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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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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, see the FAQ.
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The golden ratio was defined by the ancients with a geometric meaning, e.g.: $\hskip 2.3in$ 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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