PhiLambda

SUBGRUPO DE LIE: TEOREMA DE RAÍCES COMPLEJAS ÁUREAS PRIMAS Y TEOREMA MATRICIAL DE ROTACIÓN ÁUREA PRIMA

JAVIER GRISALES HERRERA

13/03/21

La verdad es la intersección entre los hechos y las teorías.

La verdad es hija del tiempo (Veritas filia temporis), no de la autoridad. Francis Bacon

https://www.alertatolima.com/noticias/tolima/ibague/ibaguereno-es-autor-de-uno-de-los-descubrimientos-matematicos-mas#:~:text=Javier%20Grisales%20de%2037%20a%C3%B1os,propiedades%20aritm%C3%A9ticas%2C%20geom%C3%A9tricas%2C%20trigonom%C3%A9tricas%2C

Una matriz ortogonal A es una matriz cuadrada cuya matriz inversa es igual a su matriz traspuesta. El conjunto de matrices ortogonales es un conjunto de transformaciones isométricas (invariantes) de espacios vectoriales reales o espacios de Hilbert reales. Este conjunto es la representación del grupo ortogonal O(n). Cada rotación se describe mediante una matriz ortogonal nxn con entradas reales, tal que al hacer el producto con su traspuesta da como resultado la matriz identidad y con determinante 1.

El grupo real SO(n) es un grupo de Lie y es un subgrupo del grupo ortogonal O(n). Este subgrupo SO(n) se puede identificar con el grupo de rotaciones del espacio Rⁿ. Veamos el caso para una rotación r en R²

Una simetría es una función biyectiva de la permutación de los vértices de un polígono regular que preserva las distancias y los ángulos entre ellos, entre cada par de puntos, esto es una isometría. La simetría áurea prima son 8 familias de rotaciones áureas primas (reales o complejas) que preservan la estructura para la suma y el producto algebraico de mi primera función aurea prima para todo p primo mayor que 5. Así, se crean 8 isometrías del pentágono regular en la circunferencia unitaria para los números reales y complejos. Estas isometrías, son raíces quintas de potencias impares: 1, 3, 7, 9 de i en el plano complejo. Y son las siguientes:

{ i^1/5, i^3/5, i^7/5, i^9/5 }

https://www.wolframalpha.com/input/?i=e%5E%28ipi%2F10%29+%3D+i%5E%281%2F5%29

https://www.wolframalpha.com/input/?i=e%5E%283i%CF%80%2F10%29+%3D+i%5E%283%2F5%29

https://www.wolframalpha.com/input/?i=e%5E%287i%CF%80%2F10%29+%3D+i%5E%287%2F5%29

https://www.wolframalpha.com/input/?i=e%5E%289i%CF%80%2F10%29+%3D+i%5E%289%2F5%29

Ahora veamos la forma trigonométrica de dichas raíces quintas de i

https://www.wolframalpha.com/input/?i=+i%5E%281%2F5%29+%3D+cos18+%2B+i+sin18

https://www.wolframalpha.com/input/?i=+i%5E%283%2F5%29+%3D+cos54+%2B+i+sin54

https://www.wolframalpha.com/input/?i=+i%5E%287%2F5%29+%3D+cos126+%2B+i+sin126

https://www.wolframalpha.com/input/?i=+i%5E%289%2F5%29+%3D+cos162+%2B+i+sin162

En el tercer y cuarto cuadrante:

https://www.wolframalpha.com/input/?i=+i%5E%28-1%2F5%29+%3D+cos%28-18%29%2B+i+sin%28-18%29

https://www.wolframalpha.com/input/?i=+i%5E%28-3%2F5%29+%3D+cos%28-54%29%2B+i+sin%28-54%29

https://www.wolframalpha.com/input/?i=+i%5E%28-7%2F5%29+%3D+cos%28-126%29%2B+i+sin%28-126%29

https://www.wolframalpha.com/input/?i=+i%5E%28-9%2F5%29+%3D+cos%28-162%29%2B+i+sin%28-162%29

https://www.wolframalpha.com/input/?i=-i%5E%283%2F5%29+%3D++cos%28-7%CF%80%2F10%29+%2B+isin%28-7%CF%80%2F10%29

ISOMETRÍAS PRIMAS EN RADIANES:

18° ↔ π/10

54° ↔ 3π/10

126° ↔ 7π/10

162° ↔ 9π/10

198° ↔ 11π/10 ↔ -162° ↔ -9π/10

234° ↔ 13π/10 ↔ -126° ↔ -7π/10

306° ↔ 17π/10 ↔ -54° ↔ -3π/10

342° ↔ 19π/10 ↔ -18° ↔ -π/10

IGUALDAD ENTRE LAS RAÍCES COMPLEJAS

i^-1/5= -i^9/5

i^-3/5= -i^7/5

i^-7/5= -i^3/5

i^-9/5= -i^1/5

https://www.wolframalpha.com/input/?i=i%5E%28-1%2F5%29+%3D+-i%5E%289%2F5%29

https://www.wolframalpha.com/input/?i=i%5E%28-3%2F5%29+%3D+-i%5E%287%2F5%29

https://www.wolframalpha.com/input/?i=i%5E%28-7%2F5%29+%3D+-i%5E%283%2F5%29

https://www.wolframalpha.com/input/?i=i%5E%28-9%2F5%29+%3D+-i%5E%281%2F5%29

PROPOSICIÓN PARA LOS COSENOS:

Cos18° = Cos342° = 0.9510…

Cos162° = Cos198° = -0.9510…

Y en general:

Cos [ 2 3²ⁿ ]° =

0.9510…si n es impar = [ root(2+φ) ] / 2

-0.9510…si n es par = - [ root(2+φ) ] / 2

https://www.wolframalpha.com/input/?i=%5B+root%282%2B%CF%86%29+%5D+%2F+2

https://www.wolframalpha.com/input/?i=Cos+%5B+2+3%5E%282%28671%29%29+%5D%C2%B0+%3D++%5B+root%282+%2B+%CF%86%29+%5D+%2F+2

https://www.wolframalpha.com/input/?i=Cos+%5B+2+3%5E%282%286742%29%29+%5D%C2%B0+%3D++-+%5B+root%282+%2B+%CF%86%29+%5D+%2F+2

Cos54° = Cos306° = 0.5877…

Cos126° = Cos234° = -0.5877…

Y en general:

Cos [ 2 3²ⁿ⁺¹ ]° =

0.5877…si n es impar = [ root(2 - φ^-1) ] / 2

-0.5877…si n es par = - [ root(2 - φ^-1) ] / 2

https://www.wolframalpha.com/input/?i=%5B+root%282-%CF%86%5E%28-1%29%29+%5D+%2F+2

https://www.wolframalpha.com/input/?i=Cos+%5B+2+3%5E%282%28671%29%2B1%29+%5D%C2%B0+%3D++%5B+root%282+-+%CF%86%5E%28-1%29%29+%5D+%2F+2

https://www.wolframalpha.com/input/?i=Cos+%5B+2+3%5E%282%281484%29%2B1%29+%5D%C2%B0+%3D+-+%5B+root%282+-+%CF%86%5E%28-1%29%29+%5D+%2F+2

TEOREMA 1: TEOREMA DEL POLINOMIO MINIMAL AUREO PRIMO

Las 8 isometrías áureas primas corresponden a las raíces complejas del siguiente polinomio minimal:

1 - x²+ x⁴ – x⁶ + x⁸ = 0

https://www.wolframalpha.com/input/?i=%E2%88%91+n%3D0+to+4+%28-1%29%5En+x%5E%282n%29

https://www.wolframalpha.com/input/?i=1+-+x%5E2+%2B+x%5E4+-+x%5E6+%2B+x%5E8+%3D+0

Ahora veamos las isometrías áureas primas complejas de la forma:

P₁ ↔ {18°, 198°}

P₃ ↔ {126°, 306°}

P₇ ↔ {54°, 234°}

P₉ ↔ {162°, 342°}

PRIMOS PENULT-IMPARES: (Primes congruent to 11,13,17,19 mód 20)

Son aquellos primos cuyo penúltimo dígito es un número impar. Estos primos tienen 4 clases residuales módulo 360 y son los ángulos: 18°,54°,126°,162°. Estos primos siempre dan ángulos que se encuentran en el primer y segundo cuadrante del plano cartesiano 0 < θ < π

P₁ ↔ {18°}

11,31,71,131,151,191,211,251,271,311,331,431,491,571,631,691,751,811,911, 971…

P₃ ↔ {126°}

13,53,73,113,173,193,233,293,313,353,373,433,593,613,653,673,733,773,853, 953…

P₇ ↔ {54°}

17,37,97,137,157,197,257,277,317,337,397,457,557,577,617,677,757,797,857, 877,937,977,997…

P₉ ↔ {162°}

19,59,79,139,179,199,239,359,379,419,439,479,499,599,619,659,719,739,839, 859,919…

PRIMOS PENULTI-PARES: (Primes congruent to 1,3,7,9 mód 20)

Son aquellos primos cuyo penúltimo dígito es un número par. Estos primos tienen 4 clases residuales módulo 360 y son los ángulos: 198°,234°,306°,342°. Estos primos siempre dan ángulos que se encuentran en el tercer y cuarto cuadrante del plano cartesiano π < θ < 2π.

