"FINDING THE FINE STRUCTURE CONSTANT"
"The purpose of this Mathematica Notebook is to compute the Fine Structure Constant [α] to 60+ significant digits using only [π] and the number [b=22+6(Sqrt[13])].  Note the number b*π is nearly equal to (α^(-1)).

The fundamental assumption in this computation is that at least one infinite series F[b,π,α]==0 exists which converges to zero and satisfies the Ratio Test.  Most of the terms of F are of the form (b^κ)(π^λ)(α^μ) and each successive term generally increases μ by 1.  The values of κ and λ are adjusted manually to make the sum of all terms converge to zero and make the absolute value of each term generally smaller than the previous one.  After each term is inserted, a new equation F[b,π,α]==0 can be solved for α. Using approximately 78 terms, α was computed to have the value
 
0.00729735256429999992416860274723885002506940170167664094552769

The previous best known value for α is 

0.0072973525643"

"------------------------------"
"SECTION I"
 
"The extraction,of the matrices defined below,from the first verse of Genesis has been described in the article entitled Genesis of the Reciprocal Fine Structure Constant which can be found at https://christianapologetic.org/encrypted-in-genesis/genesis-of-reciprocal-fine-structure-constant"

"The value of [b] was first detected as an eigenvalue for a 3rd order, 3 dimensional tensor based on the Spectral Theory of Tensors developed by Guimei Zhang, Shenglong Hu and others."

"------------------------------"
"A11"
A11=0
"A12"
A12=2
"A13"
A13=0
"A21"
A21=0
"A22"
A22=0
"A23"
A23=0
"A31"
A31=1
"A32"
A32=1
"A33"
A33=0
"B11"
B11=2
"B12"
B12=1
"B13"
B13=2
"B21"
B21=0
"B22"
B22=0
"B23"
B23=1
"B31"
B31=2
"B32"
B32=0
"B33"
B33=0
"C11"
C11=0
"C12"
C12=2
"C13"
C13=0
"C21"
C21=0
"C22"
C22=0
"C23"
C23=0
"C31"
C31=1
"C32"
C32=1
"C33"
C33=0
"------------------------------"
"Ayz"
Ayz={{A11,A12,A13},{A21,A22,A23},{A31,A32,A33}}
MatrixForm[Ayz]
"Byz"
Byz={{B11,B12,B13},{B21,B22,B23},{B31,B32,B33}}
MatrixForm[Byz]
"Cyz"
Cyz={{C11,C12,C13},{C21,C22,C23},{C31,C32,C33}}
MatrixForm[Cyz]
"Azx"
Azx={{A13,B13,C13},{A23,B23,C23},{A33,B33,C33}}
MatrixForm[Azx]
"Bzx"
Bzx={{A12,B12,C12},{A22,B22,C22},{A32,B32,C32}}
MatrixForm[Bzx]
"Czx"
Czx={{A11,B11,C11},{A21,B21,C21},{A31,B31,C31}}
MatrixForm[Czx]
"Axy"
Axy={{C11,C12,C13},{B11,B12,B13},{A11,A12,A13}}
MatrixForm[Axy]
"Bxy"
Bxy={{C21,C22,C23},{B21,B22,B23},{A21,A22,A23}}
MatrixForm[Bxy]
"Cxy"
Cxy={{C31,C32,C33},{B31,B32,B33},{A31,A32,A33}}
MatrixForm[Cxy]
"Xxz"
Xxz={{C31,C32,C33},{B21,B22,B23},{A11,A12,A13}}
MatrixForm[Xxz]
"Xzx"
Xzx={{C11,C12,C13},{B21,B22,B23},{A31,A32,A33}}
MatrixForm[Xzx]
"Xzy"
Xzy={{A11,B11,C11},{A22,B22,C22},{A33,B33,C33}}
MatrixForm[Xzy]
"Xyz"
Xyz={{A31,B31,C31},{A22,B22,C22},{A13,B13,C13}}
MatrixForm[Xyz]
"Xyx"
Xyx={{A31,A21,A11},{B32,B22,B12},{C33,C23,C13}}
MatrixForm[Xyx]
"Xxy"
Xxy={{C31,C21,C11},{B32,B22,B12},{A33,A23,A13}}
MatrixForm[Xxy]
"------------------------------"
"yz=Ayz.Byz.Cyz.Xyz.Xzy"
yz=Ayz.Byz.Cyz.Xyz.Xzy
MatrixForm[yz]
"zx=Azx.Bzx.Czx.Xzx.Xxz"
zx=Azx.Bzx.Czx.Xzx.Xxz
MatrixForm[zx]
"xy=Axy.Bxy.Cxy.Xxy.Xyx"
xy=Axy.Bxy.Cxy.Xxy.Xyx
MatrixForm[xy]
"------------------------------"
"x=yz.Transpose[yz]"
x=yz.Transpose[yz]
MatrixForm[x]
"y=zx.Transpose[zx]"
y=zx.Transpose[zx]
MatrixForm[y]
"z=xy.Transpose[xy]"
z=xy.Transpose[xy]
MatrixForm[z]
"------------------------------"
"a1=x[[1,1]]"
a1=x[[1,1]]
"a2=x[[2,2]]"
a2=x[[2,2]]
"a3=x[[3,3]]"
a3=x[[3,3]]
"a4=2*x[[1,2]]"
a4=2*x[[1,2]]
"a5=2*x[[1,3]]"
a5=2*x[[1,3]]
"a6=2*x[[2,3]]"
a6=2*x[[2,3]]
"b1=y[[1,1]]"
b1=y[[1,1]]
"b2=y[[2,2]]"
b2=y[[2,2]]
"b3=y[[3,3]]"
b3=y[[3,3]]
"b4=2*y[[1,2]]"
b4=2*y[[1,2]]
"b5=2*y[[1,3]]"
b5=2*y[[1,3]]
"b6=2*y[[2,3]]"
b6=2*y[[2,3]]
"c1=z[[1,1]]"
c1=z[[1,1]]
"c2=z[[2,2]]"
