The Bloch-Wigner function $D(z)$ gives the volume of an ideal tetrahedron in the hyperbolic space $\mathbb{H}^3$. Here $z$ is the cross-ratio $(z\_1,z\_2,z\_3,z\_4)$ parametrizing the tetrahedron in $\mathbb{C}P^1$.  

Put $\tilde D(z\_1,z\_2,z\_3,z\_4) = D(z)$.


The five-term relation for the dilogarithm could be interpreted as the fact that the signed sum of some volumes of tetrahedra vanishes:


$$\sum^4\_{i=0} (-1)^i \tilde D(z\_0, ...., \hat z\_i, ... z\_4) = 0$$


Here the $z\_i$ are 5 points in $\mathbb{C}P^1$, and the notation $\hat z\_i$ means that we 
don't take the vertex $z\_i$ in account. 
The above equation looks like some function of a boundary of some 5-simplex.


But what is this 5-simplex (which, I think, corresponds to a 4-volume), and in what space this simplex exists (hyperbolic space ?) ?


Reference (Zagier) : <http://maths.dur.ac.uk/~dma0hg/dilog.pdf> (Pages 10 - 11)



Dilogarithm, tetrahedrons, and hyperbolic space
The five term relation comes from the fact that the sum of the volumes of tetrahedra $ABCD$ and $ABCE$ equals the sum of the volumes of the three tetrahedra $ABDE, ACDE, BCDE.$ One can think of $ABCDE$ as a degenerate four-dimensional simplex.


