<s>[INST] The relationship between the dilogarithm and the golden ratio Among the values for which the dilogarithm and its argument can both be given in closed form are the following four equations:
$Li_2( \frac{3 - \sqrt{5}}{2}) = \frac{\pi^2}{15} - log^2( \frac{1 +\sqrt{5}}{2} )$ (1)
$Li_2( \frac{-1 + \sqrt{5}}{2}) = \frac{\pi^2}{10} - log^2( \frac{1 +\sqrt{5}}{2} )$ (2)
$Li_2( \frac{1 - \sqrt{5}}{2}) = -\frac{\pi^2}{15} - log^2( \frac{1 +\sqrt{5}}{2} )$ (3)
$Li_2( \frac{-1 - \sqrt{5}}{2}) = -\frac{\pi^2}{10} - log^2( \frac{1 +\sqrt{5}}{2} )$ (4)
(from Zagier's The Remarkable Dilogarithm)
where the argument of the logarithm on the right hand side is the golden ratio $\phi$. The above equations all have this (loosely speaking) kind of duality, and almost-symmetry that gets broken by the fact that $Li_2(\phi)$ fails to make an appearance. Can anyone explain what is the significance of the fact that  $\phi$ appears on the right, but not on the left? Immediately one can see that the arguments on the lefthand side of (2)-(4) are related to $\phi$ as roots of a polynomial, but what other meaning does this structure have?
 [/INST] According to Maple
$$ {\rm polylog}\left(2, \dfrac{1+\sqrt{5}}{2} \right) = \dfrac{7 \pi^2}{30} - \dfrac{1}{2} \log^2 \left(\dfrac{1+\sqrt{5}}{2}\right) - \log   \left(\dfrac{1+\sqrt{5}}{2}\right) \log \left(\dfrac{1-\sqrt{5}}{2}\right)$$
Of course $\log((1-\sqrt{5})/2) = \log(-1/\phi) = - \log (\phi) + i \pi$ 
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