c2=z[[2,2]]
"c3=z[[3,3]]"
c3=z[[3,3]]
"c4=2*z[[1,2]]"
c4=2*z[[1,2]]
"c5=2*z[[1,3]]"
c5=2*z[[1,3]]
"c6=2*z[[2,3]]"
c6=2*z[[2,3]]
"------------------------------"
"M1p45={{a1-lamda,a4,a5},{b1,b4,b5},{c1,c4,c5}}"
M1p45={{a1-lamda,a4,a5},{b1,b4,b5},{c1,c4,c5}}
MatrixForm[M1p45]
"d1p45=Det[M1p45]"
d1p45=Det[M1p45]
"M1p63p={{a1-lamda,a6,a3},{b1,b6,b3},{c1,c6,c3-lamda}}"
M1p63p={{a1-lamda,a6,a3},{b1,b6,b3},{c1,c6,c3-lamda}}
MatrixForm[M1p63p]
"d1p63p=Det[M1p63p]"
d1p63p=Det[M1p63p]
"M1p2p5={{a1-lamda,a2,a5},{b1,b2-lamda,b5},{c1,c2,c5}}"
M1p2p5={{a1-lamda,a2,a5},{b1,b2-lamda,b5},{c1,c2,c5}}
MatrixForm[M1p2p5]
"d1p2p5=Det[M1p2p5]"
d1p2p5=Det[M1p2p5]
"M1p43p={{a1-lamda,a4,a3},{b1,b4,b3},{c1,c4,c3-lamda}}"
M1p43p={{a1-lamda,a4,a3},{b1,b4,b3},{c1,c4,c3-lamda}}
MatrixForm[M1p43p]
"d1p43p=Det[M1p43p]"
d1p43p=Det[M1p43p]
"M1p65={{a1-lamda,a6,a5},{b1,b6,b5},{c1,c6,c5}}"
M1p65={{a1-lamda,a6,a5},{b1,b6,b5},{c1,c6,c5}}
MatrixForm[M1p65]
"d1p65=Det[M1p65]"
d1p65=Det[M1p65]
"M1p2p3p={{a1-lamda,a2,a3},{b1,b2-lamda,b3},{c1,c2,c3-lamda}}"
M1p2p3p={{a1-lamda,a2,a3},{b1,b2-lamda,b3},{c1,c2,c3-lamda}}
MatrixForm[M1p2p3p]
"d1p2p3p=Det[M1p2p3p]"
d1p2p3p=Det[M1p2p3p]
"M1p2p5={{a1-lamda,a2,a5},{b1,b2-lamda,b5},{c1,c2,c5}}"
M1p2p5={{a1-lamda,a2,a5},{b1,b2-lamda,b5},{c1,c2,c5}}
MatrixForm[M1p2p5]
"d1p2p5=Det[M1p2p5]"
d1p2p5=Det[M1p2p5]
"M42p6={{a4,a2,a6},{b4,b2-lamda,b6},{c4,c2,c6}}"
M42p6={{a4,a2,a6},{b4,b2-lamda,b6},{c4,c2,c6}}
MatrixForm[M42p6]
"d42p6=Det[M42p6]"
d42p6=Det[M42p6]
"M1p2p6={{a1-lamda,a2,a6},{b1,b2-lamda,b6},{c1,c2,c6}}"
M1p2p6={{a1-lamda,a2,a6},{b1,b2-lamda,b6},{c1,c2,c6}}
MatrixForm[M1p2p6]
"d1p2p6=Det[M1p2p6]"
d1p2p6=Det[M1p2p6]
"M42p5={{a4,a2,a5},{b4,b2-lamda,b5},{c4,c2,c5}}"
M42p5={{a4,a2,a5},{b4,b2-lamda,b5},{c4,c2,c5}}
MatrixForm[M42p5]
"d42p5=Det[M42p5]"
d42p5=Det[M42p5]
"M42p3p={{a4,a2,a3},{b4,b2-lamda,b3},{c4,c2,c3-lamda}}"
M42p3p={{a4,a2,a3},{b4,b2-lamda,b3},{c4,c2,c3-lamda}}
MatrixForm[M42p3p]
"d42p3p=Det[M42p3p]"
d42p3p=Det[M42p3p]
"M563p={{a5,a6,a3},{b5,b6,b3},{c5,c6,c3-lamda}}"
M563p={{a5,a6,a3},{b5,b6,b3},{c5,c6,c3-lamda}}
MatrixForm[M563p]
"d563p=Det[M563p]"
d563p=Det[M563p]
"M1p63p={{a1-lamda,a6,a3},{b1,b6,b3},{c1,c6,c3-lamda}}"
M1p63p={{a1-lamda,a6,a3},{b1,b6,b3},{c1,c6,c3-lamda}}
MatrixForm[M1p63p]
"d1p63p=Det[M1p63p]"
d1p63p=Det[M1p63p]
"M463p={{a4,a6,a3},{b4,b6,b3},{c4,c6,c3-lamda}}"
M463p={{a4,a6,a3},{b4,b6,b3},{c4,c6,c3-lamda}}
MatrixForm[M463p]
"d463p=Det[M463p]"
d463p=Det[M463p]
"M52p3p={{a5,a2,a3},{b5,b2-lamda,b3},{c5,c2,c3-lamda}}"
M52p3p={{a5,a2,a3},{b5,b2-lamda,b3},{c5,c2,c3-lamda}}
MatrixForm[M52p3p]
"d52p3p=Det[M52p3p]"
d52p3p=Det[M52p3p]
"------------------------------"
"The following 6 by 6 matrix was defined by Guimei Zhang and Shenglong Hu in Characteristic Polynomial and Higher Order Traces of Third Order Three Dimensional Tensors"
"------------------------------"
"Mlamda={{a1-lamda,a2,a3,a4,a5,a6},{b1,b2-lamda,b3,b4,b5,b6},{c1,c2,c3-lamda,c4,c5,c6},{d1p45,0,d1p63p,d1p2p5,d1p43p+d1p65,d1p2p3p},{d1p2p5,d42p6,0,d1p2p6+d42p5,d1p2p3p,d42p3p},{0,d42p3p,d563p,d1p2p3p,d1p63p,d463p+d52p3p}}"
Mlamda={{a1-lamda,a2,a3,a4,a5,a6},{b1,b2-lamda,b3,b4,b5,b6},{c1,c2,c3-lamda,c4,c5,c6},{d1p45,0,d1p63p,d1p2p5,d1p43p+d1p65,d1p2p3p},{d1p2p5,d42p6,0,d1p2p6+d42p5,d1p2p3p,d42p3p},{0,d42p3p,d563p,d1p2p3p,d1p63p,d463p+d52p3p}}
MatrixForm[Mlamda]
"m=Det[Mlamda]"
m=Det[Mlamda]
"N[Solve[m==0,lamda],20]"
N[Solve[m==0,lamda],20]
"------------------------------"
"SECTION II"

"Det[Mlamda]==0 defines a characteristic equation with solutions -2-2*(Sqrt[3])I,-2+2*(Sqrt[3])I,22-6*(Sqrt[13]) and 22+6*(Sqrt[13]) These two complex and two real numbers are based on values of the four real numbers 2, 22-6*(Sqrt[13]), 2*(Sqrt[3]), and 22+6*(Sqrt[13]) which are defined below as J, K, L, M or JJ, KK, LL and MM."