P₁ ↔ {198°}

41,61,101,181,241,281,401,421,461,521,541,601,641,661,701,761,821,881,941…

P₃ ↔ {306°}

23,43,83,103,163,223,263,283,383,443,463,503,523,563,643,683,743,823,863, 883,983…

P₇ ↔ {234°}

07,47,67,107,127,167,227,307,347,367,467,487,547,587,607,647,727,787,827, 887,907,947,967…

P₉ ↔ {342°}

29,89,109,149,229,269,349,389,409,449,509,569,709,769,809,829,929…

Ver acá y acá los ejemplos en Wolfram:

Cos18° = Cos342° = 0.9510…

Cos162° = Cos198° = -0.9510…

Y en general:

Cos [ 2 3²ⁿ ]° =

0.9510…si n es impar = [ root(2+φ) ] / 2

-0.9510…si n es par = - [ root(2+φ) ] / 2

Cos54° = Cos306° = 0.5877…

Cos126° = Cos234° = -0.5877…

Y en general:

Cos [ 2 3²ⁿ⁺¹ ]° =

0.5877…si n es impar = [ root(2 - φ^(-1)) ] / 2

-0.5877…si n es par = - [ root(2 - φ^(-1)) ] / 2

PRIMOS PENULTI-IMPARES

i^(1/5) = Cos[ (2) 3^(2(67))° ] + i Sin[ 2((31))^(-1)(10^((31)-1) -1)° ]

https://www.wolframalpha.com/input/?i=i%5E%281%2F5%29+%3D+Cos%5B+%282%29+3%5E%282%2867%29%29%C2%B0+%5D+%2B+i+Sin%5B+2%28%2831%29%29%5E%28-1%29%2810%5E%28%2831%29-1%29+-1%29%C2%B0+%5D

i^(7/5) = Cos[ (2) 3^(2(52)+1)° ] + i Sin[2((13))^(-1)(10^((13)-1) -1)°]

https://www.wolframalpha.com/input/?i=i%5E%287%2F5%29+%3D+Cos%5B+%282%29+3%5E%282%2852%29%2B1%29%C2%B0+%5D+%2B+i+Sin%5B2%28%2813%29%29%5E%28-1%29%2810%5E%28%2813%29-1%29+-1%29%C2%B0%5D

i^(3/5) = Cos[ (2) 3^(2(53)+1)° ] + i Sin[2((137))^(-1)(10^((137)-1) -1)°]

https://www.wolframalpha.com/input/?i=i%5E%283%2F5%29+%3D+Cos%5B+%282%29+3%5E%282%2853%29%2B1%29%C2%B0+%5D+%2B+i+Sin%5B2%28%28137%29%29%5E%28-1%29%2810%5E%28%28137%29-1%29+-1%29%C2%B0%5D

i^(9/5) = Cos[ (2) 3^(2(52))° ] + i Sin[2((59))^(-1)(10^((59)-1) -1)°]

https://www.wolframalpha.com/input/?i=i%5E%289%2F5%29+%3D+Cos%5B+%282%29+3%5E%282%2852%29%29%C2%B0+%5D+%2B+i+Sin%5B2%28%2859%29%29%5E%28-1%29%2810%5E%28%2859%29-1%29+-1%29%C2%B0%5D

IGUALDAD ENTRE LAS RAÍCES COMPLEJAS

i^-1/5= -i^9/5

i^-3/5= -i^7/5

i^-7/5= -i^3/5

i^-9/5= -i^1/5

PRIMOS PENULTI-PARES

-i^(1/5) = Cos[ (2) 3^(2(64))° ] + i Sin[2((101))^(-1)(10^((101)-1) -1)°]

https://www.wolframalpha.com/input/?i=-i%5E%281%2F5%29+%3D+Cos%5B+%282%29+3%5E%282%2864%29%29%C2%B0+%5D+%2B+i+Sin%5B2%28%28101%29%29%5E%28-1%29%2810%5E%28%28101%29-1%29+-1%29%C2%B0%5D

-i^(3/5) = Cos[ (2) 3^(2(62)+1)° ] + i Sin[2((23))^(-1)(10^((23)-1) -1)°]

https://www.wolframalpha.com/input/?i=-i%5E%283%2F5%29+%3D+Cos%5B+%282%29+3%5E%282%2862%29%2B1%29%C2%B0+%5D+%2B+i+Sin%5B2%28%2823%29%29%5E%28-1%29%2810%5E%28%2823%29-1%29+-1%29%C2%B0%5D

-i^(7/5) = Cos[ (2) 3^(2(67)+1)° ] + i Sin[2((67))^(-1)(10^((67)-1) -1)°]

https://www.wolframalpha.com/input/?i=-i%5E%287%2F5%29+%3D+Cos%5B+%282%29+3%5E%282%2867%29%2B1%29%C2%B0+%5D+%2B+i+Sin%5B2%28%2867%29%29%5E%28-1%29%2810%5E%28%2867%29-1%29+-1%29%C2%B0%5D

-i^(9/5) = Cos[ (2) 3^(2(67))° ] + i Sin[2((109))^(-1)(10^((109)-1) -1)°]

https://www.wolframalpha.com/input/?i=-i%5E%289%2F5%29+%3D+Cos%5B+%282%29+3%5E%282%2867%29%29%C2%B0+%5D+%2B+i+Sin%5B2%28%28109%29%29%5E%28-1%29%2810%5E%28%28109%29-1%29+-1%29%C2%B0%5D

TEOREMA DE RAÍCES COMPLEJAS ÁUREAS PRIMAS

En el plano complejo existen 8 raíces quintas que son isomorfas a 8 clases primas áureas

Estas imágenes son:

i^(1/5)

i^(3/5)

i^(7/5)

i^(9/5)

i^(-1/5)

i^(-3/5)

i^(-7/5)

i^(-9/5)

Cos(18)° + i Sin(18)° = i^(1/5) P_{1 impar}

Cos(54)° + i Sin(54)° = i^(3/5) P_{7 impar}

Cos(126)° + i Sin(126)° = i^(7/5) P_{3 impar}

Cos(162)° + i Sin(162)° = i^(9/5) P_{9 impar}

Cos(198)° + i Sin(198)° = i^(-9/5) P_{1 par}

Cos(234)° + i Sin(234)° = i^(-7/5) P_{7 par}

Cos(306)° + i Sin(306)° = i^(-3/5) P_{3 par}

Cos(342)° + i Sin(342)° = i^(-1/5) P_{9 par}

i^(7/5) = Cos[ 2((673))^(-1)(10^((673)-1) -1) ]° + i Sin[ 2((577))^(-1)(10^((577)-1) -1) ]°

https://www.wolframalpha.com/input/?i=i%5E%287%2F5%29+%3D++Cos%5B+2%28%28673%29%29%5E%28-1%29%2810%5E%28%28673%29-1%29+-1%29+%5D%C2%B0+%2B+i+Sin%5B+2%28%28577%29%29%5E%28-1%29%2810%5E%28%28577%29-1%29+-1%29+%5D%C2%B0

i^(1/5) = Cos[ 2((971))^(-1)(10^((971)-1) -1) ]° + i Sin[ 2((859))^(-1)(10^((859)-1) -1) ]°

https://www.wolframalpha.com/input/?i=i%5E%281%2F5%29+%3D++Cos%5B+2%28%28971%29%29%5E%28-1%29%2810%5E%28%28971%29-1%29+-1%29+%5D%C2%B0+%2B+i+Sin%5B+2%28%28859%29%29%5E%28-1%29%2810%5E%28%28859%29-1%29+-1%29+%5D%C2%B0

-i^(7/5) = Cos[ 2((137))^(-1)(10^((137)-1) -1) ]° + i Sin[ 2((43))^(-1)(10^((43)-1) -1) ]°

https://www.wolframalpha.com/input/?i=-i%5E%287%2F5%29+%3D++Cos%5B+2%28%28137%29%29%5E%28-1%29%2810%5E%28%28137%29-1%29+-1%29+%5D%C2%B0+%2B+i+Sin%5B+2%28%2843%29%29%5E%28-1%29%2810%5E%28%2843%29-1%29+-1%29+%5D%C2%B0

Cos[ 2((311))^(-1)(10^((311)-1) -1) ]° + i Sin[ 2((271))^(-1)(10^((271)-1) -1) ]° = i^(1/5)

https://www.wolframalpha.com/input/?i=Cos%5B+2%28%28311%29%29%5E%28-1%29%2810%5E%28%28311%29-1%29+-1%29+%5D%C2%B0+%2B+i+Sin%5B+2%28%28271%29%29%5E%28-1%29%2810%5E%28%28271%29-1%29+-1%29+%5D%C2%B0+%3D+i%5E%281%2F5%29+

Cos[ 2((53))^(-1)(10^((53)-1) -1) ]° + i Sin[ 2((233))^(-1)(10^((233)-1) -1) ]° = i^(7/5)

https://www.wolframalpha.com/input/?i=Cos%5B+2%28%2853%29%29%5E%28-1%29%2810%5E%28%2853%29-1%29+-1%29+%5D%C2%B0+%2B+i+Sin%5B+2%28%28233%29%29%5E%28-1%29%2810%5E%28%28233%29-1%29+-1%29+%5D%C2%B0+%3D+i%5E%287%2F5%29+

i^(-9/5) = Cos[ 2((59))^(-1)(10^((59)-1) -1) ]° + i Sin[ 2((29))^(-1)(10^((29)-1) -1) ]°

https://www.wolframalpha.com/input/?i=i%5E%28-9%2F5%29+%3D++Cos%5B+2%28%2859%29%29%5E%28-1%29%2810%5E%28%2859%29-1%29+-1%29+%5D%C2%B0+%2B+i+Sin%5B+2%28%2829%29%29%5E%28-1%29%2810%5E%28%2829%29-1%29+-1%29+%5D%C2%B0

i^(3/5) = Cos[ 2((977))^(-1)(10^((977)-1) -1) ]° + i Sin[ 2((613))^(-1)(10^((613)-1) -1) ]°

https://www.wolframalpha.com/input/?i=i%5E%283%2F5%29+%3D++Cos%5B+2%28%28977%29%29%5E%28-1%29%2810%5E%28%28977%29-1%29+-1%29+%5D%C2%B0+%2B+i+Sin%5B+2%28%28613%29%29%5E%28-1%29%2810%5E%28%28613%29-1%29+-1%29+%5D%C2%B0

TEOREMA 2: TEOREMA DE MATRIZ DE ROTACIÓN ÁUREA PRIMA

Existen matrices de rotación de simetría áurea prima, tal que su determinante es siempre 1. Esto solo se cumple para primos de la misma familia y para las potencias enteras de 2 y 3 en los cosenos.

