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.