"J=2.0"
J=2.0
"K=0.3666923"
K=0.3666923
"L=3.4641016"
L=3.4641016
"M=43.6333077"
M=43.6333077
"------------------------------"
"SetPrecision[JJ,20]"
SetPrecision[JJ,20]
"JJ=2."
JJ=2.
"SetPrecision[KK,20]"
SetPrecision[KK,20]
"KK=22.-6.*(Sqrt[13.])"
KK=22.-6.*(Sqrt[13.])
"SetPrecision[LL,20]"
SetPrecision[LL,20]
"LL=2.*(Sqrt[3.])"
LL=2.*(Sqrt[3.])
"SetPrecision[MM,20]"
SetPrecision[MM,20]
"MM=22.+6.*(Sqrt[13.])"
MM=22.+6.*(Sqrt[13.])
"------------------------------"
" In the remainder of Section II, it is shown that various unitless numbers, of physics and mathematics, can be expressed in terms of JJ, KK, LL and MM to any required level of precision"  
"1-----------------------------"
"Proton-Electron Gravitational to Electrical Force Ratio"
"u=43.6333-2*(3.4641+2*.3667+.3667/2)"
u=43.6333-2*(3.4641+2*.3667+.3667/2)
"v=43.6333-3.4641+2*.3667"
v=43.6333-3.4641+2*.3667
"u*(10^(-v))"
u*(10^(-v))
"Bondi gives a value of 4.35*(10^(-40))"
"D1=(M-J*(L+J*K+K/J))*(10^(-(M - L + (J*K))))"
D1=(M-J*(L+J*K+K/J))*(10^(-(M-L+(J*K))))
"DD1=(MM-JJ*(LL+JJ*KK+KK/JJ))*(10^(-(MM - LL + (JJ*KK))))"
DD1=(MM-JJ*(LL+JJ*KK+KK/JJ))*(10^(-(MM-LL+(JJ*KK))))
"2-----------------------------"
"Characteristic Length of Universe: (Light Velocity divided by Hubble's Constant) divided by Classical Electron Radius"
"p=43.6333-3.4641+2*.3667"
p=43.6333-3.4641+2*.3667
"(10^p)/2"
(10^p)/2
"Bondi gives a value of 4*(10^40)"
"D2=(10^((M - L + J*K)))/J"
D2=(10^((M-L+J*K)))/J
"DD2=(10^((MM - LL + JJ*KK)))/JJ"
DD2=(10^((MM-LL+JJ*KK)))/JJ
"3-----------------------------"
"Number of Heavy Elementary Particles in Universe"
"s=2*(43.6333-3.4641)-.3667/2"
s=2*(43.6333-3.4641)-.3667/2
"(10^s)"
(10^s)
"Bondi gives a value of (10^80)"
"D3=(10^(J*(M - L) - K/J))"
D3=(10^(J*(M-L)-K/J))
"DD3=(10^(JJ*(MM - LL) - KK/JJ))"
DD3=(10^(JJ*(MM-LL)-KK/JJ))
"4-----------------------------"
"Reciprocal Fine Structure Constant"
"43.6333*3.4641-(3.4641^2)-3.4641/2-.3667-2*.3667/43.6333"
43.6333*3.4641-(3.4641^2)-3.4641/2-.3667-2*.3667/43.6333
"Measured Value is 137.035999"
"D4=M*L-(L^J)-L/J-K-J*K/M"
D4=M*L-(L^J)-L/J-K-J*K/M
"DD4=MM*LL-(LL^JJ)-LL/JJ-KK-JJ*KK/MM"
DD4=MM*LL-(LL^JJ)-LL/JJ-KK-JJ*KK/MM
"------------------"
"DDD4=MM*LL-(LL^JJ)-LL/JJ-KK-(JJ-KK)*KK/MM-KK/(MM*(JJ^JJ))+(LL+KK+LL/JJ-1)*(KK/MM)^JJ+KK/(JJ*(MM^JJ))-KK/(JJ*MM)^JJ-(JJ^JJ)/(MM-(JJ^(JJ*JJ))*KK)^LL"
DDD4=MM*LL-(LL^JJ)-LL/JJ-KK-(JJ-KK)*KK/MM-KK/(MM*(JJ^JJ))+(LL+KK+LL/JJ-1)*(((KK/MM))^JJ)+KK/(JJ*(MM^JJ))-KK/(((JJ*MM))^JJ)-(JJ^JJ)/(((MM-(JJ^(JJ*JJ))*KK))^LL)
"SetPrecision[DDD4,12]"
SetPrecision[DDD4,12]
"5-----------------------------"
"π"
"Estimated Reciprocal Fine Structure Constant Divided by 22+6*(Sqrt[13])"
"(43.6333*3.4641-(3.4641^2)-3.4641/2-.3667-2*.3667/43.6333)/43.6333"
(43.6333*3.4641-(3.4641^2)-3.4641/2-.3667-2*.3667/43.6333)/43.6333
"Defined Value of π is 3.141592653..."
