 2018 Planck constants. Cosmological/ scales from Planck 2018 $\mathrm{\Lambda }$CDM fits.

The key point is that a notation depth on the order of $n\approx 200$ spans the gap from Planck units to cosmological scales with a compact binary indexing rule. In that sense, the model provides a discrete analogue of logarithmic scale separation: lower notations describe refined geometric structure, while higher notations organize macroscopic length and time scales. The numerical coincidence at $n=202$ is not presented as a dynamical derivation of cosmic expansion, but as a scale-matching feature of the base-2 hierarchy.

This scaling section will support the later cosmological interpretation without claiming that the model replaces $\mathrm{\Lambda }$CDM or general relativity. Instead, it frames the observed hierarchy of scales as a finite depth within a binary refinement structure rooted in Planck units.

Figure 3. Base-2 notation chain spanning Planck scale ($n=0$) to cosmological scales ($n\approx 202$). Logarithmic plot of $L_n=2^n{L}_{\mathrm{P}}$​ (solid) and $T_n=2^n{t}_{\mathrm{P}}$​ (dashed) against notation depth $n$. Dotted lines mark standard cosmological scales for reference.

4. Cosmological interpretation

The base-2 notation chain (Table 1, Fig. 3) naturally spans from Planck scales to cosmological scales over approximately 200 steps. At notation depth $n=202$, the characteristic length $L_{202}\sim {10}^{26}$m approximates the radius of the observable universe, while $T_{202}\sim 4.4\times {10}^{17}$s (approximately 13.8 billion years) matches the Planck 2018 age estimate.

This numerical correspondence arises because ~60 orders of magnitude using base-10 separate Planck units from cosmological scales, requiring n ≈ 202 binary doublings (base 2) to bridge that gap: log₂(10⁶⁰) ≈ 200. Notation $n\approx 200$ thus demarcates discrete geometric refinement of lower notations and the effective continuum descriptions for large-scale cosmology.

The 81018 model makes no claim to derive cosmic expansion dynamics. Instead, it offers a combinatorial perspective: if spacetime admits a binary scale hierarchy from Planck units, the observable universe corresponds to finite depth $n\sim 200$ within that hierarchy. This provides a discrete geometric analog to standard logarithmic cosmological plotting.

The construction suggests three heuristic organizing principles:

• Finite notation depth corresponds to finite cosmic age/size.
• Binary refinement mirrors the hierarchical structure of physical scales.
• Planck anchoring provides a natural ultraviolet boundary.

These principles position 81018 as a scale-separation framework rather than a dynamical theory, potentially useful for organizing cosmological data analysis or exploring effective descriptions across the Planck-to-cosmic hierarchy.

5. Limitations and outlook

The 81018 model is explicitly a toy combinatorial framework with well-defined limitations that merit discussion.

Primary limitations:

1. No dynamics: The model specifies a static scale hierarchy but provides no equations of motion, field equations, or evolution laws. It cannot predict expansion rates, density parameters, or observables beyond order-of-magnitude scale matching.
2. Geometric idealization: The tetrahedral-octahedral refinement assumes perfect regularity and infinite divisibility down to Planck scales. Real spacetime geometry involves quantum fluctuations, curvature, and topology changes not captured here.
3. Numerological risk: The near-match at $n=202$ is suggestive but not predictive. Without independent physical input, the specific notation depth risks appearing as a tuned parameter rather than a derived result.
4. No quantum gravity: The construction treats Planck units as fixed anchors without incorporating quantum gravitational effects, loop corrections, or ultraviolet completion.

Outlook and potential extensions:

Despite these limitations, the framework suggests several constructive directions:

• Data organization: The fixed binary notation ladder provides a universal logarithmic coordinate system for tabulating physical scales across all regimes, potentially useful for multi-scale simulations or phenomenology.
• Effective field theory interface: Notation depth $n$ could serve as a natural renormalization group “flow parameter,” mapping ultraviolet (low $n$) to infrared (high $n$) physics within effective field theory constructions.
• Geometric quantization: The discrete polyhedral lattice might provide a basis for geometric quantization or spin network-like structures, with notation depth encoding combinatorial complexity.
• Scale-separation phenomenology: The clean separation between “discrete geometric” (low $n$) and “continuum cosmological” (high $n$) regimes suggests applications in understanding effective theories across the hierarchy.

The 81018 model thus occupies a niche as a scale catalog rather than a dynamical theory — a discrete geometric template for organizing the 60-decade span from Planck to cosmic scales within a single coherent framework. Future work might explore whether this combinatorial structure suggests novel organizing principles for multi-scale physics or provides useful heuristics for quantum gravity phenomenology.

References

[1] CODATA Recommended Values of the Fundamental Physical Constants: 2018.    
    E. Tiesinga *et al.*, Reviews of Modern Physics **93**, 025010 (2021).
[2] Planck 2018 results. VI. Cosmological parameters.  
    N. Aghanim *et al.* (Planck Collaboration), Astronomy & Astrophysics **641**, A6 (2020).
[3] Observable universe.  
    Wikipedia, accessed March 2026. [For radius/age estimates]
[4] Tetrahedral-octahedral honeycomb.  
    Wikipedia, accessed March 2026. [Space-filling properties]
[5] Dual polyhedra.  
    T. Banchoff, Brown University Geometry Center materials.
[6] Planck units.  
    Wikipedia + CODATA 2018 values.
[7] 81018 science education project.  
    81018.com, ongoing work 2011–2026.    

Acknowledgements

The author thanks Perplexity AI for extensive assistance in refining this work through iterative synthetic peer review. Special thanks to the free tier of Perplexity and ChatGPT, which made this documentation of LLM-assisted research possible for an independent researcher without institutional support.

This work originated in a high school geometry class exploring tetrahedral-octahedral constructions, later formalized through the 81018 project (81018.com). The author acknowledges the patience of readers who engage with speculative frameworks developed outside traditional academic channels.

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