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?



The relationship between the dilogarithm and the golden ratio
The identities $L\_2(\frac{\sqrt{5}-1}{2}) = \frac{\pi^2}{10}$ and $L\_2(\frac{3-\sqrt{5}}{2}) = \frac{\pi^2}{15}$ are due to J. Landen. The rest of the identities you wrote, I suppose, could be obtained by using some other ones, like $L\_2(1) = \frac{\pi^2}{6}$, $L\_2(-1) = -\frac{\pi^2}{12}$, $L\_2(\frac{1}{2}) = \frac{\pi^2}{12}$, due to L. Euler, and the so-called five-term relation for the Wigner-Bloch dilogarithm function, which uses an analytic continuation of the dilogarithm initially defined by the known infinite series. This Wigner-Bloch's dilogarithm function combines the dilogarithm (the imaginary part of its analytic continuations, to be more precise) and the usual logarithm. I think that the paper by [A. Kirillov](http://arxiv.org/pdf/hep-th/9408113v2.pdf) could serve you as a good reference. In short: I suggest that one of the relations comes from some computation, the others come from the five-term relation. 


Also one may try the following way (in the case of the above identities): use the five-term relation for the usual dilogarithm $L\_2$, and then apply [A. Kirillov, 1.6. Exercises to Section 1.](http://arxiv.org/pdf/hep-th/9408113v2.pdf) where identity (iii) involves both dilogarithm and logarithm squared.


I hope that the paper by Kirillov that I'm referring to, has a good background on the dilogarithm function and dilogarithm identities. Also, this paper has quite a rich reference list, worth seeing if one needs any additional bibliography on the subject. 