TEOREMA DE GRISALES

Las clases áureas primas son invariantes bajo isometrías locales.

PRIMOS PENULTI-IMPARES:

det [ { Cos[ 2 3^(2(67)) ]°, -Sin[ (2((911))^(-1)(10^((911)-1) -1)) ] }, { Sin[ (2((311))^(-1)(10^((311)-1) -1)) ] , Cos[ 2 3^(2(67)) ]° } ]

https://www.wolframalpha.com/input/?i=det+%5B+%7B+Cos%5B+2+3%5E%282%2867%29%29+%5D%C2%B0%2C+-Sin%5B+%282%28%28911%29%29%5E%28-1%29%2810%5E%28%28911%29-1%29+-1%29%29+%5D+%7D%2C++%7B+Sin%5B+%282%28%28311%29%29%5E%28-1%29%2810%5E%28%28311%29-1%29+-1%29%29+%5D+%2C+Cos%5B+2+3%5E%282%2867%29%29+%5D%C2%B0+%7D+%5D

det [ { Cos[ 2 3^(2(67)+1) ]°, -Sin[ (2((113))^(-1)(10^((113)-1) -1)) ] }, { Sin[ (2((733))^(-1)(10^((733)-1) -1)) ] , Cos[ 2 3^(2(767)+1) ]° } ]

https://www.wolframalpha.com/input/?i=det+%5B+%7B+Cos%5B+2+3%5E%282%2867%29%2B1%29+%5D%C2%B0%2C+-Sin%5B+%282%28%28113%29%29%5E%28-1%29%2810%5E%28%28113%29-1%29+-1%29%29+%5D+%7D%2C++%7B+Sin%5B+%282%28%28733%29%29%5E%28-1%29%2810%5E%28%28733%29-1%29+-1%29%29+%5D+%2C+Cos%5B+2+3%5E%282%28767%29%2B1%29+%5D%C2%B0+%7D+%5D

det [ { Cos[ 2 3^(2(83)+1) ]°, -Sin[ (2((137))^(-1)(10^((137)-1) -1)) ] }, { Sin[ (2((997))^(-1)(10^((997)-1) -1)) ] , Cos[ 2 3^(2(87)+1) ]° } ]

https://www.wolframalpha.com/input/?i=det+%5B+%7B+Cos%5B+2+3%5E%282%2883%29%2B1%29+%5D%C2%B0%2C+-Sin%5B+%282%28%28137%29%29%5E%28-1%29%2810%5E%28%28137%29-1%29+-1%29%29+%5D+%7D%2C++%7B+Sin%5B+%282%28%28997%29%29%5E%28-1%29%2810%5E%28%28997%29-1%29+-1%29%29+%5D+%2C+Cos%5B+2+3%5E%282%2887%29%2B1%29+%5D%C2%B0+%7D+%5D

det [ { Cos[ 2 3^(2(186)) ]°, -Sin[ (2((599))^(-1)(10^((599)-1) -1)) ] }, { Sin[ (2((499))^(-1)(10^((499)-1) -1)) ] , Cos[ 2 3^(2(186)) ]° } ]

https://www.wolframalpha.com/input/?i=det+%5B+%7B+Cos%5B+2+3%5E%282%28186%29%29+%5D%C2%B0%2C+-Sin%5B+%282%28%28599%29%29%5E%28-1%29%2810%5E%28%28599%29-1%29+-1%29%29+%5D+%7D%2C++%7B+Sin%5B+%282%28%28499%29%29%5E%28-1%29%2810%5E%28%28499%29-1%29+-1%29%29+%5D+%2C+Cos%5B+2+3%5E%282%28186%29%29+%5D%C2%B0+%7D+%5D

PRIMOS PENULTI-PARES

det [ { Cos[ 2 3^(2(186)) ]°, -Sin[ (2((941))^(-1)(10^((941)-1) -1)) ] }, { Sin[ (2((541))^(-1)(10^((541)-1) -1)) ] , Cos[ 2 3^(2(186)) ]° } ]

https://www.wolframalpha.com/input/?i=det+of+%5B+%7B+Cos%5B+2+3%5E%282%28186%29%29+%5D%C2%B0%2C+-Sin%5B+%282%28%28941%29%29%5E%28-1%29%2810%5E%28%28941%29-1%29+-1%29%29+%5D+%7D%2C++%7B+Sin%5B+%282%28%28541%29%29%5E%28-1%29%2810%5E%28%28541%29-1%29+-1%29%29+%5D+%2C+Cos%5B+2+3%5E%282%28186%29%29+%5D%C2%B0+%7D+%5D

det [ { Cos[ 2 3^(2(186)+1) ]°, -Sin[ (2((443))^(-1)(10^((443)-1) -1)) ] }, { Sin[ (2((883))^(-1)(10^((883)-1) -1)) ] , Cos[ 2 3^(2(186)+1) ]° } ]

https://www.wolframalpha.com/input/?i=det+%5B+%7B+Cos%5B+2+3%5E%282%28186%29%2B1%29+%5D%C2%B0%2C+-Sin%5B+%282%28%28443%29%29%5E%28-1%29%2810%5E%28%28443%29-1%29+-1%29%29+%5D+%7D%2C++%7B+Sin%5B+%282%28%28883%29%29%5E%28-1%29%2810%5E%28%28883%29-1%29+-1%29%29+%5D+%2C+Cos%5B+2+3%5E%282%28186%29%2B1%29+%5D%C2%B0+%7D+%5D

det [ { Cos[ 2 3^(2(176)+1) ]°, -Sin[ (2((727))^(-1)(10^((727)-1) -1)) ] }, { Sin[ (2((967))^(-1)(10^((967)-1) -1)) ] , Cos[ 2 3^(2(186)+1) ]° } ]

https://www.wolframalpha.com/input/?i=det+%5B+%7B+Cos%5B+2+3%5E%282%28176%29%2B1%29+%5D%C2%B0%2C+-Sin%5B+%282%28%28727%29%29%5E%28-1%29%2810%5E%28%28727%29-1%29+-1%29%29+%5D+%7D%2C++%7B+Sin%5B+%282%28%28967%29%29%5E%28-1%29%2810%5E%28%28967%29-1%29+-1%29%29+%5D+%2C+Cos%5B+2+3%5E%282%28186%29%2B1%29+%5D%C2%B0+%7D+%5D

det [ { Cos[ 2 3^(2(346)) ]°, -Sin[ (2((409))^(-1)(10^((409)-1) -1)) ] }, { Sin[ (2((929))^(-1)(10^((929)-1) -1)) ] , Cos[ 2 3^(2(8714)) ]° } ]

https://www.wolframalpha.com/input/?i=det+%5B+%7B+Cos%5B+2+3%5E%282%28346%29%29+%5D%C2%B0%2C+-Sin%5B+%282%28%28409%29%29%5E%28-1%29%2810%5E%28%28409%29-1%29+-1%29%29+%5D+%7D%2C++%7B+Sin%5B+%282%28%28929%29%29%5E%28-1%29%2810%5E%28%28929%29-1%29+-1%29%29+%5D+%2C+Cos%5B+2+3%5E%282%288714%29%29+%5D%C2%B0+%7D+%5D

MATRIZ DE ROTACIÓN EN R3

[ { 1, 0, 0 }, { 0, Cos[ 2 3^(2(78)) ]°, -Sin[ (2((311))^(-1)(10^((311)-1) -1)) ] }, { 0, Sin[ (2((131))^(-1)(10^((131)-1) -1)) ] , Cos[ 2 3^(2(788))]° } ]

https://www.wolframalpha.com/input/?i=%5B+%7B+1%2C+0%2C+0+%7D%2C++%7B+0%2C+Cos%5B+2+3%5E%282%2878%29%29+%5D%C2%B0%2C+-Sin%5B+%282%28%28311%29%29%5E%28-1%29%2810%5E%28%28311%29-1%29+-1%29%29+%5D+%7D%2C+%7B+0%2C+Sin%5B+%282%28%28131%29%29%5E%28-1%29%2810%5E%28%28131%29-1%29+-1%29%29+%5D+%2C+Cos%5B+2+3%5E%282%28788%29%29%5D%C2%B0+%7D+%5D

