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I was reading about phytollaxis in plants and Fermat's spirals when I came across the Wikipedia article on golden angles. Surprisingly, the radian golden angle is very nearly approximated by a simple rational number, $2.4$. In the spirit of near-integers, is there any mathematically 'deep' reason for this curiosity? or is it merely a mathematical coincidence? What about the irrational numbers $\pi$ and $3-\sqrt5$ would hint to their product being so nearly rational?

Equivalently, why is it that $\pi\approx\dfrac{12}{15-5\sqrt5}$ so closely? Are there other 'nice' algebraic approximations for $\pi$?

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    $\begingroup$ Just like any approximate equality only valid in the decimal system, $(1-1/\phi)2\pi\approx 2.4$ is purely coincidental. It's just because there are three nines in a row very early in the decimal expansion of the number. You can find similar series of nines in other numbers as well (like $\pi$) only not quite as quickly. $\endgroup$ – 5xum Jan 6 '15 at 7:29
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    $\begingroup$ It's not "closer" to a rational number any more than any other irrational number being "close" to some rational number. $\endgroup$ – barak manos Jan 6 '15 at 7:31
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    $\begingroup$ Here are some other coincidences. en.wikipedia.org/wiki/Mathematical_coincidence I don't know if the one you mention has an explanation. $\endgroup$ – user204305 Jan 6 '15 at 7:33
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    $\begingroup$ What about $e^{\pi\sqrt{163}}\approx262537412640768743.999999999999250072\cdots$ ? $\endgroup$ – Yves Daoust Jan 6 '15 at 8:37
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Good rational approximations of Pi are obtained from its continued fraction expansion.

Among them, $\dfrac{22}{7}$ and $\dfrac{355}{113}$.

Due to the way these fractions are computed, they are more accurate than other fractions with denominators of the same order of magnitude, so they are the most "efficient". This is in connection with Dirichlet's theorem.

Anyway, they provide no real gain as the number of their digits is similar to the number of correct decimals they generate.

There are certainly more efficient approximations if you allow integer square roots, just because you enable more freedom.

It doesn't seem that $\dfrac{12}{15-5\sqrt5}$ be extremely efficient. Among all approximations of this form with integers below $20$ and square root of a single digit (there's more than $250000$ such combinations, no surprise some are good ones), it is the best. But if you allow a radical at the numerator it is by far beaten by $\dfrac{4\sqrt7-2}{\sqrt3+1}$.

Also in relation to the Golden section, $\dfrac{23}{41\sqrt5-99}$, $\dfrac{909}{240\sqrt5-826}$, $\dfrac65\dfrac{\sqrt5+1}{\sqrt5-1}^*$, $\dfrac{92\sqrt5+47}{40\sqrt5-9}$, $\dfrac{263\sqrt5+261}{124\sqrt5-7}$ ($^*$ same as in the OP).

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