The Archimedean Basis:
$(\alpha^{-1})$ via Recursive Operators
Notes: In my environment, the formulas that were displayed correctly in the preview are displayed distorted, so I have attached the same PDF as the post.
This is my first post in the Wolfram Community. Around last November, I noticed that Python was integrated into Gemini Pro, which sparked my interest in numbers, puzzles, and geometry. Although I am just an amateur without a professional background in mathematics, I hope to contribute to the community by sharing the interesting discoveries I've made through my explorations with AI.
This approach derives the inverse fine structure constant ( $\alpha^{-1}$) using a recursive fractal operator. While this is a heuristic geometric model rather than a derivation from first principles of QFT, the numerical result matches CODATA 2022 values with an accuracy comparable to experimental uncertainty, suggesting a fascinating "geometric resonance."
1. The Recursive Geometric Operator & Heuristic Model
We define a recursive operator $\Omega(f_0, N)$ as the fixed point (limit) of the sequence:
$$ x{n+1} = f0 \left( 1 + \frac{1}{N + x_n} \right) $$
Mathematically, this corresponds to the positive root of the quadratic equation:
$$ x^2 + (N - f0)x - f0(N + 1) = 0 $$
We use the normalized area difference between a square and a circle as the geometric seed ( $f_{vac}$) and apply a self-interaction feedback loop to the symmetry base (N = 154). By solving this self-consistent equation (where the output $\alpha$ regulates the input N), we determine $\alpha^{-1}$:
- Geometric Seed: $f_{vac} = \frac{4 - \pi}{24}$
- Effective Symmetry: $N_{eff} = 154 - \frac{\alpha}{2}$
- Equation: $$ \alpha^{-1} = 137 + \Omega(f{vac}, N{eff}) $$
Result:
- Calculated: $137.035\,999\,177...$
- CODATA 2022: $137.035\,999\,177(21)$
- Precision: $|\Delta| \approx 10^{-11}$
2. The Geometric Foundation: Archimedean Basis
Why does the model rely on the integer base 154? The answer lies in the classical Archimedean Method of Exhaustion.
Archimedes established $22/7$ not just as an approximation of $\pi$, but as a strict Upper Bound. Based on this Archimedean upper bound ( $22/7$), we normalize the geometry of a square and its inscribed circle:
- Square ( $S_s = 4$): Scaled as $28/7$. Normalized ( $\times 1/2$) $\rightarrow$ Integer Base 14
- Circle ( $S_c \approx 22/7$): Scaled as $22/7$. Normalized ( $\times 1/2$) $\rightarrow$ Integer Base 11
- Residue ( $S_r = 4 - \pi$): Scaled as $6/7$. Normalized ( $\times 1/2$) $\rightarrow$ Integer Base 3
This quantization suggests a fundamental symmetry breaking ratio of $14 : 11 : 3$. From this, our Symmetry Base ( $N$) emerges naturally: $$ N = Ss \times Sc = 14 \times 11 = 154 $$
The "Integer Core" of the fine structure constant ( $137$) is then derived by subtracting the geometric boundaries (Square + Residue) from this bulk symmetry:
$$ 137 = 154 - (Ss + Sr) = 154 - (14 + 3) = 154 - 17 $$
3. Numerical Closure and Symmetry (154 : 137)
The relationship between the Symmetry Base ( $154$) and the Integer Core ( $137$) satisfies a self-consistent scaling mediated by the factor 17:
- Scaling Factor: $154 = (17 \times 9) + 1$
- Integer Core: $137 = (17 \times 8) + 1$
- Ratio: $\frac{137}{154} \approx \frac{8}{9}$
Structure: N = $17 \times 9 + 1$ and $\text{Core} = 17 \times 8 + 1$.
4. Geometric Substructure and the 137 Node
Scaling the fundamental Archimedean ratio ( $14:11:3$) by a factor of 11 to align with our Symmetry Base ( $N=154$) reveals a deeper substructure:
$$ 14(\times 11) : 11(\times 11) : 3(\times 11) \rightarrow \mathbf{154 : 121 : 33} $$
Here, 121 represents the bulk area of the inscribed circle, and 33 represents the total area of the four corner residues ( $4 - \pi$).
Remarkably, if we take the circle and exactly half of the residue (geometrically corresponding to a selection of two out of the four corners, or a chiral pair), we arrive at a value strikingly close to the integer core:
$$ S{circle} + \frac{S{residue}}{2} = 121 + \frac{33}{2} \approx 121 + 16 = \mathbf{137} $$
This suggests that the integer core ( $\approx 137$) is not arbitrary but is geometrically rooted in the inscribed circle plus a semi-residual component.
5. Visualization
Below is the Wolfram Language code to reproduce the derivation and visualize the Geometric-Arithmetic Schema.
