17 July 2019 AD; St Alexis the Beggar; Bl Martyrs of Compiegne

Again, I use the same transformation into the positive Real half-plane:

The transformation is as follows:

first quadrant: theta final = theta

second quadrant: theta final = 180 degrees - absolute value |theta|

third quadrant: theta final = 180 degrees + absolute value |theta|

fourth quadrant: theta final = 360 degrees - absolute value |theta|

This is a boring part - just calculations but it has to be done so you know where the Final Angles come from.

There will be three different angles, with the same quantum numbers as before - for "regular"  graviton mixing angles.

First final angle:

These printouts are the results of substituting for Real value X constant C_M_4, C_M_5 and C_M_6 in the main program (more about it later):

The first Graviton mixing final angle is:

These printouts are the results of substituting for Real value X constant C_M_4 in the main program (more about it later):

ALFA C_M_4 INVISIBLE
A + Bi
-0.38080927536072116911D-02 -0.17501519652516662768D-02
MAGNITUDE & THETA IN RADIANS
0.41910144740325136134D-02 -0.27107943954127136088D+01
MAGNITUDE & THETA IN DEGREES
0.41910144740325136134D-02 -0.15531707798486613115D+03
MAGNITUDE & THETA FINAL IN RADIANS
0.41910144740325136134D-02 0.58523870490025071689D+01
MAGNITUDE & THETA FINAL IN DEGREES
0.41910144740325136134D-02 0.33531707798486615957D+03

ALFA C_M_4 VISIBLE
C + Di
0.21824754228784004617D+01 0.15902488461974433331D+01
MAGNITUDE & THETA IN RADIANS
0.27003870767540996134D+01 0.62969295440730754798D+00
MAGNITUDE & THETA IN DEGREES
0.27003870767540996134D+01 0.36078748676662492301D+02
MAGNITUDE & THETA FINAL IN RADIANS
0.27003870767540996134D+01 0.62969295440730754798D+00
MAGNITUDE & THETA FINAL IN DEGREES
0.27003870767540996134D+01 0.36078748676662492301D+02

PRODUCT
( A + Bi ) * ( C + Di )
-0.55278916993774214819D-02 -0.98754787581008078579D-02
MAGNITUDE & THETA IN RADIANS
0.11317361324166779965D-01 -0.20811014410054058388D+01
MAGNITUDE & THETA IN DEGREES
0.11317361324166779965D-01 -0.11923832930820363174D+03
MAGNITUDE & THETA FINAL IN RADIANS
0.11317361324166779965D-01 0.52226940945951998430D+01
MAGNITUDE & THETA FINAL IN DEGREES
0.11317361324166779965D-01 0.29923832930820367437D+03

DIVISION
( A + Bi ) / ( C + Di )
-0.15214081877693334203D-02 0.30665438104278180866D-03
MAGNITUDE & THETA IN RADIANS
0.15520050847933131773D-02 -0.33404873498200213788D+01
MAGNITUDE & THETA IN DEGREES
0.15520050847933131773D-02 -0.19139582666152864476D+03
MAGNITUDE & THETA FINAL IN RADIANS
0.15520050847933131773D-02 0.19889469623022781875D+00
MAGNITUDE & THETA FINAL IN DEGREES
0.15520050847933131773D-02 0.11395826661528619894D+02

SUM
( A + Bi ) + ( C + Di )
0.21786673301247931889D+01 0.15884986942321916992D+01
MAGNITUDE & THETA IN RADIANS
0.26962788129068684384D+01 0.63000007603606933415D+00
MAGNITUDE & THETA IN DEGREES
0.26962788129068684384D+01 0.36096345449787726523D+02 This is angle θ13  from before the transformation
MAGNITUDE & THETA FINAL IN RADIANS
0.26962788129068684384D+01 0.63000007603606933415D+00
MAGNITUDE & THETA FINAL IN DEGREES
0.26962788129068684384D+01 0.36096345449787726523D+02 This is angle θ13 after the transformation

θ13 = ( 0.36096345449787726523D+02 ) * [ (1 / 3.5 ) * ( 3 / 5 )] = 6.1879 degrees