[ { 1, 0, 0 }, { 0, Cos[ 2 3^(2(246)+1) ]°, -Sin[ (2((613))^(-1)(10^((613)-1) -1)) ] }, { 0, Sin[ (2((593))^(-1)(10^((593)-1) -1)) ] , Cos[ 2 3^(2(666)+1)]° } ]

https://www.wolframalpha.com/input/?i=+%5B+%7B+1%2C+0%2C+0+%7D%2C++%7B+0%2C+Cos%5B+2+3%5E%282%28246%29%2B1%29+%5D%C2%B0%2C+-Sin%5B+%282%28%28613%29%29%5E%28-1%29%2810%5E%28%28613%29-1%29+-1%29%29+%5D+%7D%2C+%7B+0%2C+Sin%5B+%282%28%28593%29%29%5E%28-1%29%2810%5E%28%28593%29-1%29+-1%29%29+%5D+%2C+Cos%5B+2+3%5E%282%28666%29%2B1%29%5D%C2%B0+%7D+%5D

[ { 1, 0, 0 }, { 0, Cos[ 2 3^(2(4678)+1) ]°, -Sin[ (2((577))^(-1)(10^((577)-1) -1)) ] }, { 0, Sin[ (2((677))^(-1)(10^((677)-1) -1)) ] , Cos[ 2 3^(2(6678)+1)]° } ]

https://www.wolframalpha.com/input/?i=%5B+%7B+1%2C+0%2C+0+%7D%2C++%7B+0%2C+Cos%5B+2+3%5E%282%284678%29%2B1%29+%5D%C2%B0%2C+-Sin%5B+%282%28%28577%29%29%5E%28-1%29%2810%5E%28%28577%29-1%29+-1%29%29+%5D+%7D%2C+%7B+0%2C+Sin%5B+%282%28%28677%29%29%5E%28-1%29%2810%5E%28%28677%29-1%29+-1%29%29+%5D+%2C+Cos%5B+2+3%5E%282%286678%29%2B1%29%5D%C2%B0+%7D+%5D

[ { 1, 0, 0 }, { 0, Cos[ (2) 3^(2(508)) ]°, -Sin[ (2((769))^(-1)(10^((769)-1) -1)) ] }, { 0, Sin[ (2((269))^(-1)(10^((269)-1) -1)) ] , Cos[ (2) 3^(2(668))]° } ]

https://www.wolframalpha.com/input/?i=%5B+%7B+1%2C+0%2C+0+%7D%2C++%7B+0%2C+Cos%5B+%282%29+3%5E%282%28508%29%29+%5D%C2%B0%2C+-Sin%5B+%282%28%28769%29%29%5E%28-1%29%2810%5E%28%28769%29-1%29+-1%29%29+%5D+%7D%2C+%7B+0%2C+Sin%5B+%282%28%28269%29%29%5E%28-1%29%2810%5E%28%28269%29-1%29+-1%29%29+%5D+%2C+Cos%5B+%282%29+3%5E%282%28668%29%29%5D%C2%B0+%7D+%5D

MATRICES DE RAÍCES IMAGINARIAS CON RADIO PHI

determinant of [ { i^(7/5), -i^(9/5) } , { i^(9/5) , i^(7/5) } ]

https://www.wolframalpha.com/input/?i=determinant+of+%5B+%7B+i%5E%287%2F5%29%2C+-i%5E%289%2F5%29+%7D+%2C++%7B+i%5E%289%2F5%29+%2C+i%5E%287%2F5%29+%7D+%5D

https://www.wolframalpha.com/input/?i=determinant+of+%5B+%7B+i%5E%281%2F5%29%2C+-i%5E%283%2F5%29+%7D+%2C++%7B+i%5E%283%2F5%29+%2C+i%5E%281%2F5%29+%7D+%5D

https://www.wolframalpha.com/input/?i=determinant+of+%5B+%7B+i%5E%289%2F5%29%2C+-i%5E%281%2F5%29+%7D+%2C++%7B+i%5E%281%2F5%29+%2C+i%5E%289%2F5%29+%7D+%5D

https://www.wolframalpha.com/input/?i=determinant+of+%5B+%7B+i%5E%287%2F5%29%2C+-i%5E%281%2F5%29+%7D+%2C++%7B+i%5E%281%2F5%29+%2C+i%5E%287%2F5%29+%7D+%5D

https://www.wolframalpha.com/input/?i=determinant+of+%5B+%7B+i%5E%287%2F5%29%2C+-i%5E%283%2F5%29+%7D+%2C++%7B+i%5E%283%2F5%29+%2C+i%5E%287%2F5%29+%7D+%5D

https://www.wolframalpha.com/input/?i=eigenvalue+%5B+%7B+i%5E%287%2F5%29%2C+-i%5E%283%2F5%29+%7D+%2C++%7B+i%5E%283%2F5%29+%2C+i%5E%287%2F5%29+%7D+%5D

TEOREMA 3: TEOREMA DE LAS RAÍCES COMPLEJAS DE FIBONACCI

De la misma manera que existen 8 clases residuales isométricas para los numeros primos, tambien existen 4 clases residuales para los 12th numeros de Fibonacci. En este punto, cabe resaltar que las restantes 4 isometrías no corresponden a ningun numero de Fibonacci ni primo.

https://www.wolframalpha.com/input/?i=%E2%88%91+n%3D0+to+4+x%5E%28n%29

https://www.wolframalpha.com/input/?i=1+%2B+x+%2B+x%5E2+%2B+x%5E3+%2B+x%5E4+%3D+0

Dichas rotaciones se ordenan de la siguiente manera:

F_12{1,6} ≡ 144° (mód 360)

F_12{2,7} ≡ 288° (mód 360)

F_12{3,8} ≡ 72° (mód 360)

F_12{4,9} ≡ 216° (mód 360)

Son las clases residuales módulo 360 de los 12th números de Fibonacci. Y dichas clases son las isometrías del pentágono regular. El coseno de cada 12th Fibonacci es igual a –φ/2 y φ^-1/2

La notación F_12{1,6},significa que son los 12th Fibonacci cuya última cifra es 1 o 6. Y así para los demás.

Table [ ( Fib(12 n) ) mod360, { n, 1, 100, 1 } ]

https://www.wolframalpha.com/input/?i=Table+%5B+%28+Fib%2812+n%29+%29+mod360%2C+%7B+n%2C+1%2C+100%2C+1+%7D+%5D

Veamos acá y acá en Wolfram:

[ Fib(12 43) ] mód360 = 72

https://www.wolframalpha.com/input/?i=%5B+Fib%2812+43%29+%5D+mod360+%3D+72

[ Fib(12 541) ] mod360 = 144

https://www.wolframalpha.com/input/?i=%5B+Fib%2812+541%29+%5D+mod360+%3D+144

[ Fib(12 599) ] mod360 = 216

https://www.wolframalpha.com/input/?i=%5B+Fib%2812+599%29+%5D+mod360+%3D+216

[ Fib(12 47) ] mód360 = 288

https://www.wolframalpha.com/input/?i=%5B+Fib%2812+47%29+%5D+mod360+%3D+288

Table [ ( Fib(12 n) ) mod360 = 72 , { n, 3, 200, 5 } ]

https://www.wolframalpha.com/input/?i=Table+%5B+%28+Fib%2812+n%29+%29+mod360+%3D+72+%2C+%7B+n%2C+3%2C+200%2C+5+%7D+%5D

Table [ ( Fib(12 n) ) mod360 = 144 , { n, 1, 200, 5 } ]

https://www.wolframalpha.com/input/?i=Table+%5B+%28+Fib%2812+n%29+%29+mod360+%3D+144+%2C+%7B+n%2C+1%2C+200%2C+5+%7D+%5D

Table [ ( Fib(12 n) ) mod360 = 216 , { n, 4, 200, 5 } ]

https://www.wolframalpha.com/input/?i=Table+%5B+%28+Fib%2812+n%29+%29+mod360+%3D+216+%2C+%7B+n%2C+4%2C+200%2C+5+%7D+%5D

Table [ ( Fib(12 n) ) mod360 = 288 , { n, 2, 200, 5 } ]

https://www.wolframalpha.com/input/?i=Table+%5B+%28+Fib%2812+n%29+%29+mod360+%3D+288+%2C+%7B+n%2C+2%2C+200%2C+5+%7D+%5D

POLINOMIO MINIMAL DE 12TH FIBONACCI: 72, 144, 216, 288

1 + x + x^2 + x^3 + x^4 = 0

https://www.wolframalpha.com/input/?i=1+%2B+x+%2B+x%5E2+%2B+x%5E3+%2B+x%5E4+%3D+0

∑ n=0 to 4 x^n

https://www.wolframalpha.com/input/?i=%E2%88%91+n%3D0+to+4+x%5En

POLINOMIO MINIMAL DE: 36, 108, 252, 324

1 - x + x^2 - x^3 + x^4 = 0

https://www.wolframalpha.com/input/?i=1+-+x+%2B+x%5E2+-+x%5E3+%2B+x%5E4+%3D+0

∑ n=0 to 4 (-1)^n x^n

https://www.wolframalpha.com/input/?i=%E2%88%91+n%3D0+to+4++%28-1%29%5En+x%5En

LAS 8 RAÍCES COMPLEJAS DE FIBONACCI

https://www.wolframalpha.com/input/?i=1+-+x%5E2+%2B+x%5E4+-+x%5E6+%2B+x%5E8+-+x%5E10+%2B+x%5E12+-+x%5E14+%2B+x%5E16+-+x%5E18+%3D+0

https://www.wolframalpha.com/input/?i=1+-+x%5E2+%2B+x%5E4+-+x%5E6+%2B+x%5E8+-+x%5E10+%2B+x%5E12+-+x%5E14+%2B+x%5E16+-+x%5E18+%2B+x%5E20+-+x%5E22+%2B+x%5E24+-+x%5E26+%2B+x%5E28+%3D+0+

