The Phi Phormula?

or ... "Hey, it's just an expression!"

It's been noted that phi is just one of an infinite series of numbers that can be constructed from the following expression using the square root (√) of integer numbers:

(1+√n) / 2

It just so happens that you get phi when you let n equal 5.  Let n be other integers and you get a series of numbers whose squares (see Phi2 in table in green) each exceed their root by a difference (see Δ in table in blue) that increases by 0.25 for each number in the series, as shown below.

Phi, being the 5th one in the series, just happens to be the one that produces a difference of 1 with its square, leading to the unique property that it shares with no other number:

Phi + 1 = Phi ²
 

So does this demystify phi, making it just one of a series of phi-like numbers?  Not necessarily, as this is only one aspect of phi's unique properties.  Phi is also the only number that produces a difference of 1 with its reciprocal:

Phi - 1 = 1 / Phi

This is the key to its relationship to the golden section, which is based on sectioning a line in a way that fulfills two requirements:

A = B + C

and

A/B = B/C
 

A is to B as B is to C, where
A is 161.8% of B and B is 161.8% of C, and
B is 61.8% of A and C is 61.8% of B

Let n be any integer other than 5 and you won't find the same pattern of consistent differences as shown above or the unique reciprocal and additive properties of phi.

Insights on phi's formula in the table above contributed by Joseph Conklin