θ13 Final = ( 0.36096345449787726523D+02 ) * [ (1 / 3.5 ) * ( 3 / 5 )] = 6.1879 degrees

i.e ("ordinary" angle) * (quantum number) = final angle

The second final angle for Graviton is:

These printouts are the results of substituting for Real value X constant C_M_5 in the main program (more about it later):

ALFA C_M_5 INVISIBLE
A + Bi
0.26377875161943746818D+01 -0.25974945732101939466D+00
MAGNITUDE & THETA IN RADIANS
0.26505457477978855430D+01 -0.98156020909732591417D-01
MAGNITUDE & THETA IN DEGREES
0.26505457477978855430D+01 -0.56239257319255369083D+01
MAGNITUDE & THETA FINAL IN RADIANS
0.26505457477978855430D+01 0.61850292862698532659D+01
MAGNITUDE & THETA FINAL IN DEGREES
0.26505457477978855430D+01 0.35437607426807443289D+03

ALFA C_M_5 VISIBLE
C + Di
0.51958635378203501531D+01 0.11265467523896126600D+01
MAGNITUDE & THETA IN RADIANS
0.53165877674473298597D+01 0.21351134923185283165D+00
MAGNITUDE & THETA IN DEGREES
0.53165877674473298597D+01 0.12233299189128958417D+02
MAGNITUDE & THETA FINAL IN RADIANS
0.53165877674473298597D+01 0.21351134923185283165D+00
MAGNITUDE & THETA FINAL IN DEGREES
0.53165877674473298597D+01 0.12233299189128958417D+02

PRODUCT
( A + Bi ) * ( C + Di )
0.13998203883492015720D+02 0.16219682255997276421D+01
MAGNITUDE & THETA IN RADIANS
0.14091859099801773425D+02 0.11535532832212024024D+00
MAGNITUDE & THETA IN DEGREES
0.14091859099801773425D+02 0.66093734572034215091D+01
MAGNITUDE & THETA FINAL IN RADIANS
0.14091859099801773425D+02 0.11535532832212022636D+00
MAGNITUDE & THETA FINAL IN DEGREES
0.14091859099801773425D+02 0.66093734572034215091D+01

DIVISION
( A + Bi ) / ( C + Di )
0.47452465899718043740D+00 -0.15287616101948242608D+00
MAGNITUDE & THETA IN RADIANS
0.49854264873172599026D+00 -0.31166737014158540919D+00
MAGNITUDE & THETA IN DEGREES
0.49854264873172599026D+00 -0.17857224921054495326D+02
MAGNITUDE & THETA FINAL IN RADIANS
0.49854264873172599026D+00 0.59715179370380004897D+01
MAGNITUDE & THETA FINAL IN DEGREES
0.49854264873172599026D+00 0.34214277507894547625D+03

SUM
( A + Bi ) + ( C + Di )
0.78336510540147248349D+01 0.86679729506859326538D+00
MAGNITUDE & THETA IN RADIANS
0.78814609297264324184D+01 0.11020219031595600290D+00
MAGNITUDE & THETA IN DEGREES
0.78814609297264324184D+01 0.63141203982017515273D+01 This is angle θ2 from before the transformation
MAGNITUDE & THETA FINAL IN RADIANS
0.78814609297264324184D+01 0.11020219031595600290D+00
MAGNITUDE & THETA FINAL IN DEGREES
0.78814609297264324184D+01 0.63141203982017515273D+01 This is angle θ23 after the transformation

θ23  = ( 0.63141203982017515273D+01 ) * [ 1 / ( 3.52 ) ] = 0.5154 degrees

θ23 Final = ( 0.63141203982017515273D+01 ) * [ 1 / ( 3.52 ) ] = 0.5154 degrees

i.e ("ordinary" angle) * (quantum number) = final angle

And the third final angle is:

These printouts are the results of substituting for Real value X constant C_M_6 in the main program (more about it later):