ISOMETRÍAS DE 12TH FIBONACCI:

72° ↔ 2π/5

144° ↔ 4π/5

216° ↔ 6π/5 ↔ -144° ↔ -4π/10

288° ↔ 8π/5 ↔ -72° ↔ -2π/10

36° ↔ π/5

108° ↔ 3π/5

252° ↔ 7π/5 ↔ -108° ↔ -3π/5

324° ↔ 9π/5 ↔ -36° ↔ -π/5

i^2/5 = Cos36 + i Sin36

i^4/5 = Cos72 + i Sin72

i^6/5 = Cos108 + i Sin108

i^8/5 = Cos144 + i Sin144

i^-2/5 = - i^8/5 = Cos324 + i Sin324

i^-4/5 = - i^6/5 = Cos288 + i Sin288

i^-6/5 = - i^4/5 = Cos252 + i Sin252

i^-8/5 = - i^2/5 = Cos216 + i Sin216

i^(4/5) = Cos(72) + iSin (72)

https://www.wolframalpha.com/input/?i=i%5E%284%2F5%29+%3D++Cos%2872%29+%2B+iSin+%2872%29

i^(8/5) = Cos(144) + iSin(144)

https://www.wolframalpha.com/input/?i=+i%5E%288%2F5%29+%3D+Cos%28144%29+%2B+iSin%28144%29

- i^(6/5) = Cos(288) + iSin (288)

https://www.wolframalpha.com/input/?i=-+i%5E%286%2F5%29+%3D+Cos%28288%29+%2B+iSin+%28288%29

- i^(2/5) = Cos(216) + iSin (216)

https://www.wolframalpha.com/input/?i=-+i%5E%282%2F5%29+%3D+Cos%28216%29+%2B+iSin+%28216%29+

TEOREMA DE RAÍCES QUINTAS DE FIBONACCI

Existen 4 ángulos áureos que están a 180 grados de las isometrías de Fibonacci. Estos ángulos son: 36°, 108°, 252°, 324°. Y corresponden a los vértices de un pentágono rotado 180° en sentido horario con respecto al pentágono de los Fibonacci.

216° - 180° = 36°

288° - 180° = 108°

72° + 180° = 252°

144° + 180° = 324°

F_12k{1,6} ≡ 144° (mód 360)

F_12k{2,7} ≡ 288° (mód 360)

F_12k{3,8} ≡ 72° (mód 360)

F_12k{4,9} ≡ 216° (mód 360)

Cos(36)° + i Sin(36)° = i^(2/5) F_12k(6)-pi

Cos(72)° + i Sin(72)° = i^(4/5) F_12k(2)

Cos(108)° + i Sin(108)° = i^(6/5) F_12k(8)-pi

Cos(144)° + i Sin(144)° = i^(8/5) F_12k(4)

Cos(216)° + i Sin(216)° = i^(-8/5) F_12k(6)

Cos(252)° + i Sin(252)° = i^(-6/5) F_12k(2)-pi

Cos(288)° + i Sin(288)° = i^(-4/5) F_12k(8)

Cos(324° + i Sin(324)° = i^(-1/5) F_12k(4)-pi

Ángulos Fibonacci

i^(-4/5) = Cos [ Fib (12 43) ]° + i Sin [ Fib (12 48) - 180 ]°

https://www.wolframalpha.com/input/?i=+i%5E%28-4%2F5%29+%3D+Cos+%5B+Fib+%2812+43%29++%5D%C2%B0+%2B+i+Sin+%5B+Fib+%2812+48%29+-+180+%5D%C2%B0+

i^(6/5) = Cos [ Fib (12 43) -180 ]° + i Sin [ Fib (12 48) ]°

https://www.wolframalpha.com/input/?i=+i%5E%286%2F5%29+%3D+Cos+%5B+Fib+%2812+43%29+-180++%5D%C2%B0+%2B+i+Sin+%5B+Fib+%2812+48%29++%5D%C2%B0+

i^(-8/5) = Cos [ Fib (12 44) ]° + i Sin [ Fib (12 49) ]°

https://www.wolframalpha.com/input/?i=+i%5E%28-8%2F5%29+%3D+Cos+%5B+Fib+%2812+44%29+%5D%C2%B0+%2B+i+Sin+%5B+Fib+%2812+49%29++%5D%C2%B0+

i^(-2/5) = Cos [ Fib (12 49) -180 ]° + i Sin [ Fib (12 46) -180 ]°

https://www.wolframalpha.com/input/?i=+i%5E%28-2%2F5%29+%3D+Cos+%5B+Fib+%2812+49%29+-180+%5D%C2%B0+%2B+i+Sin+%5B+Fib+%2812+46%29+-180+%5D%C2%B0+

i^(2/5) = Cos(36) + iSin (36)

https://www.wolframalpha.com/input/?i=i%5E%282%2F5%29+%3D++Cos%2836%29+%2B+iSin+%2836%29

i^(6/5) = Cos(108) + iSin (108)

https://www.wolframalpha.com/input/?i=+i%5E%286%2F5%29+%3D+Cos%28108%29+%2B+iSin+%28108%29

i^(-6/5) = Cos (252) + i Sin (252)

https://www.wolframalpha.com/input/?i=+i%5E%28-6%2F5%29+%3D+Cos%28252%29+%2B+iSin+%28252%29

i^(-2/5) = Cos (324) + i Sin (324)

https://www.wolframalpha.com/input/?i=+i%5E%28-2%2F5%29+%3D+Cos+%28324%29+%2B+i+Sin+%28324%29

i^(2/5) = cos36 + i sin144

https://www.wolframalpha.com/input/?i=+i%5E%282%2F5%29+%3D+cos36+%2B+i+sin144

i^(-2/5) = cos36 + i sin216

https://www.wolframalpha.com/input/?i=+i%5E%28-2%2F5%29+%3D+cos36+%2B+i+sin216

i^(-2/5) = cos36 + i sin324

https://www.wolframalpha.com/input/?i=+i%5E%28-2%2F5%29+%3D+cos36+%2B+i+sin324

i^(6/5) = cos108 + i sin72

https://www.wolframalpha.com/input/?i=+i%5E%286%2F5%29+%3D+cos108+%2B+i+sin72

Ángulo 36

i^(2/5) = Cos [ Fib (12 44) - 180 ]° + i Sin [ Fib (12 49) - 180 ]°

https://www.wolframalpha.com/input/?i=i%5E%282%2F5%29+%3D+Cos+%5B+Fib+%2812+44%29+-+180+%5D%C2%B0+%2B+i+Sin+%5B+Fib+%2812+49%29+-+180+%5D%C2%B0

Ángulo 108

i^(6/5) = Cos [ Fib (12 42) - 180 ]° + i Sin [ Fib (12 47) - 180 ]°

https://www.wolframalpha.com/input/?i=i%5E%286%2F5%29+%3D+Cos+%5B+Fib+%2812+42%29+-+180+%5D%C2%B0+%2B+i+Sin+%5B+Fib+%2812+47%29+-+180+%5D%C2%B0

Ángulo 252

i^(-6/5) = Cos [ Fib (12 43) - 180 ]° + i Sin [ Fib (12 48) - 180 ]°

https://www.wolframalpha.com/input/?i=i%5E%28-6%2F5%29+%3D+Cos+%5B+Fib+%2812+43%29+-+180+%5D%C2%B0+%2B+i+Sin+%5B+Fib+%2812+48%29+-+180+%5D%C2%B0

Ángulo 324

i^(-2/5) = Cos [ Fib (12 41) - 180 ]° + i Sin [ Fib (12 46) - 180 ]°

https://www.wolframalpha.com/input/?i=i%5E%28-2%2F5%29+%3D+Cos+%5B+Fib+%2812+41%29+-+180+%5D%C2%B0+%2B+i+Sin+%5B+Fib+%2812+46%29+-+180+%5D%C2%B0

