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One way to define the Golden Ratio (or, its reciprocal) is that positive number $x$ such that $1 + x = \frac{1}{x}$. However, unknown to the ancients, $1 + x$ is just the first part of the Maclaurin expansion for $\exp(x)$. So, perhaps the defining equation should actually be $\exp(x) = \frac{1}{x}$.

Note: The literature is somewhat ambiguous as to whether 0.618… or 1.618… should be considered the Golden Ratio. I have chosen 0.618… for this question.

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closed as not a real question by Brian M. Scott, Pete L. Clark, J. M., Sasha, t.b. Sep 30 '11 at 10:21

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

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There are a huge number of simple functions with Maclaurin expansion beginning $1+x$; it's just a not-too-improbably coincidence. The real solution to $e^x=x^{-1}$ is $x=W(1)=0.567\dots$ $\ne0.618\dots=\phi$ - this number does not have any direct connection to the Fibonacci numbers that give the golden ratio its characteristic aesthetic and recursive properties, nor is it the "most irrational number" through its simple continued fraction (which is the explanation for why Fibonacci numbers are the worst-case scenario for the Euclidean algorithm). –  anon Sep 20 '11 at 21:23
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I think you will find that the golden ratio, which can be related to geometric structures like the pentagon in Euclid, is conventionally defined by the equation which gives 1.618... . Personally I don't quite understand why anyone would want to confuse the situation by proposing something different, especially involving an exponential as you propose. –  Mark Bennet Sep 20 '11 at 21:23
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You can solve exp(x) = 1/x if you like. You get a completely different number, though. I don't see what makes you want to 'complete' the 1+x - it's a legitimate function in its own right. "The ancients" liked this because it was geometrically very interesting, and geometry was how they worked! –  Billy Sep 20 '11 at 21:33
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I don't understand the point of this question. The golden ratio is a certain quadratic irrational number classically defined in terms of a geometric proportionality condition. It has various algebraic and arithmetic properties. You are suggesting that some completely different number be called the Golden Ratio... why? (Your question makes as much sense to me as "Should we perhaps call the head of the Athens, GA post office 'Barack Obama' instead of the President of the United States?") –  Pete L. Clark Sep 20 '11 at 21:51
    
So far as I can see, this is neither a real question nor constructive; I’ve voted to close. –  Brian M. Scott Sep 20 '11 at 22:27

1 Answer 1

up vote 13 down vote accepted

The golden ratio was defined by the ancients with a geometric meaning, e.g.:

$\hskip 2.3in$ golden rectangle

so that (in the above) $$\frac{a+b}{a}=\frac{a}{b}$$ hence $1+1/\phi=\phi$, where $\varphi=a/b$ (here we use $\varphi=1.618\dots$, which is the standard convention, while $1/\varphi=\varphi-1=\Phi$ is the conjugate). This gives the quadratic equation $\varphi^2=\varphi+1$ with positive solution $(1+\sqrt{5})/2$. It is also the characteristic equation of the linear recurrence satisfied by the Fibonacci numbers, i.e. $$F_{n+2}=F_{n+1}+F_n.$$ This is why Binet's formula exists, why all integer powers of $\varphi$ can be expressed as linear combinations of $\varphi$ and $1$ with Fibonacci coefficients (allowing negative integer index). It is also the connection between $\varphi$ being the "most irrational number" through its simple continued fraction expansion $[1;1,1,\dots]=1+\mathbb{K}\frac{1}{1}$ and Fibonacci pairs $(F_{n+1},F_n)$ (as numerator and denominator in the golden ratio's convergents) being the worst-case scenarios in the Euclidean algorithm.

These properties have been cited speculatively as the reason the golden ratio and Fibonacci numbers appear in nature in certain branching, arranging or other patterned phenomena: it might optimize some structural features that, for example, most benefit a plant's sunlight-gathering, and at the same time is simple enough for relatively unsophisticated mechanisms to create recursively. Even the clock cycle in the human brainwaves involves the golden ratio!

The bottom line is this: the inherent power-linear recursive nature of the golden ratio is responsible for all of its "golden" properties, a lot of them pertaining to ratios intrinsic to very basic geometric figures. New theoretical properties are still being discovered every once in a while (my favorite one I recently read about: the Beraha constants $B_n = 2(1+\cos\frac{2\pi}{n})$ and Tutte's identities on chromatic polynomials for spherical triangulations).

The equation $\exp(x)=1/x$ doesn't have such a natural recursion or its numerous consequences, nor does it have a direct geometric meaning and subsequent aesthetic quality as the Greeks designated for it. The fact that $\exp$ has an initial Maclaurin series $1+x$ is essentially a not-too-improbable coincidence (given it's only the first two terms and there are myriad basic functions that start out $1+x+\cdots$), and the exponential captures none of the golden properties of the golden ratio for which it was given the title in the first place.

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Fair enough. I have up-voted and accepted your answer. I was pretty sure I was opening a can of worms, but I had wondered about this for a long time, and so, given MSE, I just HAD to ask. Case closed. –  Mike Jones Sep 21 '11 at 2:32

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