21. Book 7 - Graphs of General Form of Transcendental Function Alpha

29 September 2017 AD; Feast of St Michael the Archangel

30 September 2017 AD; Feast of St Jerome


Andrew Yanthar-Wasilik


A couple of thoughts on the parameters of the equation

ExpM = ( A / B )C (Eqn. M)

This is the equation for Main Exponent ExpM.

Parts “A”, “B”, and “C” contain some fractional exponents and terms which were a matter of choice.


After many tests, I have chosen denominator equal to 24, in Equation “C”:


Cx = ( ( C0) ( π/ e )x ) * ((x − 8)/(24))        (Eqn. C)



The axis between Real and Complex parts falls on constant C8 = π

In other cases, they would fall in between the whole numbers or some other whole constants, which would “undermine” the role of C8π as a first constant, the parent constant.

Choosing other parameters would give the only exact value of fine structure constant alpha, αE, not the other coupling constants such as weak, gravity, and strong nuclear force (checked).

Neutrino mixing angles would not show up as properties of the integer values for constants (probably). I might come back to this topic later on and run some extra tests for other possibilities of the values of the denominator in Eqn. C.


The other problem is the choice of the numerator in the same equation, and parameters in other Equations, such as Equation 1 from Book 3:


 (C0 ⁄ ((C16 ⁄ 10)(11 ⁄ 3)))(C16 ⁄ 3)       (Eqn. 1)


The exponent could be run differently, for example, using whole numbers instead of Cx. This was also checked, and I will not return to this issue.

Nagoya University team, in my opinion, got the most precise value of fine structure constant, alpha, αE:

 α( − 1) = 137.035 999 181 727 13        (Res. 1)

However, using the General Formulas gives a slightly different result starting on the 12th decimal position - it may be the calculator error; I have to run it on the FORTRAN later on.

Let us move to the main topic of this article - Graphs of coupling constants versus constant and Graphs of mixing angles for Neutrinos versus constant. 

As can be seen from Figure 1 and Figure 3 - Graph of α and Graph of θ versus x (constant), there is a division between the Real and Complex section at constant C8π.

In the Real part are three coupling constants (marked in red on graphs):

alpha electromagnetic, αE, (aka fine structure constant) at C16 x-coordinates,

alpha weak, αW, weak force coupling constant at C17 x-coordinates,

alpha gravity,αG, gravity force coupling constant at C17 + 8 ⁄ 11 x-coordinates and, at C17 + 13 ⁄ 18 x- coordinates two candidates. (2 March 2019 - Gravity should fall at integer value 18, i.e., C18 gives weaker force than the present educated guess).

In the Complex part, there are:

alpha strong, αS, strong nuclear force coupling constant at Cx-coordinate.

The approximate values of the constants are:

αE = 7.3× 10 − 3

αW = 4.4 × 10 − 7

αG = 2.2 × 10 − 39 or 8.8 × 10 − 39

αS = 1.07 × 100

Theses numbers agree quite well with official guesses:

αE = exact

αW = 10 − 6 - 10 − 7

αG = 5.9 × 10 − 39

αS approx = 1

(See References in Narrative for Book 5.)

As one can see (especially in Figure 3), the shape of the graph resembles the letter M, with two hills.

As I wrote in the narrative for Book 7, the left hill may be an invisible universe (Heaven in religious terms), or it may be just dark matter and dark energy and other “dark” components of the Universe.


This blog is “the work in progress”, and it will be clear later on what the left hill is.


Neutrino Oscillations Angles and Weinberg Angle (marked in green on graphs).


(In Narrative for Book 7, in References, you can get the idea of what these angles are).


All Neutrino Angles (θ) are in the Complex part, which in polar coordinates gives the length of the Vector (Modulus) and Angle called Argument.


Weinberg Angle describes some properties of electro-weak force and equals to from best estimates from 2004:

θW= 28.74 ±0.7 degrees

Now, constant C − 7 gives the following results:


90.0 degrees - ANGLE THETA OF COMPLEX NUMBER (ARGUMENT) IN DEGREES 61.148536003448903 = 28.85 degrees

It agrees with an experimental result.


Neutrino Mixing Angle Solar Oscillations angle is:

θ23 = 36.78 with 90% confidence

Now, this angle can be found at constant C − 4:


θ23 = -36.078748676662492 degrees

For Atmospheric Neutrinos with an angle of:

θ23ATM = 45 ±7.1 degrees

Constant C5 has an angle of:



For Neutrino θ13 from recent experiments :

θ13 approx = 9.0 degrees

Now, this angle can be found at constant C − 2:



Neutrino mixing angle θ12 equals:

θ12 = 34.22±1degrees

Now, the angle of constant C6  is:

θ12 = ANGLE THETA OF COMPLEX NUMBER (ARGUMENT) IN DEGREES 124.26162879014910 = 90 + 34.26 degrees.

(2 March 2019 - The angles are calculated here, not quite according to the complex number system, but they work nicely. Future articles will look into that, and they are just around the corner. Please remember that this is still a work in progress until the final formula is derived).


How do you explain that?



As to quarks, things are more complicated, and they will be presented later.

Andrew Yanthar-Wasilik


I have some difficulties posting the graphs. They will be posted after the issue is solved. - See "Book 7 - Graphs only."

Links with graph paper:

Print a Ruler

Dr. Philippe Marquis

Thanks to:



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