18. Book 6 - Derivation of General Formula for Constants of the Cosmology and Quantum Mechanics - part I.

On 22 September 2017 AD, the feast of St Maurice.

Andrew Yanthar-Wasilik

 

We are finding General Formula for Equations from the previous book, "Book 5 Integer Formula for Dimensionless Coupling Constants of Fundamental Forces".

All Coupling Constants use the same equations, from which constants can be calculated.

✠Part A of the equation (A/B)^C

 

Integer Formula for fine structure constant alpha, α_E, ruling electromagnetic force is

A₁₆ = ( C₀)^{(24 ⁄ 24)}

Integer Formula for the weak force, α_W, ruling force of decays is

A₁₇ = ( C₀)^{(27 ⁄ 24)}

Integer Formula for the strong nuclear force, α_S, ruling quarks, and nucleons is

A₁ = ( C₀)^{( − 21 ⁄ 24)}

From this partial sequence can be obtained a complete sequence.

At n=8, there is an axis of symmetry. Axis at n=8 divide (as it will be shown on the Graphs of the Function) values between Real and Complex. This can be achieved only when the denominator equals 24.

For n=8, the nominator equals 0.

For an explanation of how to find formulae for some sequences, see the webpage of Brian Kell⁽¹⁾

So we have in the first-row number of a constant, in the second-row value of the exponent of "A,"

and in the third row, the difference between the two adjoining values of the exponent “A”. 

 

n =                11                            12                          13                       14                     15                         16                         17

E(n) =       (9)/(24)               (12)/(24)              (15)/(24)            (18)/(24)          (21)/(24)             (24)/(24)             (27)/(24)

Δ(E(n)) =  (3)/(24)                (3)/(24)               (3)/(24)               (3)/(24)              (3)/(24)              (3)/(24)

 

Constant difference between the values of the Exponents Δ(E(n)) = (3)/(24) and this is the slope ( ((Δy)/(Δx))= a ) of the function (y = a*x + b) we are trying to find. Y-intercept equals b. Y-intercept is at C₀ and equals b = -(24)/(24).

 

So, the exponent can be written as a linear function:

 

y = ((3)/(24))*x + (-(24)/(24)) = (3*x − 24)/(24) = (x − 8)/(8) 

 

Now, the “A” part of the equation can be written as:

 

✠A_x = ( C₀)^{((x − 8)/(8))}     (Eqn. A)

 

✠Part B

 

The following sequence is the "B" part of the equation (A/B)^C

B₁₆ = ( C₁₆ / ( 8 + 2 * (24/24) ) )^{(88 ⁄ 24)}

B₁₇ = ( C₁₇ / ( 9 + 2 * ( 24/27 ) ) )^{(99 ⁄ 24)}

B₁ = ( C₁ / ( -7 + 2 * ( -24/21 ) ) )^{( − 77 ⁄ 24)}

Now, taking under consideration the Exponent part - it can be seen that this sequence is the one in "A" multiplied by a factor of 11, giving:

n =                          6                               7                           8                                 9                        10                        11

E (n ) =           -(22)/(24)               -(11)/(24)               (0)/(24)                   (11)/(24)           (22)/(24)             (33)/(24)

Δ(E(n)) =         (11)/(24)                (11)/(24)               (11)/(24)                 (11)/(24)           (11)/(24)

Slope, a = (Δy)/(Δx) = (11)/(24)

Y-intercept, at C₀, equals b = -(88)/(24)

 

And the formula for the Exponent E_B sequence is:

 

E_B= ((11*x − 88)/(24))

 

The sequence of the base of the power is:

Constant C₁₆ is ( 8 + 2 * (24/24)

Constant C₁₇ is ( 9 + 2 * ( 24/27 )

Constant C₁ is ( -7 + 2 * ( -24/21 )

It can be seen that in the 2 * d part, the “d” equals to reciprocal of the value of the exponent from part A, which we calculated right at the beginning so that this sequence will be equal to:

 

E_partB = 2 * ((8)/(x − 8)) = ((16)/(x − 8))

 

Remaining factors from the sum:

8, 9, -7 gives sequence = (x-8)

So, the base of the power equals:

Base of the power = (x-8) + ((16)/(x − 8)) = (x² − 16x + 80)/(x − 8)

If we take the reciprocal of this function, we will have the product of C_x, and this reciprocal is raised to the Exponent E_B. Constants in the nominator of the base of the power C₁₆, C₁₇, and C1 are just the number of the constant, "n", which we replaced by "x", and each constant can be calculated from the formula from "Book 1".

 

C_x = ( C₀) ( π/ e )^x     (Eqn. I)

 

The formula for a sequence of the part "B" is then:

 

✠B_x = {[(C₀)(π ⁄ e)^x][(x − 8)/(x² − 16x + 80)]}^{[(11x − 88)/(24)]}     (Eqn. B)

 

✠Part C of the equation (A/B)^C

 

The Exponents of the equation (A/B)^C are:

The exponent of the constant C₁₆ is ( C₁₆ ) * (8/24)

The exponent of the constant C₁₇ is ( C₁₇ ) * (9/24)

The exponent of the constant C₁ is ( C₁ ) * (-7/24)

 

The sequence : -(7)/(24),..., (8)/(24), (9)/(24),... has an equation equal to ((1)/(3)) of the first one in this article, i.e.,

y_c = ((1)/(3)) * ( (x − 8)/(8) ) = (x − 8)/(24)

And C_Ex = ( C₀) ( π/ e )^x is the same as Eqn. I

(It should be C_x instead of C_Ex. However, that would be mixed up with the final result.)

Product of ( C_Ex) and ( y_c = (x − 8)/(24) ) is the Exponent C_xwe are looking for:

 

✠C_x = ( ( C₀) ( π/ e )^x ) * ((x − 8)/(24))      (Eqn. C)

 

And the final result of Exponent Main, ExpM is ExpM = ( A / B )^C

 

Substituting Equations A, B, C, and we get the long formula for ExpM, which is way too long to write; therefore, I will use parts A, B, C of the Equation

ExpM = ( A / B )^C      (Eqn. EM)

In part II, I will derive the General Formula for any value of the constant alpha and similar to alpha. 

 

Andrew Yanthar-Wasilik

 

(1)Brian Kell, CMU

 

Next article (part II)   >>>   19. Book 6 - Derivation of General Formula for Constants of the Cosmology and Quantum Mechanics - part II

 

Return to Syllabus   >>>   132. Syllabus of the course – “Voyage through God’s Universe (Cosmos and beyond) according to St. Hildegard von Bingen and others, with the help of Mathematics, Cosmology and Quantum Mechanics."