Base-2 notation, irrational numbers, and the muon
This webpage presents an exploratory numerical survey analyzing potential geometric patterns in physics using a base-2 Planck notation coordinate system (\(n = \log_2(L/\ell_P)\)). The most critical finding is a numerical near-coincidence (\(\Delta = \mathbf{0.000126}\)) between the fractional coordinate of the muon reduced Compton wavelength and a rational fraction of the golden ratio (\(\mathbf{2\phi /5}\mathinner{\;\left(\mod \,\mathbf{1}\right)}\)). However, the paper explicitly notes that statistical significance has not been established, given an illustrative independent chance probability of approximately 42%.
📊 Methodology & Search Space
The study cross-examines 36 physical, defined, or model-dependent length scales against four irrational constants (\(\pi \), \(\phi \), \(e\), and \(\sqrt{2}\)) associated with the geometry of a regular octahedron. It breaks down into two search families across 12,240 total comparisons:
- Type A (Integer Multiples): Checks if the full coordinate matches integer multiples (\(k = 1\) to \(70\)) of the constants (\(n \approx k \times c\)).
- Type B (Fractional Alignment): Checks if the fractional part of the coordinate aligns with small rational fractions (\(p/q\)) of the constants.
🏆 Top 3 Numerical Matches
All top results fell under the Type B (fractional) search family:
| Rank | Physical Scale | Coordinate (\(n\)) | Match Expression | Residual (\(\Delta \)) | Constant |
|---|---|---|---|---|---|
| 1 | Muon reduced Compton wavelength | 66.6471 | \(\text{frac} \approx (2/5)\phi \pmod 1\) | 0.000126 | \(\phi \) (Golden Ratio) |
| 2 | Earth mean radius | 138.1779 | \(\text{frac} \approx (3/8)\pi \pmod 1\) | 0.000178 | \(\pi \) |
| 3 | 1 Ångström | 82.3555 | \(\text{frac} \approx (3/4)\pi \pmod 1\) | 0.000647 | \(\pi \) |
(Note: The authors mention that macroscopic scales like the Earth’s radius are likely entirely coincidental.)
⚠️ Key Caveats & Limitations
- No statistical significance: The observed “hits” closely track what would be expected by a uniform random distribution (null model).
- Exponent-offset relationship: The alignment indicates an exponent-offset relationship (\(L \approx \ell_P \cdot 2^{66+2\phi/5}\)), not a direct spatial multiplication of the Planck length by \(\phi \) .
- Data vetting: A source reconciliation is still underway, meaning some underlying physical length inputs require further verification.