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[I just realized my response had a random link, so it may have been filtered. This is a direct copy without the link.]
I agree with some of your points at a high level, but at the very least they do have the equation correct. It's not an approximation -- the irrationality is derived from the square root of 5. I was caught off guard myself when I first saw such a simply-defined expression being equated to phi.
But the math works out to the same expression you provided.
Let x = (1 + sqrt(5)) / 2
x = 1 + 1 / x
= 1 / ((1 + sqrt(5)) / 2)
= 2 / (1 + sqrt(5)) : [1/(a/b) = b/a]
= 2*(1 - sqrt(5)) / (1 - 5) : [multiply by (1 - sqrt(5))/(1 - sqrt(5))]
= 2*(1 - sqrt(5)) / (-4) : [simplify denominator by addition]
= (1 - sqrt(5)) / (-2) : [cancel out factor of 2]
= (sqrt(5) - 1) / 2 : [bring negative sign up to the numerator]
That's 1/x, so now add 1 to get (1 + 1/x):
= 2/2 + (sqrt(5) - 1) / 2 : [2/2 = 1 and 1/x = (sqrt(5) - 1) / 2]
= (2 + sqrt(5) - 1) / 2 : [combine fractions with same denominator]
= (1 + sqrt(5)) / 2 : [combine terms in numerator, 2 - 1 = 1]
Thus, x = 1 + 1/x if x = (1+sqrt(5))/2
Personally, the significance of the square root of 5 reminds me of the dodecahedron (made of pentagons) which Plato thought to represent the cosmos.