The "Golden Ratio" is well-known as the proportion humans visually perceive as the most beautiful.
Found in the arrangement of sunflower seeds, nautilus shells, and the Parthenon, this ratio is generated by the mathematical rules of the "Fibonacci sequence" and appears everywhere in nature.
Because of this, many have pondered, "The ultimate blueprint of the universe must surely be constructed using the Golden Ratio."
However, the answer derived by the Intergeometric Theory was different.
To shape its space, the universe chose not the Golden Ratio, but the "Silver Ratio," which possesses more perfect symmetry.
(Note: The Silver Ratio here refers not to the \(1:\sqrt{2}\) ratio used in A4 paper sizes, but to the mathematical "Second Noble Number," \(1:1+\sqrt{2}\), derived from the Pell sequence.)
Why did the universe discard the Golden Ratio and choose the Silver Ratio?
The reason becomes clear when we unravel the geometry of space's "self-replication (inflation)."
1. The "Asymmetry" of the Golden Ratio and the "Symmetry" of the Silver Ratio
Imagine a process where a block of space grows by copying itself.
[ Growth by the Golden Ratio (Addition to One Side) ]
Starting with a square, a new square with a side length equal to the longer edge of the previous rectangle is continuously added to "one side." Repeating this follows the Fibonacci sequence, creating the "Golden Rectangle" and the beautiful "Golden Spiral."
However, there is a fatal flaw in this growth. Because blocks are only ever added to one side, when viewed as a whole, it becomes "asymmetric."
[ Growth by the Silver Ratio (Addition to Both Sides) ]
On the other hand, when cosmic space grows (inflates), it cannot afford to expand with a bias in only one specific direction. This is because the universe prefers perfect symmetry known as "isotropy" (looking the same in all directions).
To maintain symmetry when copying blocks, they must be added equally to "both sides," not just one.
Adding blocks to both sides of a central block, and then adding blocks to both sides again based on the new longer edge... Repeating this rule of "addition to both sides" (the Pell sequence) naturally causes the spatial proportions to converge to the "Silver Ratio ( \(\sqrt{2}+1\) )."
If the Golden Ratio is "asymmetrical beauty," the Silver Ratio is "the perfect beauty of mirror symmetry (bilateral symmetry)."
2. Space is Divided into "81" Blocks
A space hierarchized by repeating copies on both sides according to the rules of the Silver Ratio is ultimately perceived as a block divided into 3 main sections: the "original central part (1)" and the "additional parts on both sides (totaling \(\sqrt{2}\))."
Our universe is a "4-dimensional spacetime" comprising 3 spatial dimensions (X, Y, Z) and time (T).
Because this "self-replication into 3 sections" occurs in all 4 of these dimensions, the entire space becomes finely divided like a Rubik's Cube.
The total number of these grid cells (degrees of freedom) N is determined as follows:
\[N=3\times 3\times 3\times 3=81\]
The universe arranges all its energy and physical laws based on these "81 degrees of freedom." (This number 81 also played a decisive role in solving the [120-order-of-magnitude Cosmological Constant Problem].)
3. Calculating the Remaining "Asymmetry ( \(r_*\) )"
Now, while the space created by the Silver Ratio possesses beautiful mirror symmetry, it is not a completely flat, uniform space. A "proportional asymmetry" of \(1:\sqrt{2}+1\) derived from the Silver Ratio inevitably remains between the "short side" and the "long side."
Let's calculate the "magnitude of asymmetry per degree of freedom" when this fundamental asymmetry remaining in the universe is distributed across all 81 degrees of freedom of space.
We call this the 0th-order asymmetry constant ( \(r_{*0}\) ):
\[r_{*0}=81\times \frac{1}{\sqrt{2}+1}\approx 33.5507...\]
However, the fractal (self-similar) hierarchy of the universe does not end here.
Below the space we observe, microscopic hierarchies continue infinitely, constantly exerting minuscule influences (corrections) like facing mirrors.
When we sum the influence ( \(\Delta r_*\) ) from these infinite hierarchies using an infinite series, a minute correction of about 0.254% is applied.
\[\Delta r_*=\sum_{k=1}^{\infty}\frac{1}{(2\times 81+1)^{2k+1}(\sqrt{2}+1)^k}=0.254\,122\,722\%\]
Multiplying these together yields the ultimate geometric constant that governs our universe: the "Hierarchical Asymmetry Constant ( \(r_*\) )."
\[r_*=r_{*0}(1+\Delta r_*)=33.636\,560\,021\,92\]
4. The Magic Constant \(r_*\) Generates Elementary Particles
This seemingly arbitrary number, 33.636..., produced as a result of the universe choosing the "Silver Ratio" over the Golden Ratio and dividing space into "81", is of profound importance.
In fact, this \(r_*\) functions as the "geometric effective radius" when elementary particles collide with the Higgs field to acquire mass.
Simply by calculating the volume ratios of a "point, line, and area" using this constant as the radius, one of the greatest mysteries in modern physics—"the mass differences between the electron, muon, and tauon"—can be derived with terrifying precision.
Next time, using this magic constant \(r_*\), we will uncover the true nature of "Koide's Mass Formula," which has fascinated physicists for years, and prove that the masses of elementary particles are nothing but pure geometry.