ALFA C_M_6 INVISIBLE
A + Bi
-0.82011219380003803963D+04 -0.24703309114555013366D+04
MAGNITUDE & THETA IN RADIANS
0.85650998741429575603D+04 -0.28490182032569002502D+01
MAGNITUDE & THETA IN DEGREES
0.85650998741429575603D+04 -0.16323671880256532063D+03
MAGNITUDE & THETA FINAL IN RADIANS
0.85650998741429575603D+04 0.59906108568466933662D+01
MAGNITUDE & THETA FINAL IN DEGREES
0.85650998741429575603D+04 0.34323671880256529221D+03

ALFA C_M_6 VISIBLE
C + Di
0.58081415808837846271D+01 -0.28861111112701247627D+01
MAGNITUDE & THETA IN RADIANS
0.64856877792712213093D+01 -0.46117082591115249324D+00
MAGNITUDE & THETA IN DEGREES
0.64856877792712213093D+01 -0.26423141959271465140D+02
MAGNITUDE & THETA FINAL IN RADIANS
0.64856877792712213093D+01 0.58220144812684333502D+01
MAGNITUDE & THETA FINAL IN DEGREES
0.64856877792712213093D+01 0.33357685804072849578D+03

PRODUCT
( A + Bi ) * ( C + Di )
-0.54762926830063996022D+05 0.93213174647768410068D+04
MAGNITUDE & THETA IN RADIANS
0.55550563581966453057D+05 -0.33101890291680526879D+01
MAGNITUDE & THETA IN DEGREES
0.55550563581966453057D+05 -0.18965986076183676801D+03
MAGNITUDE & THETA FINAL IN RADIANS
0.55550563581966453057D+05 0.16859637557825957188D+00
MAGNITUDE & THETA FINAL IN DEGREES
0.55550563581966453057D+05 0.96598607618367768879D+01

DIVISION
( A + Bi ) / ( C + Di )
-0.96290146659909362370D+03 -0.90379538104625260075D+03
MAGNITUDE & THETA IN RADIANS
0.13206155099722352588D+04 -0.23878473773457478124D+01
MAGNITUDE & THETA IN DEGREES
0.13206155099722352588D+04 -0.13681357684329384483D+03
MAGNITUDE & THETA FINAL IN RADIANS
0.13206155099722352588D+04 0.55294400309355413725D+01
MAGNITUDE & THETA FINAL IN DEGREES
0.13206155099722352588D+04 0.31681357684329390167D+03

SUM
( A + Bi ) + ( C + Di )
-0.81953137964194975211D+04 -0.24732170225667714476D+04
MAGNITUDE & THETA IN RADIANS
0.85603721100485945499D+04 -0.28484996932553334759D+01
MAGNITUDE & THETA IN DEGREES
0.85603721100485945499D+04 -0.16320701036784021198D+03 This is angle θ12 before the transformation
MAGNITUDE & THETA FINAL IN RADIANS
0.85603721100485945499D+04 0.59900923468451265919D+01
MAGNITUDE & THETA FINAL IN DEGREES
0.85603721100485945499D+04 0.34320701036784021198D+03 This is angle θ12 after the transformation

θ12  = ( - 0.16320701036784021198D+03 ) * [ 1 / 3.5 ) * ( 3 / 2 ) ] = - 69.9459 degrees

θ12 Final = ( 0.34320701036784021198D+03 ) * [ 1 / 3.5 ) * ( 3 / 2 ) ] = 147.0887 degrees

i.e ("ordinary" angle) * (quantum number) = final angle

The sum of all six angles for Graviton (three "ordinary" and three final angles) is:

θ SUM OF GRAVITON ANGLES = ( 6.1879 + 6.1879 ) + ( 0.5154 + 0.5154 ) + ( -69.9459 + 147.0887 ) = 90.5494 degrees

When this sum of Graviton angles is added to the sum of Quark angles ( 90.4706 degrees) we get a spherical triangle, with the total sum just a bit bigger than 180 degrees.

Again, the Quarks are tied up to the Gravitons - they form one family.

When we add 180+ degrees (spherical triangle) to 360+ degrees (spherical quadrangle) the sum is 540+ degrees - a spherical pentagon.