GENERAL FORMULAE FOR PHIN
 
 
 
A remarkably simple (circular) type equation shows the general value for phin.

(Phin)n = 1/(2 - phin)

  • Thus when n = 1,
  • Phi1 = 1/(2 - phi1). Thus here, phi1 = 1.
     
     
  • When n = 2,
  • (Phi2)2 = 1/(2 - phi2). Apart from the trivial solution i.e. 1 (which satisfies the equation for all values of n), the real valued solution for phi2 (i.e. phi) is 1.6180339887..,
     
     
  • When n = 3,
  •   (Phi3)3 = 1/(2 - phi3).

    As we have seen the value for phi3 = 1.839286755
     
     

  • When n = 4,
  • (Phi4)4 = 1/(2 - phi4).

    The value here for phi4 = 1.9275619
     
     

    So it goes for any value of n.
     
     
     
     

    In terms of connections as between the different ratios, a fascinating approximating type relationship exists.

    In general terms it can be expressed as
     
     

    Phin/Phi2n = Phin +1 - 1

    Thus when n = 1, we get Phi1/Phi2 =1/1.6180339887 = .6180339887

    = Phi1+1 - 1 (exactly in this case).
     
     

    When n = 2, we get Phi2/Phi4 = 1.6180339887/1.9275619..

    = .83942.

    This reasonably approximates the value of Phi2 +1 - 1, i.e. Phi3 - 1 = .839286755...
     
     

    When n= 3, we get Phi3/Phi6 = 1.839286755../1.98358..

    = .92726

    This reasonably approximates the value of Phi3 +1 - 1 i.e. Phi4 - 1 = .9275619
     
     

    With n = 4, we get Phi4/Phi8 = 1.9275619../ 1.99604..

    = .96569

    This reasonably approximates the value of Phi4 +1 - 1, i.e. Phi5 - 1

    = .96594..
     
     

    With n = 5, we get Phi5/Phi10 = 1.9659482../ 1.99902

    = .98345..

    This reasonably approximates the value of Phi5 +1 - 1, i.e. Phi6 - 1

    = .98358..
     
     

    Finally - to illustrate - with n = 6, we get Phi6/Phi12 = 1.98358../ 1.999756

    = .99191..

    Once again, this reasonably approximates the value of Phi6 +1 - 1, i.e. Phi7 - 1

    .9919
     

    So the approximation tends to improve for larger values of n.
     
     
     
     

    I will give one more example here of a precise and remarkable connection as between phi (i.e. phi2) and phin.

    (Phin - 1)/(Phi2 - 1) - 1 divided by (Phin/Phi2) - 1 = Phi22 (for n > 2).

    Thus when n = 3,

    (Phin - 1)/(Phi2 - 1) - 1 = (1.839286755. - 1)/(1.6180339887. - 1) - 1 = (.839286755./.6180339887) - 1 = .357994.

    (Phin/Phi2) - 1 = (1.839286755.)/(1.6180339887.) - 1

    = .1367417.

    Dividing both results we get .357994/.13674 = 2.618034 = Phi22.
     
     

    For example when n = 5,

    (Phin - 1)/(Phi2 - 1) - 1 = (1.9659482. - 1)/(1.6180339887. - 1) - 1 =

    (.9659482./.6180339887.) - 1 = .562937.

    (Phin/Phi2) - 1 = (1.9659482..)/(1.6180339887.) - 1

    = .2150228.

    Dividing both results we get .562937/.2150228 = 2.618033..= Phi22
     
     

    Another remarkable general result can also be given as follows
     
     

    (Phin2 - 1)/(Phi22 - 1) - 1 divided by (Phin2/Phi22) - 1 = Phi2

    For example when n = 3,

    (Phin2 - 1)/(Phi22 - 1) - 1 = (3.382975767. - 1)/(2.6180339887 - 1) - 1

    = (2.382975767/1.6180338887) - 1 = .47276..
     
     

    (Phin2/Phi22) - 1 = (3.382975767.. )/(2.6180339887..) - 1

    = .292182

    Dividing both results we get .47276/.292182 = 1.618033 = Phi2
     
     

    Similar - though less elegant - general results can be found for powers of Phin and Phi2 >2.