"D5=D4/M"
D5=D4/M
"DD5=DD4/MM"
DD5=DD4/MM
"DDD5=DDD4/MM"
DDD5=DDD4/MM
"SetPrecision[DDD5,12]"
SetPrecision[DDD5,12]
"------------------"
"DDD5P=DDD5+LL/(JJ*(MM^JJ))"
DDD5P=DDD5+LL/(JJ*(MM^JJ))
"SetPrecision[DDD5P,12]"
SetPrecision[DDD5P,12]
"6-----------------------------"
"Golden Ratio"
"(3.4641-2*.3667)^(1/2)-2/(43.6333+2*.3667)+.3667/43.6333+.3667/(43.6333*(2^2))"
(((3.4641-2*.3667))^(1/2))-2/(43.6333+2*.3667)+.3667/43.6333+.3667/(43.6333*(2^2))
"Derived value of Golden Ratio is 1.618033988..."
"D6=(L-J*K)^(1/J)-J/(M+J*K)+K/M+K/(M*(J^J))"
D6=(((L-J*K))^(1/J))-J/(M+J*K)+K/M+K/(M*(J^J))
"DD6=(LL-JJ*KK)^(1/JJ)-JJ/(MM+JJ*KK)+KK/MM+KK/(MM*(JJ^JJ))"
DD6=(((LL-JJ*KK))^(1/JJ))-JJ/(MM+JJ*KK)+KK/MM+KK/(MM*(JJ^JJ))
"7-----------------------------"
"Mass of Proton compared to Electron"
"(43.6333-2*.3667)^2-3.4641-2.0*.3667+3.464/43.6333-.3667/3.4641-3.4641/(43.6333*3.4641)"
(((43.6333-2*.3667))^2)-3.4641-2.0*.3667+3.464/43.6333-.3667/3.4641-3.4641/(43.6333*3.4641)
"Measured Value is 1836.15"
"D7=(M-J*K)^J-L-J*K+L/M-K/L-L/(M*L)"
D7=(((M-J*K))^J)-L-J*K+L/M-K/L-L/(M*L)
"DD7=(MM-JJ*KK)^JJ-LL-JJ*KK+LL/MM-KK/LL-LL/(MM*LL)"
DD7=(((MM-JJ*KK))^JJ)-LL-JJ*KK+LL/MM-KK/LL-LL/(MM*LL)
"SetPrecision[DD7,12]"
SetPrecision[DD7,12]
"8-----------------------------"
"Fraction of Rest Mass Energy Needed to Disperse Galaxies"
"d=2*(3.4641-2*0.3667)"
d=2*(3.4641-2*0.3667)
"2*(10^(-d))"
2*(10^(-d))
"Rees gives a range of Q=(10^(-5)) to (10^(-6))"
"D8=J*(10^(-J*(L - J*K)))"
D8=J*(10^(-J*(L-J*K)))
"DD8=JJ*(10^(-JJ*(LL - JJ*KK)))"
DD8=JJ*(10^(-JJ*(LL-JJ*KK)))
"9-----------------------------"
"Fraction of Mass converted to energy during Creation of Helium by Fusion in Stars"
"1/((43.6333-3.4641)(3.4641))"
1/((43.6333-3.4641)(3.4641))
"Rees gives a value of .007"
"D9=1/((M-L)*L)"
D9=1/((M-L)*L)
"DD9=1/((MM-LL)*LL)"
DD9=1/((MM-LL)*LL)
"10----------------------------"
"Weak Force Coupling Constant"
"r=2*3.4641"
r=2*3.4641
"3.4641*(10^(-r))"
3.4641*(10^(-r))
"Predicted to be in the Range of (10^(-6)) to (10^(-7))"
"D10=L*(10^(-(J*L)))"
D10=L*(10^(-(J*L)))
"DD10=LL*(10^(-(JJ*LL)))"
DD10=LL*(10^(-(JJ*LL)))
"11----------------------------"
"Mass of Higgs Boson compared to that of Electron"
"(43.6333)^2*(43.6333-3.4641+(2*2*.3667))*(3.4641-.3667)+43.6333*(3.4641^2)"
(((43.6333))^2)*(43.6333-3.4641+(2*2*.3667))*(3.4641-.3667)+43.6333*(3.4641^2)
"Measured value is 246037"
"D11=(M^J)*(M-L+((J^J)*K))*(L-K)+M*(L^J)"
D11=(M^J)*(M-L+((J^J)*K))*(L-K)+M*(L^J)
"DD11=(MM^JJ)*(MM-LL+((JJ^JJ)*KK))*(LL-KK)+MM*(LL^JJ)"
DD11=(MM^JJ)*(MM-LL+((JJ^JJ)*KK))*(LL-KK)+MM*(LL^JJ)
"12----------------------------"
"Mass of Majorana Neutrino compared to that of Electron"
"r=2*3.4641"
r=2*3.4641
"(3.4641+.3667+.3667/3.4641)*(10^(-r))"
(3.4641+.3667+.3667/3.4641)*(10^(-r))
"Measured value is 4.7*(10^(-7))"
"D12=(L+K+K/L)*(10^(-J*L))"
D12=(L+K+K/L)*(10^(-J*L))
"DD12=(LL+KK+KK/LL)*(10^(-JJ*LL))"
DD12=(LL+KK+KK/LL)*(10^(-JJ*LL))
"13-----------------------------"
"The 3 Diminsions of Space (Rees)"
"DD13=(LL/JJ)^JJ"
DD13=(((LL/JJ))^JJ)
"14-----------------------------"
"The early Universe Expanded at Precisely the Rate Required to avoid Gravitational Collapse.  Ω=1 in the Early Universe but is not currently less than 0.3 (Rees)"
"KK=22-6*(Sqrt[13]) may be a good lower limit"
"Ω=KK"
Ω=KK
"15------------------------------"
"Cosmological Constant Driving Expansion of the Universe.  λ May be about 0.7 (Rees)"
"λ=JJ*KK"
λ=JJ*KK
"16------------------------------"
"e"
"(2*3.4641*.3667)+(.3667)^2+2*(.3667)^(2*2)+.3667/43.6333-.3667/(2*43.6333*3.4641)-2/((2^2)*43.6333)^2"
(2*3.4641*.3667)+(((.3667))^2)+2*(((.3667))^(2*2))+.3667/43.6333-.3667/(2*43.6333*3.4641)-2/((((2^2)*43.6333))^2)
"Value of e is 2.718281828"
"D16=(J*L*K)+(K^J)+J*(K^(J*J))+K/M-K/(J*M*L)-J/((J^J)*M)^J"
D16=(J*L*K)+(K^J)+J*(K^(J*J))+K/M-K/(J*M*L)-J/((((J^J)*M))^J)
"DD16=(JJ*LL*KK)+(KK^JJ)+JJ*(KK^(JJ*JJ))+KK/MM-KK/(JJ*MM*LL)-JJ/((JJ^JJ)*MM)^JJ"
DD16=(JJ*LL*KK)+(KK^JJ)+JJ*(KK^(JJ*JJ))+KK/MM-KK/(JJ*MM*LL)-JJ/((((JJ^JJ)*MM))^JJ)
"SetPrecision[DD16,12]"
SetPrecision[DD16,12]
"-------------------------------"
"SECTION III"
"In this Section, the fine structure constant is computed per the process described at the beginning of this program. It was motivated by the fact that MM*π=137.078 which is nearly the value of (α^(-1)).  The initial part of this Section makes estimates for successive terms in the infinite series."
