  Motivated by the cluster structure of two-loop scattering amplitudes in N=4
Yang-Mills theory we define "cluster polylogarithm functions". We find that all
such functions of weight 4 are made up of a single simple building block
associated to the A_2 cluster algebra. Adding the requirement of locality on
generalized Stasheff polytopes, we find that these A_2 building blocks arrange
themselves to form a unique function associated to the A_3 cluster algebra.
This A_3 function manifests all of the cluster algebraic structure of the
two-loop n-particle MHV amplitudes for all n, and we use it to provide an
explicit representation for the most complicated part of the n=7 amplitude as
an example.