RAÍCES COMPLEJAS ÁUREAS DE 12TH FIBONACCI

- i^(2/5) = Cos [ Fib (12 44) ]° + i Sin [ Fib (12 44) ]°

https://www.wolframalpha.com/input/?i=-+i%5E%282%2F5%29+%3D+Cos+%5B+Fib+%2812+44%29+%5D%C2%B0+%2B+i+Sin+%5B+Fib+%2812+44%29+%5D%C2%B0+

i^(-8/5) = Cos [ Fib (12 4554) ]° + i Sin [ Fib (12 4474) ]°

https://www.wolframalpha.com/input/?i=i%5E%28-8%2F5%29+%3D+Cos+%5B+Fib+%2812+4554%29+%5D%C2%B0+%2B+i+Sin+%5B+Fib+%2812+4474%29+%5D%C2%B0+

i^(-8/5) = Cos [ Fib (12 459) ]° + i Sin [ Fib (12 479) ]°

https://www.wolframalpha.com/input/?i=i%5E%28-8%2F5%29+%3D+Cos+%5B+Fib+%2812+459%29+%5D%C2%B0+%2B+i+Sin+%5B+Fib+%2812+479%29+%5D%C2%B0+

i^(4/5) = Cos [ Fib (12 13) ]° + i Sin [ Fib (12 23) ]°

https://www.wolframalpha.com/input/?i=i%5E%284%2F5%29+%3D+Cos+%5B+Fib+%2812+13%29+%5D%C2%B0+%2B+i+Sin+%5B+Fib+%2812+23%29+%5D%C2%B0+

i^(8/5) = Cos [ Fib (12 101) ]° + i Sin [ Fib (12 551) ]°

https://www.wolframalpha.com/input/?i=i%5E%288%2F5%29+%3D+Cos+%5B+Fib+%2812+101%29+%5D%C2%B0+%2B+i+Sin+%5B+Fib+%2812+551%29+%5D%C2%B0+

-i^(6/5) = Cos [ Fib (12 532) ]° + i Sin [ Fib (12 22) ]°

https://www.wolframalpha.com/input/?i=-i%5E%286%2F5%29+%3D+Cos+%5B+Fib+%2812+532%29+%5D%C2%B0+%2B+i+Sin+%5B+Fib+%2812+22%29+%5D%C2%B0+

i^(-4/5) = Cos [ Fib (12 532) ]° + i Sin [ Fib (12 32) ]°

https://www.wolframalpha.com/input/?i=i%5E%28-4%2F5%29+%3D+Cos+%5B+Fib+%2812+532%29+%5D%C2%B0+%2B+i+Sin+%5B+Fib+%2812+32%29+%5D%C2%B0+

Table [ ( 2^(2n-1) 3^(2n+1) ) mod 360 = 144 , { n, 2, 100, 1 } ]

https://www.wolframalpha.com/input/?i=Table+%5B+%28+2%5E%282n-1%29+3%5E%282n%2B1%29+%29+mod+360+%3D+144+%2C+%7B+n%2C+2%2C+100%2C+1+%7D+%5D

MATRIZ DE ROTACIÓN DE FIBONACCI 12TH

El determinante de las matrices de 12th Fibonacci es 1.

det [ { Cos [ Fib (12 31) ]°, -Sin [ Fib (12 31)] }, { Sin [ Fib (12 76) ] , Cos [ Fib (12 76) ] } ]

https://www.wolframalpha.com/input/?i=det+%5B+%7B+Cos+%5B+Fib+%2812+31%29+%5D%C2%B0%2C+-Sin+%5B+Fib+%2812+31%29%5D+%7D%2C++%7B+Sin+%5B+Fib+%2812+76%29+%5D+%2C+Cos+%5B+Fib+%2812+76%29+%5D+%7D+%5D+

det [ { Cos [ Fib (12 32) ]°, -Sin [ Fib (12 37)] }, { Sin [ Fib (12 72) ] , Cos [ Fib (12 77) ] } ]

https://www.wolframalpha.com/input/?i=det+%5B+%7B+Cos+%5B+Fib+%2812+32%29+%5D%C2%B0%2C+-Sin+%5B+Fib+%2812+37%29%5D+%7D%2C++%7B+Sin+%5B+Fib+%2812+72%29+%5D+%2C+Cos+%5B+Fib+%2812+77%29+%5D+%7D+%5D+

det [ { Cos [ Fib (12 43) ]°, -Sin [ Fib (12 18)] }, { Sin [ Fib (12 733) ] , Cos [ Fib (12 118) ] } ]

https://www.wolframalpha.com/input/?i=det+%5B+%7B+Cos+%5B+Fib+%2812+43%29+%5D%C2%B0%2C+-Sin+%5B+Fib+%2812+18%29%5D+%7D%2C++%7B+Sin+%5B+Fib+%2812+733%29+%5D+%2C+Cos+%5B+Fib+%2812+118%29+%5D+%7D+%5D+

det [ { Cos [ Fib (12 94) ]°, -Sin [ Fib (12 19)] }, { Sin [ Fib (12 74) ] , Cos [ Fib (12 29) ] } ]

https://www.wolframalpha.com/input/?i=det+%5B+%7B+Cos+%5B+Fib+%2812+94%29+%5D%C2%B0%2C+-Sin+%5B+Fib+%2812+19%29%5D+%7D%2C++%7B+Sin+%5B+Fib+%2812+74%29+%5D+%2C+Cos+%5B+Fib+%2812+29%29+%5D+%7D+%5D+

LA TRAZA ES EL NÚMERO ÁUREO Y SU INVERSO MULTIPLICATIVO:

[ { Cos [ Fib (12 91) ]°, -Sin [ Fib (12 11)] }, { Sin [ Fib (12 76) ] , Cos [ Fib (12 26) ] } ]

https://www.wolframalpha.com/input/?i=%5B+%7B+Cos+%5B+Fib+%2812+91%29+%5D%C2%B0%2C+-Sin+%5B+Fib+%2812+11%29%5D+%7D%2C++%7B+Sin+%5B+Fib+%2812+76%29+%5D+%2C+Cos+%5B+Fib+%2812+26%29+%5D+%7D+%5D+

[ { Cos [ Fib (12 94) ]°, -Sin [ Fib (12 19)] }, { Sin [ Fib (12 74) ] , Cos [ Fib (12 29) ] } ]

https://www.wolframalpha.com/input/?i=%5B+%7B+Cos+%5B+Fib+%2812+94%29+%5D%C2%B0%2C+-Sin+%5B+Fib+%2812+19%29%5D+%7D%2C++%7B+Sin+%5B+Fib+%2812+74%29+%5D+%2C+Cos+%5B+Fib+%2812+29%29+%5D+%7D+%5D+

[ { Cos [ Fib (12 93) ]°, -Sin [ Fib (12 13)] }, { Sin [ Fib (12 78) ] , Cos [ Fib (12 28) ] } ]

https://www.wolframalpha.com/input/?i=%5B+%7B+Cos+%5B+Fib+%2812+93%29+%5D%C2%B0%2C+-Sin+%5B+Fib+%2812+13%29%5D+%7D%2C++%7B+Sin+%5B+Fib+%2812+78%29+%5D+%2C+Cos+%5B+Fib+%2812+28%29+%5D+%7D+%5D+

[ { Cos [ Fib (12 92) ]°, -Sin [ Fib (12 922)] }, { Sin [ Fib (12 77) ] , Cos [ Fib (12 27) ] } ]

https://www.wolframalpha.com/input/?i=%5B+%7B+Cos+%5B+Fib+%2812+92%29+%5D%C2%B0%2C+-Sin+%5B+Fib+%2812+922%29%5D+%7D%2C++%7B+Sin+%5B+Fib+%2812+77%29+%5D+%2C+Cos+%5B+Fib+%2812+27%29+%5D+%7D+%5D+

MATRIZ DE ROTACIÓN DE ÁNGULOS NO ASOCIADOS:

det [ { Cos [ 36 ]°, -Sin [ 36 ] }, { Sin [ 36 ]° , Cos [ 36 ]° } ]

https://www.wolframalpha.com/input/?i=det+%5B+%7B+Cos+%5B+36+%5D%C2%B0%2C+-Sin+%5B+36+%5D+%7D%2C++%7B+Sin+%5B+36+%5D%C2%B0+%2C+Cos+%5B+36+%5D%C2%B0+%7D+%5D

det [ { Cos [ 108 ]°, -Sin [ 108 ] }, { Sin [ 108 ]° , Cos [ 108 ]° } ]

https://www.wolframalpha.com/input/?i=det+%5B+%7B+Cos+%5B+108+%5D%C2%B0%2C+-Sin+%5B+108+%5D+%7D%2C++%7B+Sin+%5B+108+%5D%C2%B0+%2C+Cos+%5B+108+%5D%C2%B0+%7D+%5D

det [ { Cos [ 252 ]°, -Sin [ 252 ] }, { Sin [ 252 ]° , Cos [ 252 ]° } ]

https://www.wolframalpha.com/input/?i=det+%5B+%7B+Cos+%5B+252+%5D%C2%B0%2C+-Sin+%5B+252+%5D+%7D%2C++%7B+Sin+%5B+252+%5D%C2%B0+%2C+Cos+%5B+252+%5D%C2%B0+%7D+%5D

det [ { Cos [ 324 ]°, -Sin [ 324 ] }, { Sin [ 324 ]° , Cos [ 324 ]° } ]

https://www.wolframalpha.com/input/?i=det+%5B+%7B+Cos+%5B+324+%5D%C2%B0%2C+-Sin+%5B+324+%5D+%7D%2C++%7B+Sin+%5B+324+%5D%C2%B0+%2C+Cos+%5B+324+%5D%C2%B0+%7D+%5D

TEOREMA DEL COCIENTE HIPERBÓLICO ÁUREO PRIMO

El cociente del seno hiperbólico entre el coseno hiperbólico de mi función áurea prima da como resultado 8 imágenes exponenciales áureas que son las distintas combinaciones de las 8 familias disjuntas de números primos.

Haciendo una tabla 8x8 para las imágenes, de tal manera, que las clases primas de la columna sean el numerador y las clases primas de las filas sean el denominador, se obtienen los siguientes resultados generales.