"SetPrecision[α,10]"
SetPrecision[α,10]
"SetPrecision[a,60]"
SetPrecision[a,60]
"SetPrecision[MM,60]"
SetPrecision[MM,60]
"SetPrecision[b,60]"
SetPrecision[b,60]
"b=MM"
b=MM
"SetPrecision[π,60]"
SetPrecision[π,60]
"SetPrecision[deltad,60]"
SetPrecision[deltad,60]
"SetPrecision[delta1d,60]"
SetPrecision[delta1d,60]
"SetPrecision[delta2d,60]"
SetPrecision[delta2d,60]
"SetPrecision[δd,60]"
SetPrecision[δd,60]
"SetPrecision[δ1d,60]"
SetPrecision[δ1d,60]
"SetPrecision[δ2d,60]"
SetPrecision[δ2d,60]
"SetPrecision[Δd,60]"
SetPrecision[Δd,60]
"---------------------------"
"α=.0072973525643"
α=.0072973525643
"MM*π"
MM*π
"-(α^(-1))"
-(α^(-1))
"-2*π*α"
-2*π*α
"α/2"
α/2
"2*(α^2)"
2*(α^2)
"MM*(α^3)"
MM*(α^3)
"-MM*(π^2)*(α^4)"
-MM*(π^2)*(α^4)
"(MM^2)*π*(α^5)"
(MM^2)*π*(α^5)
"-(MM^3)*(α^6)"
-(MM^3)*(α^6)
"-(MM^2)*(π^6)*(α^7)"
-(MM^2)*(π^6)*(α^7)
"-2*(π^5)*(MM^3)*(α^8)"
-2*(π^5)*(MM^3)*(α^8)
"-(π^8)*(MM^3)*(α^9)"
-(π^8)*(MM^3)*(α^9)
"-π*(MM^6)*(α^10)"
-π*(MM^6)*(α^10)
"+(π^6)*(MM^5)*(α^11)"
+(π^6)*(MM^5)*(α^11)
"+(π^2)*(MM^7)*(α^12)"
+(π^2)*(MM^7)*(α^12)
"+(π^2)*(MM^8)*(α^13)"
+(π^2)*(MM^8)*(α^13)
"+(π^4)*(MM^8)*(α^14)"
+(π^4)*(MM^8)*(α^14)
"-(π^7)*(MM^8)*(α^15)"
-(π^7)*(MM^8)*(α^15)
"-(π^7)*(MM^8)*(α^16)"
-(π^7)*(MM^8)*(α^16)
"+(π^9)*(MM^8)*(α^17)"
+(π^9)*(MM^8)*(α^17)
"-(π^3)*(MM^11)*(α^18)"
"-(π^3)*(MM^11)*(α^18)"
"+(π^3)*(MM^12)*(α^19)"
+(π^3)*(MM^12)*(α^19)
"-(π^7)*(MM^12)*(α^20)"
-(π^7)*(MM^12)*(α^20)
"+π*(MM^15)*(α^21)"
+π*(MM^15)*(α^21)
"-(π^2)*(MM^16)*(α^22)"
-(π^2)*(MM^16)*(α^22)
"+(π^3)*(MM^17)*(α^23)"
+(π^3)*(MM^17)*(α^23)
"-(π^4)*(MM^18)*(α^24)"
-(π^4)*(MM^18)*(α^24)
"+π*(MM^20)*(α^25)"
+π*(MM^20)*(α^25)
"-π*(MM^21)*(α^26)"
-π*(MM^21)*(α^26)
"+(π^2)*(MM^22)*(α^27)"
+(π^2)*(MM^22)*(α^27)
"+(π^2)*(MM^23)*(α^28)"
+(π^2)*(MM^23)*(α^28)
"+(π^2)*(MM^24)*(α^29)"
+(π^2)*(MM^24)*(α^29)
"-(π^3)*(MM^25)*(α^30)"
-(π^3)*(MM^25)*(α^30)
"+(MM^27)*(α^31)"
+(MM^27)*(α^31)
"-π*(MM^28)*(α^32)"
-π*(MM^28)*(α^32)
"+(π^2)*(MM^28)*(α^33)"
+(π^2)*(MM^28)*(α^33)
"+(π^3)*(MM^29)*(α^34)"
+(π^3)*(MM^29)*(α^34)
"+(π^4)*(MM^30)*(α^35)"
+(π^4)*(MM^30)*(α^35)
"-(MM^32)*(α^36)"
-(MM^32)*(α^36)
"-(MM^33)*(α^37)"
-(MM^33)*(α^37)
"-(π^2)*(MM^33)*(α^38)"
-(π^2)*(MM^33)*(α^38)
"-(π^2)*(MM^34)*(α^39)"
-(π^2)*(MM^34)*(α^39)
"-(π^2)*(MM^35)*(α^40)"
-(π^2)*(MM^35)*(α^40)
"-(π^2)*(MM^36)*(α^41)"
-(π^2)*(MM^36)*(α^41)
"+π*(MM^37)*(α^42)"
+π*(MM^37)*(α^42)
"+(MM^38)*(α^43)"
+(MM^38)*(α^43)
"+(MM^39)*(α^44)"
+(MM^39)*(α^44)
"+(π^2)*(MM^39)*(α^45)"
+(π^2)*(MM^39)*(α^45)
"+(π^2)*(MM^39)*(α^46)"
+(π^2)*(MM^39)*(α^46)
"+(π^2)*(MM^40)*(α^47)"
+(π^2)*(MM^40)*(α^47)
"+(MM^42)*(α^48)"
