I have heard much about the numerous appearances of the ratio found in nature: 1.6180339887.
Are there any actual mathematical uses that have been found of this number? What are its advantages? Just curiosity, really.
Answer
I'm not sure if you mean "man-made" as in real world applications or as in mathematical applications, but one of my favorite things about the golden ratio is that it is somehow the hardest number to approximate by rationals.
For one thing, it has the slowest converging simple continued fraction expansion. This is reflected in the fact that if you apply Euclid's algorithm to two successive Fibonacci numbers, you get a quotient of 1 at every step.
There is also Hurwitz's theorem, which says that for any irrational number $\xi$, there are infinitely rationals $\frac{n}{m}$ such that$$\left| \xi - \frac{n}{m} \right| < \frac{1}{\sqrt{5}m^2}.$$ The constant on the RHS can't be improved since, if we replace $\sqrt{5}$ with some larger number, then the statement of the theorem doesn't work when we let $\xi$ be the golden ratio. (Informally, we can't get too close, in a number theoretic way, to the golden ratio.)
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