Las 8 imágenes son:

[ e^(φ) - e^(-φ) ] / [ e^(φ) + e^(-φ) ]

[ e^(-φ) - e^(φ) ] / [ e^(φ) + e^(-φ) ]

[ e^(-φ) - e^(φ) ] / [ e^(1/φ) + e^(-1/φ) ]

[ e^(φ) - e^(-φ) ] / [ e^(1/φ) + e^(-1/φ) ]

[ e^(1/φ) - e^(-1/φ) ] / [ e^(φ) + e^(-φ) ]

[ e^(-1/φ) - e^(1/φ) ] / [ e^(φ) + e^(-φ) ]

[ e^(1/φ) - e^(-1/φ) ] / [ e^(1/φ) + e^(-1/φ) ]

[ e^(-1/φ) - e^(1/φ) ] / [ e^(1/φ) + e^(-1/φ) ]

	P_{1 impar}	P_3impar	P_7impar	P_{9 impar}	P_{1 par}	P_{3 par}	P_{7 par}	P_{9 par}
P_{1 impar}	[ e^(1/φ) - e^(-1/φ) ] / [ e^(1/φ) + e^(-1/φ) ]	[e^(1/φ) - e^(-1/φ) ] / [e^(φ) + e^(-φ)]	[e^(1/φ) - e^(-1/φ) ] / [e^(φ) + e^(-φ)]	[ e^(1/φ) - e^(-1/φ) ] / [ e^(1/φ) + e^(-1/φ) ]	[ e^(1/φ) - e^(-1/φ) ] / [ e^(1/φ) + e^(-1/φ) ]	[e^(1/φ) - e^(-1/φ) ] / [e^(φ) + e^(-φ)]	[e^(1/φ) - e^(-1/φ) ] / [e^(φ) + e^(-φ)]	[ e^(1/φ) - e^(-1/φ) ] / [ e^(1/φ) + e^(-1/φ) ]
P_{3 impar}	[ e^(φ) - e^(-φ) ] / [e^(-1/φ) + e^(1/φ)]	[ e^(φ) - e^(-φ) ] / [ e^(φ) + e^(-φ) ]	[ e^(φ) - e^(-φ) ] / [ e^(φ) + e^(-φ) ]	[ e^(φ) - e^(-φ) ] / [e^(-1/φ) + e^(1/φ)]	[ e^(φ) - e^(-φ) ] / [e^(-1/φ) + e^(1/φ)]	[ e^(φ) - e^(-φ) ] / [ e^(φ) + e^(-φ) ]	[ e^(φ) - e^(-φ) ] / [ e^(φ) + e^(-φ) ]	[ e^(φ) - e^(-φ) ] / [e^(-1/φ) + e^(1/φ)]
P_{7 impar}	[ e^(φ) - e^(-φ) ] / [e^(-1/φ) + e^(1/φ)]	[ e^(φ) - e^(-φ) ] / [ e^(φ) + e^(-φ) ]	[ e^(φ) - e^(-φ) ] / [ e^(φ) + e^(-φ) ]	[ e^(φ) - e^(-φ) ] / [e^(-1/φ) + e^(1/φ)]	[ e^(φ) - e^(-φ) ] / [e^(-1/φ) + e^(1/φ)]	[ e^(φ) - e^(-φ) ] / [ e^(φ) + e^(-φ) ]	[ e^(φ) - e^(-φ) ] / [ e^(φ) + e^(-φ) ]	[ e^(φ) - e^(-φ) ] / [e^(-1/φ) + e^(1/φ)]
P_{9 impar}	[ e^(1/φ) - e^(-1/φ) ] / [ e^(1/φ) + e^(-1/φ) ]	[e^(1/φ) - e^(-1/φ) ] / [e^(φ) + e^(-φ)]	[e^(1/φ) - e^(-1/φ) ] / [e^(φ) + e^(-φ)]	[ e^(1/φ) - e^(-1/φ) ] / [ e^(1/φ) + e^(-1/φ) ]	[ e^(1/φ) - e^(-1/φ) ] / [ e^(1/φ) + e^(-1/φ) ]	[e^(1/φ) - e^(-1/φ) ] / [e^(φ) + e^(-φ)]	[e^(1/φ) - e^(-1/φ) ] / [e^(φ) + e^(-φ)]	[ e^(1/φ) - e^(-1/φ) ] / [ e^(1/φ) + e^(-1/φ) ]
P_{1 par}	[ e^(-1/φ) - e^(1/φ) ] / [ e^(1/φ) + e^(-1/φ) ]	[ e^(-1/φ) - e^(1/φ) ] / [e^(φ) + e^(-φ)]	[ e^(-1/φ) - e^(1/φ) ] / [e^(φ) + e^(-φ)]	[ e^(-1/φ) - e^(1/φ) ] / [ e^(1/φ) + e^(-1/φ) ]	[ e^(-1/φ) - e^(1/φ) ] / [ e^(1/φ) + e^(-1/φ) ]	[ e^(-1/φ) - e^(1/φ) ] / [e^(φ) + e^(-φ)]	[ e^(-1/φ) - e^(1/φ) ] / [e^(φ) + e^(-φ)]	[ e^(-1/φ) - e^(1/φ) ] / [ e^(1/φ) + e^(-1/φ) ]
P_{3 par}	[ e^(-φ) - e^(φ) ] / [e^(-1/φ) + e^(1/φ)]	[ e^(-φ) - e^(φ) ] / [ e^(φ) + e^(-φ) ]	[ e^(-φ) - e^(φ) ] / [ e^(φ) + e^(-φ) ]	[ e^(-φ) - e^(φ) ] / [e^(-1/φ) + e^(1/φ)]	[ e^(-φ) - e^(φ) ] / [e^(-1/φ) + e^(1/φ)]	[ e^(-φ) - e^(φ) ] / [ e^(φ) + e^(-φ) ]	[ e^(-φ) - e^(φ) ] / [ e^(φ) + e^(-φ) ]	[ e^(-φ) - e^(φ) ] / [e^(-1/φ) + e^(1/φ)]
P_{7 par}	[ e^(-φ) - e^(φ) ] / [e^(-1/φ) + e^(1/φ)]	[ e^(-φ) - e^(φ) ] / [ e^(φ) + e^(-φ) ]	[ e^(-φ) - e^(φ) ] / [ e^(φ) + e^(-φ) ]	[ e^(-φ) - e^(φ) ] / [e^(-1/φ) + e^(1/φ)]	[ e^(-φ) - e^(φ) ] / [e^(-1/φ) + e^(1/φ)]	[ e^(-φ) - e^(φ) ] / [ e^(φ) + e^(-φ) ]	[ e^(-φ) - e^(φ) ] / [ e^(φ) + e^(-φ) ]	[ e^(-φ) - e^(φ) ] / [e^(-1/φ) + e^(1/φ)]
P_{9 par}	[ e^(-1/φ) - e^(1/φ) ] / [ e^(1/φ) + e^(-1/φ) ]	[ e^(-1/φ) - e^(1/φ) ] / [e^(φ) + e^(-φ)]	[ e^(-1/φ) - e^(1/φ) ] / [e^(φ) + e^(-φ)]	[ e^(-1/φ) - e^(1/φ) ] / [ e^(1/φ) + e^(-1/φ) ]	[ e^(-1/φ) - e^(1/φ) ] / [ e^(1/φ) + e^(-1/φ) ]	[ e^(-1/φ) - e^(1/φ) ] / [e^(φ) + e^(-φ)]	[ e^(-1/φ) - e^(1/φ) ] / [e^(φ) + e^(-φ)]	[ e^(-1/φ) - e^(1/φ) ] / [ e^(1/φ) + e^(-1/φ) ]

Sinh[ 2sin[ (2((11))^(-1)(10^((11)-1) -1)) ]° ] / Cosh[ 2sin[ (2((31))^(-1)(10^((31)-1) -1)) ]° ] = [ e^(1/φ) - e^(-1/φ) ] / [ e^(1/φ) + e^(-1/φ) ]

https://wolframalpha.com/input/?i=Sinh%5B+2sin%5B+%282%28%2811%29%29%5E%28-1%29%2810%5E%28%2811%29-1%29+-1%29%29+%5D%C2%B0+%5D+%2F+Cosh%5B+2sin%5B+%282%28%2831%29%29%5E%28-1%29%2810%5E%28%2831%29-1%29+-1%29%29+%5D%C2%B0+%5D+%3D++%5B+e%5E%281%2F%CF%86%29+-+e%5E%28-1%2F%CF%86%29+%5D+%2F+%5B+e%5E%281%2F%CF%86%29+%2B+e%5E%28-1%2F%CF%86%29+%5D

Sinh[ 2sin[ (2((11))^(-1)(10^((11)-1) -1)) ]° ] / Cosh[ 2sin[ (2((29))^(-1)(10^((29)-1) -1)) ]° ] = [ e^(1/φ) - e^(-1/φ) ] / [e^(1/φ) + e^(-1/φ)]

https://www.wolframalpha.com/input/?i=Sinh%5B+2sin%5B+%282%28%2811%29%29%5E%28-1%29%2810%5E%28%2811%29-1%29+-1%29%29+%5D%C2%B0+%5D+%2F+Cosh%5B+2sin%5B+%282%28%2829%29%29%5E%28-1%29%2810%5E%28%2829%29-1%29+-1%29%29+%5D%C2%B0+%5D+%3D++%5B+e%5E%281%2F%CF%86%29+-+e%5E%28-1%2F%CF%86%29+%5D+%2F+%5Be%5E%281%2F%CF%86%29+%2B+e%5E%28-1%2F%CF%86%29%5D

Sinh[ 2sin[ (2((13))^(-1)(10^((13)-1) -1)) ]° ] / Cosh[ 2sin[ (2((53))^(-1)(10^((53)-1) -1)) ]° ] = [ e^(φ) - e^(-φ) ] / [ e^(φ) + e^(-φ) ]

https://www.wolframalpha.com/input/?i=Sinh%5B+2sin%5B+%282%28%2813%29%29%5E%28-1%29%2810%5E%28%2813%29-1%29+-1%29%29+%5D%C2%B0+%5D+%2F+Cosh%5B+2sin%5B+%282%28%2853%29%29%5E%28-1%29%2810%5E%28%2853%29-1%29+-1%29%29+%5D%C2%B0+%5D+%3D++%5B+e%5E%28%CF%86%29+-+e%5E%28-%CF%86%29+%5D+%2F+%5B+e%5E%28%CF%86%29+%2B+e%5E%28-%CF%86%29+%5D

Sinh[ 2sin[ (2((103))^(-1)(10^((103)-1) -1)) ]° ] / Cosh[ 2sin[ (2((13))^(-1)(10^((13)-1) -1)) ]° ] = [ e^(-φ) - e^(φ) ] / [ e^(φ) + e^(-φ) ]

https://www.wolframalpha.com/input/?i=Sinh%5B+2sin%5B+%282%28%28103%29%29%5E%28-1%29%2810%5E%28%28103%29-1%29+-1%29%29+%5D%C2%B0+%5D+%2F+Cosh%5B+2sin%5B+%282%28%2813%29%29%5E%28-1%29%2810%5E%28%2813%29-1%29+-1%29%29+%5D%C2%B0+%5D+%3D++%5B+e%5E%28-%CF%86%29+-+e%5E%28%CF%86%29+%5D+%2F+%5B+e%5E%28%CF%86%29+%2B+e%5E%28-%CF%86%29+%5D

Sinh[ 2sin[ (2((103))^(-1)(10^((103)-1) -1)) ]° ] / Cosh[ 2sin[ (2((11))^(-1)(10^((11)-1) -1)) ]° ] = [ e^(-φ) - e^(φ) ] / [ e^(-1/φ) + e^(1/φ) ]

https://www.wolframalpha.com/input/?i=Sinh%5B+2sin%5B+%282%28%28103%29%29%5E%28-1%29%2810%5E%28%28103%29-1%29+-1%29%29+%5D%C2%B0+%5D+%2F+Cosh%5B+2sin%5B+%282%28%2811%29%29%5E%28-1%29%2810%5E%28%2811%29-1%29+-1%29%29+%5D%C2%B0+%5D+%3D++%5B+e%5E%28-%CF%86%29+-+e%5E%28%CF%86%29+%5D+%2F+%5B+e%5E%28-1%2F%CF%86%29+%2B+e%5E%281%2F%CF%86%29+%5D

Sinh[ 2sin[ (2((109))^(-1)(10^((109)-1) -1)) ]° ] / Cosh[ 2sin[ (2((17))^(-1)(10^((17)-1) -1)) ]° ] = [ e^(-1/φ) - e^(1/φ) ] / [e^(φ) + e^(-φ)]

https://www.wolframalpha.com/input/?i=Sinh%5B+2sin%5B+%282%28%28109%29%29%5E%28-1%29%2810%5E%28%28109%29-1%29+-1%29%29+%5D%C2%B0+%5D+%2F+Cosh%5B+2sin%5B+%282%28%2817%29%29%5E%28-1%29%2810%5E%28%2817%29-1%29+-1%29%29+%5D%C2%B0+%5D+%3D+%5B+e%5E%28-1%2F%CF%86%29+-+e%5E%281%2F%CF%86%29+%5D+%2F+%5Be%5E%28%CF%86%29+%2B+e%5E%28-%CF%86%29%5D

TEOREMA EXPONENCIAL ÁUREO PRIMO

La funcion exponencial e^x se puede expresar como la suma del seno hiperbólico y del coseno hiperbólico

Sinh(x) + Cosh(x) = e^x

Si reemplazamos la variable x, en mi función áurea prima se obtienen 8 imágenes áureas unicas para cada una de las 8 clases residuales o familias disjuntas primas.

e^(φ)

e^(-φ)

e^(1/φ)

e^(-1/φ)

(1/2) [ -e^(φ) + e^(-φ) + e^(1/φ) + e^(-1/φ) ]

(1/2) [ e^(φ) - e^(-φ) + e^(1/φ) + e^(-1/φ) ]

(1/2) [ e^(φ) + e^(-φ) - e^(1/φ) + e^(-1/φ) ]

(1/2) [ e^(φ) + e^(-φ) + e^(1/φ) - e^(-1/φ) ]

	P_{1 impar}	P_3impar	P_7impar	P_{9 impar}	P_{1 par}	P_{3 par}	P_{7 par}	P_{9 par}
P_{1 impar}	e^(1/φ)	(1/2) [ e^(φ) + e^(-φ) + e^(1/φ) - e^(-1/φ) ]	(1/2) [ e^(φ) + e^(-φ) + e^(1/φ) - e^(-1/φ) ]	e^(1/φ)	e^(1/φ)	(1/2) [ e^(φ) + e^(-φ) + e^(1/φ) - e^(-1/φ) ]	(1/2) [ e^(φ) + e^(-φ) + e^(1/φ) - e^(-1/φ) ]	e^(1/φ)
P_{3 impar}	(1/2) [ e^(φ) - e^(-φ) + e^(1/φ) + e^(-1/φ) ]	e^(φ)	e^(φ)	(1/2) [ e^(φ) - e^(-φ) + e^(1/φ) + e^(-1/φ) ]	(1/2) [ e^(φ) - e^(-φ) + e^(1/φ) + e^(-1/φ) ]	e^(φ)	e^(φ)	(1/2) [ e^(φ) - e^(-φ) + e^(1/φ) + e^(-1/φ) ]
P_{7 impar}	(1/2) [ e^(φ) - e^(-φ) + e^(1/φ) + e^(-1/φ) ]	e^(φ)	e^(φ)	(1/2) [ e^(φ) - e^(-φ) + e^(1/φ) + e^(-1/φ) ]	(1/2) [ e^(φ) - e^(-φ) + e^(1/φ) + e^(-1/φ) ]	e^(φ)	e^(φ)	(1/2) [ e^(φ) - e^(-φ) + e^(1/φ) + e^(-1/φ) ]
P_{9 impar}	e^(1/φ)	(1/2) [ e^(φ) + e^(-φ) + e^(1/φ) - e^(-1/φ) ]	(1/2) [ e^(φ) + e^(-φ) + e^(1/φ) - e^(-1/φ) ]	e^(1/φ)	e^(1/φ)	(1/2) [ e^(φ) + e^(-φ) + e^(1/φ) - e^(-1/φ) ]	(1/2) [ e^(φ) + e^(-φ) + e^(1/φ) - e^(-1/φ) ]]	e^(1/φ)
P_{1 par}	e^(-1/φ)	(1/2) [ e^(φ) + e^(-φ) - e^(1/φ) + e^(-1/φ) ]	(1/2) [ e^(φ) + e^(-φ) - e^(1/φ) + e^(-1/φ) ]	e^(-1/φ)	e^(-1/φ)	(1/2) [ e^(φ) + e^(-φ) - e^(1/φ) + e^(-1/φ) ]	(1/2) [ e^(φ) + e^(-φ) - e^(1/φ) + e^(-1/φ) ]	e^(-1/φ)
P_{3 par}	(1/2) [ -e^(φ) + e^(-φ) + e^(1/φ) + e^(-1/φ) ]	e^(-φ)	e^(-φ)	(1/2) [ -e^(φ) + e^(-φ) + e^(1/φ) + e^(-1/φ) ]	(1/2) [ -e^(φ) + e^(-φ) + e^(1/φ) + e^(-1/φ) ]	e^(-φ)	e^(-φ)	(1/2) [ -e^(φ) + e^(-φ) + e^(1/φ) + e^(-1/φ) ]
P_{7 par}	(1/2) [ -e^(φ) + e^(-φ) + e^(1/φ) + e^(-1/φ) ]	e^(-φ)	e^(-φ)	(1/2) [ -e^(φ) + e^(-φ) + e^(1/φ) + e^(-1/φ) ]	(1/2) [ -e^(φ) + e^(-φ) + e^(1/φ) + e^(-1/φ) ]	e^(-φ)	e^(-φ)	(1/2) [ -e^(φ) + e^(-φ) + e^(1/φ) + e^(-1/φ) ]
P_{9 par}	e^(-1/φ)	(1/2) [ e^(φ) + e^(-φ) - e^(1/φ) + e^(-1/φ) ]	(1/2) [ e^(φ) + e^(-φ) - e^(1/φ) + e^(-1/φ) ]	e^(-1/φ)	e^(-1/φ)	(1/2) [ e^(φ) + e^(-φ) - e^(1/φ) + e^(-1/φ) ]	(1/2) [ e^(φ) + e^(-φ) - e^(1/φ) + e^(-1/φ) ]	e^(-1/φ)