+(MM^42)*(α^48)
"+(MM^43)*(α^49)"
+(MM^43)*(α^49)
"+(π^3)*(MM^43)*(α^50)"
+(π^3)*(MM^43)*(α^50)
"-(π^2)*(MM^44)*(α^51)"
-(π^2)*(MM^44)*(α^51)
"-(π^2)*(MM^45)*(α^52)"
-(π^2)*(MM^45)*(α^52)
"-(π^6)*(MM^45)*(α^53)"
-(π^6)*(MM^45)*(α^53)
"-(MM^47)*(α^54)"
-(MM^47)*(α^54)
"-(π^3)*(MM^47)*(α^55)"
-(π^3)*(MM^47)*(α^55)
"-(MM^49)*(α^56)"
-(MM^49)*(α^56)
"-π*(MM^50)*(α^57)"
-π*(MM^50)*(α^57)
"-(π^2)*(MM^50)*(α^58)"
-(π^2)*(MM^50)*(α^58)
"-(MM^52)*(α^59)"
-(MM^52)*(α^59)
"-(MM^53)*(α^60)"
-(MM^53)*(α^60)
"-(π^4)*(MM^53)*(α^61)"
-(π^4)*(MM^53)*(α^61)
"-π*(MM^52)*(α^62)"
-π*(MM^52)*(α^62)
"+π*(MM^53)*(α^63)"
+π*(MM^53)*(α^63)
"-π*(MM^54)*(α^64)"
-π*(MM^54)*(α^64)
"-(π^2)*(MM^54)*(α^65)"
-(π^2)*(MM^54)*(α^65)
"+π*(MM^55)*(α^66)"
+π*(MM^55)*(α^66)
"-(π^3)*(MM^55)*(α^67)"
-(π^3)*(MM^55)*(α^67)
"+(π^3)*(MM^56)*(α^68)"
+(π^3)*(MM^56)*(α^68)
"-(π^2)*(MM^57)*(α^69)"
-(π^2)*(MM^57)*(α^69)
"-(π^4)*(MM^57)*(α^70)"
-(π^4)*(MM^57)*(α^70)
"-(π^3)*(MM^58)*(α^71)"
-(π^3)*(MM^58)*(α^71)
"-(π^3)*(MM^59)*(α^72)"
-(π^3)*(MM^59)*(α^72)
"+(MM^61)*(α^73)"
+(MM^61)*(α^73)
"+(π^3)*(MM^61)*(α^74)"
+(π^3)*(MM^61)*(α^74)
"-(MM^63)*(α^75)"
-(MM^63)*(α^75)
"-(π^2)*(MM^63)*(α^76)"
-(π^2)*(MM^63)*(α^76)
"-π*(MM^64)*(α^77)"
-π*(MM^64)*(α^77)
"-(π^2)*(MM^64)*(α^78)"
-(π^2)*(MM^64)*(α^78)
"-------------------------------"
"deltad=MM*π-(α^(-1))-2*π*α+α/2+2*(α^2)+MM*(α^3)-MM*(π^2)*(α^4)+(MM^2)*π*(α^5)-(MM^3)*(α^6)-(MM^2)*(π^6)*(α^7)-2*(π^5)*(MM^3)*(α^8)-(π^8)*(MM^3)*(α^9)-π*(MM^6)*(α^10)+(π^6)*(MM^5)*(α^11)+(π^2)*(MM^7)*(α^12)+(π^2)*(MM^8)*(α^13)+(π^4)*(MM^8)*(α^14)-(π^7)*(MM^8)*(α^15)-(π^7)*(MM^8)*(α^16)+(π^9)*(MM^8)*(α^17)+(π^3)*(MM^12)*(α^19)-(π^7)*(MM^12)*(α^20)+π*(MM^15)*(α^21)-(π^2)*(MM^16)*(α^22)+(π^3)*(MM^17)*(α^23)-(π^4)*(MM^18)*(α^24)+π*(MM^20)*(α^25)-π*(MM^21)*(α^26)+(π^2)*(MM^22)*(α^27)+(π^2)*(MM^23)*(α^28)+(π^2)*(MM^24)*(α^29)-(π^3)*(MM^25)*(α^30)+(MM^27)*(α^31)-π*(MM^28)*(α^32)+(π^2)*(MM^28)*(α^33)+(π^3)*(MM^29)*(α^34)"
deltad=MM*π-(α^(-1))-2*π*α +α/2+2*(α^2)+MM*(α^3)-MM*(π^2)*(α^4)+(MM^2)*π*(α^5)-(MM^3)*(α^6)-(MM^2)*(π^6)*(α^7)-2*(π^5)*(MM^3)*(α^8)-(π^8)*(MM^3)*(α^9)-π*(MM^6)*(α^10)+(π^6)*(MM^5)*(α^11)+(π^2)*(MM^7)*(α^12)+(π^2)*(MM^8)*(α^13)+(π^4)*(MM^8)*(α^14)-(π^7)*(MM^8)*(α^15)-(π^7)*(MM^8)*(α^16)+(π^9)*(MM^8)*(α^17)+(π^3)*(MM^12)*(α^19)-(π^7)*(MM^12)*(α^20)+π*(MM^15)*(α^21)-(π^2)*(MM^16)*(α^22)+(π^3)*(MM^17)*(α^23)-(π^4)*(MM^18)*(α^24)+π*(MM^20)*(α^25)-π*(MM^21)*(α^26)+(π^2)*(MM^22)*(α^27)+(π^2)*(MM^23)*(α^28)+(π^2)*(MM^24)*(α^29)-(π^3)*(MM^25)*(α^30)+(MM^27)*(α^31)-π*(MM^28)*(α^32)+(π^2)*(MM^28)*(α^33)+(π^3)*(MM^29)*(α^34)
"-------------------"