(* ========================================================== *)
(* Recursive Geometric Derivation of Inverse Fine Structure Constant *)
(* ========================================================== *)
(* 1. Define the Recursive Geometric Operator (The Fractal Limit) *)
(* Mathematically defined as the positive root of: x^2 + (N - f0)x - f0(N + 1) = 0 *)
GeometricOp[f0_, Nval_] := Root[#^2 + (Nval - f0) # - f0 (Nval + 1) &, 2]
(* 2. Define Parameters based on Archimedean Basis *)
(* Geometric Seed (f_vac): Normalized Area Difference between Square and Inscribed Circle *)
fvac = (4 - Pi)/24;
(* Effective Symmetry Base (N_eff): Integer Base 154 with Self-Interaction Feedback *)
(* The feedback term corresponds to -alpha/2 *)
Neff[alphaInv_] := 154 - (1/alphaInv)/2;
(* 3. Solve the Self-Consistent Equation *)
(* The value is derived as: alpha^-1 = 137 + GeometricOp(f_vac, N_eff) *)
solLowEnergy = FindRoot[
x == 137 + GeometricOp[fvac, Neff[x]],
{x, 137.036},
WorkingPrecision -> 20
];
alphaInvCalc = x /. solLowEnergy;
(* 4. Output the Results *)
Print[Style["=== Geometric Derivation Results ===", 16, Bold]];
(* Calculated Value *)
Print[Style[Row[{
"Calculated \!\(\*SuperscriptBox[\(\[Alpha]\), \(-1\)]\): ",
NumberForm[alphaInvCalc, 12]
}], 14, Blue]];
(* CODATA 2022 Standard *)
Print[Style["CODATA 2022 Value: 137.035999177(21)", 12]];
(* Precision Check *)
Print["----------------------------------------"];
Print[Style[Row[{
"Difference: ",
NumberForm[alphaInvCalc - 137.035999177, 10]
}], 12, Red]];
(* ========================================== *)
(* Geometric Visualization of Alpha Inverse *)
(* Structure: Circle + 2 Active Corners *)
(* ========================================== *)
Graphics[{
(* ---------------------------------------- *)
(* 1. The Excluded Vacuum (Residue) *)
(* ---------------------------------------- *)
(* Represents the 'virtual' or 'subtracted' geometry *)
{
FaceForm[Opacity[0.15, Red]],
EdgeForm[Directive[Thin, Dashed, Red]],
(* Top-Left Corner *)
Rectangle[{-1, 0}, {0, 1}],
(* Bottom-Right Corner *)
Rectangle[{0, -1}, {1, 0}]
},
(* ---------------------------------------- *)
(* 2. The 137 Node (Active Entity) *)
(* ---------------------------------------- *)
(* Represents the fusion of the Circle and 2 Spikes *)
{
FaceForm[RGBColor[0.2, 0.5, 0.9]], (* Solid Blue *)
EdgeForm[None], (* No internal edges to show fusion *)
(* The Active Corners (Spikes) *)
Rectangle[{0, 0}, {1, 1}], (* Top-Right *)
Rectangle[{-1, -1}, {0, 0}], (* Bottom-Left *)
(* The Core Circle *)
Disk[{0, 0}, 1]
},
(* ---------------------------------------- *)
(* 3. Outline of the 137 Node *)
(* ---------------------------------------- *)
(* Optional: Adds a clean border around the fused blue shape *)
{
FaceForm[None],
EdgeForm[Directive[Thickness[0.005], RGBColor[0.1, 0.3, 0.7]]],
Disk[{0, 0}, 1],
Rectangle[{0, 0}, {1, 1}],
Rectangle[{-1, -1}, {0, 0}]
},
(* ---------------------------------------- *)
(* 4. The Symmetry Base (Frame) *)
(* ---------------------------------------- *)
{
FaceForm[None],
EdgeForm[Directive[Thin, Black]],
Rectangle[{-1, -1}, {1, 1}]
}
},
(* Plot Settings *)
PlotRange -> {{-1.2, 1.2}, {-1.2, 1.2}},
ImageSize -> 500,
PlotLabel -> Style["Geometric-Arithmetic Schema of \!\(\*SuperscriptBox[\(\[Alpha]\), \(-1\)]\)", 18, Bold, FontFamily -> "Arial"]
]
6. Zenodo link
Here is a link to Zenod's report with almost the same content. Please refer to it if you like.
[https://doi.org/10.5281/zenodo.18676185]
7. Background
Last year, I first encountered bijective decimal notation. I was struck by how "cool" the notation looked, which led me to examine various numbers—such as primes and Kaprekar’s constants—through this lens. Seeing transformations like 7A82(8082) and 3AAAA1(411101) felt incredibly fresh.
Following that thread, I turned my attention to the fine-structure constant. During my analysis, I treated protons and electrons as "bijective information systems," each possessing its own base system. While I don't necessarily believe this aligns with physical reality, I found the perspective intriguing enough to adopt. This path eventually led me to the number 17.
Along the way, I discovered historical efforts by figures like Arthur Eddington. Given my focus on bijective decimals at the time, I felt that Eddington’s "136 + 1" might have been a result of "overlooking the zero" (or failing to account for it)—though I realize that might be an unconventional way to put it. Gemini Pro provided a much more elegant framing of this idea using information theory.
I incorporated the circular residual into this formula, inspired by Gemini Pro's suggestion that deriving the fine-structure constant resembles the squaring of the circle in being an insoluble problem. The denominator of 24 appeared by chance when Gemini Pro repeatedly performed a brute-force search based on completely erroneous and contradictory instructions.
After layering these insights, I arrived at the formula I am presenting today. While a more mathematically precise formula may exist, I feel I’ve taken this as far as I can, so I’ve decided to share it. I hope it sparks your curiosity!
Attachments:
