Why is the radian golden angle $(1-1/\varphi)\cdot2\pi=\pi(3-\sqrt5)\approx2.39996$ so close to a 'nice' rational number? 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 $\pi$? 
 A: 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). 