"delta1d=deltad+(π^4)*(MM^30)*(α^35)-(MM^32)*(α^36)-(MM^33)*(α^37)-(π^2)*(MM^33)*(α^38)-(π^2)*(MM^34)*(α^39)-(π^2)*(MM^35)*(α^40)-(π^2)*(MM^36)*(α^41)+π*(MM^37)*(α^42)+(MM^38)*(α^43)+(MM^39)*(α^44)+(π^2)*(MM^39)*(α^45)+(π^2)*(MM^39)*(α^46)+(π^2)*(MM^40)*(α^47)+(MM^42)*(α^48)+(MM^43)*(α^49)+(π^3)*(MM^43)*(α^50)-(π^2)*(MM^44)*(α^51)-(π^2)*(MM^45)*(α^52)-(π^6)*(MM^45)*(α^53)-(MM^47)*(α^54)-(π^3)*(MM^47)*(α^55)-(MM^49)*(α^56)-π*(MM^50)*(α^57)-(π^2)*(MM^50)*(α^58)-(MM^52)*(α^59)-(MM^53)*(α^60)-(π^4)*(MM^53)*(α^61)"
delta1d=deltad+(π^4)*(MM^30)*(α^35)-(MM^32)*(α^36)-(MM^33)*(α^37)-(π^2)*(MM^33)*(α^38)-(π^2)*(MM^34)*(α^39)-(π^2)*(MM^35)*(α^40)-(π^2)*(MM^36)*(α^41)+π*(MM^37)*(α^42)+(MM^38)*(α^43)+(MM^39)*(α^44)+(π^2)*(MM^39)*(α^45)+(π^2)*(MM^39)*(α^46)+(π^2)*(MM^40)*(α^47)+(MM^42)*(α^48)+(MM^43)*(α^49)+(π^3)*(MM^43)*(α^50)-(π^2)*(MM^44)*(α^51)-(π^2)*(MM^45)*(α^52)-(π^6)*(MM^45)*(α^53)-(MM^47)*(α^54)-(π^3)*(MM^47)*(α^55)-(MM^49)*(α^56)-π*(MM^50)*(α^57)-(π^2)*(MM^50)*(α^58)-(MM^52)*(α^59)-(MM^53)*(α^60)-(π^4)*(MM^53)*(α^61)
"-------------------"
"delta2d=delta1d-π*(MM^52)*(α^62)+π*(MM^53)*(α^63)-π*(MM^54)*(α^64)-(π^2)*(MM^54)*(α^65)+π*(MM^55)*(α^66)-(π^3)*(MM^55)*(α^67)+(π^3)*(MM^56)*(α^68)-(π^2)*(MM^57)*(α^69)-(π^4)*(MM^57)*(α^70)-(π^3)*(MM^58)*(α^71)-(π^3)*(MM^59)*(α^72)+(MM^61)*(α^73)+(π^3)*(MM^61)*(α^74)-(MM^63)*(α^75)-(π^2)*(MM^63)*(α^76)-π*(MM^64)*(α^77)-(π^2)*(MM^64)*(α^78)"
delta2d=delta1d-π*(MM^52)*(α^62)+π*(MM^53)*(α^63)-π*(MM^54)*(α^64)-(π^2)*(MM^54)*(α^65)+π*(MM^55)*(α^66)-(π^3)*(MM^55)*(α^67)+(π^3)*(MM^56)*(α^68)-(π^2)*(MM^57)*(α^69)-(π^4)*(MM^57)*(α^70)-(π^3)*(MM^58)*(α^71)-(π^3)*(MM^59)*(α^72)+(MM^61)*(α^73)+(π^3)*(MM^61)*(α^74)-(MM^63)*(α^75)-(π^2)*(MM^63)*(α^76)-π*(MM^64)*(α^77)-(π^2)*(MM^64)*(α^78)
"-------------------------------"
"b=MM"
b=MM
"δd=-(a^(-1))+(b*π)-(2*π-1/2)*a+2*(a^2)+b*(a^3)-b*(π^2)*(a^4)+(b^2)*π*(a^5)-(b^3)*(a^6)-(b^2)*(π^6)*(a^7)-2*(b^3)*(π^5)(a^8)-(b^3)*(π^8)*(a^9)-(b^6)*π*(a^10)+(b^5)*(π^6)*(a^11)+(b^7)*(π^2)*(a^12)+(b^8)*(π^2)*(a^13)+(b^8)*(π^4)*(a^14)-(b^8)*(π^7)*(a^15)-(b^8)*(π^7)*(a^16)+(b^8)*(π^9)*(a^17)+(b^12)*(π^3)*(a^19)-(b^12)*(π^7)*(a^20)+(b^15)*π*(a^21)-(b^16)*(π^2)*(a^22)+(b^17)*(π^3)*(a^23)-(b^18)*(π^4)*(a^24)+(b^20)*π*(a^25)-(b^21)*π*(a^26)+(b^22)*(π^2)*(a^27)+(b^23)*(π^2)*(a^28)+(b^24)*(π^2)*(a^29)-(b^25)*(π^3)*(a^30)+(b^27)*(a^31)-(b^28)*π*(a^32)+(b^28)*(π^2)*(a^33)+(b^29)*(π^3)*(a^34)"
δd=-(a^(-1))+(b*π)-(2*π-1/2)*a+2*(a^2)+b*(a^3)-b*(π^2)*(a^4)+(b^2)*π*(a^5)-(b^3)*(a^6)-(b^2)*(π^6)*(a^7)-2*(b^3)*(π^5)(a^8)-(b^3)*(π^8)*(a^9)-(b^6)*π*(a^10)+(b^5)*(π^6)*(a^11)+(b^7)*(π^2)*(a^12)+(b^8)*(π^2)*(a^13)+(b^8)*(π^4)*(a^14)-(b^8)*(π^7)*(a^15)-(b^8)*(π^7)*(a^16)+(b^8)*(π^9)*(a^17)+(b^12)*(π^3)*(a^19)-(b^12)*(π^7)*(a^20)+(b^15)*π*(a^21)-(b^16)*(π^2)*(a^22)+(b^17)*(π^3)*(a^23)-(b^18)*(π^4)*(a^24)+(b^20)*π*(a^25)-(b^21)*π*(a^26)+(b^22)*(π^2)*(a^27)+(b^23)*(π^2)*(a^28)+(b^24)*(π^2)*(a^29)-(b^25)*(π^3)*(a^30)+(b^27)*(a^31)-(b^28)*π*(a^32)+(b^28)*(π^2)*(a^33)+(b^29)*(π^3)*(a^34)
"-------------------"
"δ1d=δd+(b^30)*(π^4)*(a^35)-(b^32)*(a^36)-(b^33)*(a^37)-(b^33)*(π^2)*(a^38)-(b^34)*(π^2)*(a^39)-(b^35)*(π^2)*(a^40)-(b^36)*(π^2)*(a^41)+(b^37)*π*(a^42)+(b^38)*(a^43)+(b^39)*(a^44)+(b^39)*(π^2)*(a^45)+(b^39)*(π^2)*(a^46)+(b^40)*(π^2)*(a^47)+(b^42)*(a^48)+(b^43)*(a^49)+(b^43)*(π^3)*(a^50)-(b^44)*(π^2)*(a^51)-(b^45)*(π^2)*(a^52)-(b^45)*(π^6)*(a^53)-(b^47)*(a^54)-(b^47)*(π^3)*(a^55)-(b^49)*(a^56)-(b^50)*π*(a^57)-(b^50)*(π^2)*(a^58)-(b^52)*(a^59)-(b^53)*(a^60)-(b^53)*(π^4)*(a^61)"
δ1d=δd+(b^30)*(π^4)*(a^35)-(b^32)*(a^36)-(b^33)*(a^37)-(b^33)*(π^2)*(a^38)-(b^34)*(π^2)*(a^39)-(b^35)*(π^2)*(a^40)-(b^36)*(π^2)*(a^41)+(b^37)*π*(a^42)+(b^38)*(a^43)+(b^39)*(a^44)+(b^39)*(π^2)*(a^45)+(b^39)*(π^2)*(a^46)+(b^40)*(π^2)*(a^47)+(b^42)*(a^48)+(b^43)*(a^49)+(b^43)*(π^3)*(a^50)-(b^44)*(π^2)*(a^51)-(b^45)*(π^2)*(a^52)-(b^45)*(π^6)*(a^53)-(b^47)*(a^54)-(b^47)*(π^3)*(a^55)-(b^49)*(a^56)-(b^50)*π*(a^57)-(b^50)*(π^2)*(a^58)-(b^52)*(a^59)-(b^53)*(a^60)-(b^53)*(π^4)*(a^61)
"-------------------"
"δ2d=δ1d-(b^52)*π*(a^62)+(b^53)*π*(a^63)-(b^54)*π*(a^64)-(b^54)*(π^2)*(a^65)+(b^55)*π*(a^66)-(b^55)*(π^3)*(a^67)+(b^56)*(π^3)*(a^68)-(b^57)*(π^2)*(a^69)-(b^57)*(π^4)*(a^70)-(b^58)*(π^3)*(a^71)-(b^59)*(π^3)*(a^72)+(b^61)*(a^73)+(b^61)*(π^3)*(a^74)-(b^63)*(a^75)-(b^63)*(π^2)*(a^76)-(b^64)*π*(a^77)-(b^64)*(π^2)*(a^78)"
δ2d=δ1d-(b^52)*π*(a^62)+(b^53)*π*(a^63)-(b^54)*π*(a^64)-(b^54)*(π^2)*(a^65)+(b^55)*π*(a^66)-(b^55)*(π^3)*(a^67)+(b^56)*(π^3)*(a^68)-(b^57)*(π^2)*(a^69)-(b^57)*(π^4)*(a^70)-(b^58)*(π^3)*(a^71)-(b^59)*(π^3)*(a^72)+(b^61)*(a^73)+(b^61)*(π^3)*(a^74)-(b^63)*(a^75)-(b^63)*(π^2)*(a^76)-(b^64)*π*(a^77)-(b^64)*(π^2)*(a^78)
"-------------------------------"
"Δd:=Evaluate[δ2d/.{b->MM,π->N[π]}]"
Δd:=Evaluate[δ2d/.{b->MM,π->N[π]}]
"-------------------------------"
"NSolve[SetPrecision[Δd,60]==0,a,Reals,WorkingPrecision->60]"
NSolve[SetPrecision[Δd,60]==0,a,Reals,WorkingPrecision->60]
"-------------------------------"
"Δprecd=Δd/. x_Real:>SetPrecision[x,60];
root=FindRoot[Δprecd==0,{a,.005`60},WorkingPrecision->60,AccuracyGoal->60]"
Δprecd=Δd/. x_Real:>SetPrecision[x,60];
root=FindRoot[Δprecd==0,{a,.005`60},WorkingPrecision->60,AccuracyGoal->60]
"N[a/.root,60]"
N[a/.root,60]